Physics 115B Lab 1: Energy, Temperature, Heat, and Power

Learning goals

“Technique”: looking for lab practices that make a measurement reliable.

Distinguish between heat and temperature.

Measure energy in different forms and demonstrate that it is conserved.

Apply these experiences to everyday life.

Part I: Thermometers from many lands

There is a collection of thermometers on the table, including an unusual Star-Trekky one (the “grey instrument”, “g.i.”) and several digital thermometers, some hooked up to computers.

1. Pick up the g.i., study it, play with it, and see if you can figure out how to use it.

2. Use it on yourself (or a partner), on the wall, on a computer screen, and at least four other interesting items. Make a table in your notebook showing the results. If you find especially interesting uses or outcomes, record them.

3. Shine the red lamp on the mirror at about a 45º angle. Use the g.i. to measure the bulb temperature. Then use it to measure the temperature of the mirror on the reflection of the bulb. Is the mirror really that hot? Discuss how the g.i. might work, and put some speculations in your notebook.

4. Fill a beaker of water and use each of the thermometers (including the g.i. and a computer one) on the water. Record your results in a table in your notebook. Does the temperature change if you gently stir the water?

5. Put your hands around the beaker and repeat Step 4 with the computer thermometer. Is the energy of the water changing? Why? Does the temperature change if you stir gently? Why?

6. Describe in one paragraph some real-world situations where the g.i. would be useful. Describe cases in which it would be inaccurate.

Part II: Liquid Nitrogen Demos

The AI’s will show you cool things with serious cold!!

Part III: Transforming Kinetic Energy to Heat Energy

1. Carefully open the bottle and see what is inside. Measure the temperature inside using a digital thermometer. Make sure the thermometer doesn’t touch the bottom of the plastic bottle. Leave the thermometer in place to see how long the temperature takes to stabilize.

2. Describe (to your lab partners and in your notebook) what would happen if you shook the bottle vigorously and then set it down. We want a description in terms of energy. (Energy in what form? From where?) Would the energy content of the bottle change? What about the temperature? What if you waited awhile afterward?

3. Put the lid on the bottle and shake it 40 times vigorously. Repeat your temperature measurement. Watch it for a while. Record what you observe, then graph the result. Compare what you observe to your expectation in Step 2. Try repeating the experiment several times. (Everyone should do it at least once.) Why might such repetition be useful? Describe several reasons.

4. There is a big brass object attached to the table. On top of it, there is a hole where you can insert a digital thermometer and measure its temperature. Using friction with the belt, try to increase its temperature by 2°C. Do you have more respect for heat energy now? Note: please don’t let the belt slip off the object – this could damage the thermometer probe.

Part IV: Heat and Temperature

Using the materials available on the table, we’ll take a look at how objects store heat. In order to get this accurately, you’ll need to make some careful measurements and record the data in your notebook. You will follow the steps listed below. Read the steps now, but, before you do them, read the rest of this section. You should understand exactly how the steps will allow you to determine the specific heat of your sample before you start making the measurements.

1. Get a metal block from an AI and measure its mass (Mmetal). Also measure the mass of an empty styrofoam cup.

2. Fill a beaker with cold water and ice. You’ll find those in the fridge/freezer.

3. Put tap water in a styrofoam cup and measure its total mass. From that determine the mass of the water (MH20). There should be just enough water in the cup just to have your block completely submerged. Measure its temperature without the block (TH20,intial). You can use two probes on the same computer, one for the ice water, one for the tap water.

4. Attach a string to the metal sample and put it in the ice water. Measure the temperature of the water (Tice water =Tmetal,initial).

5. Then, carefully place the metal sample in the styrofoam cup of water. Stir and track the temperature on the computer. When it reaches a constant value, record it: (TH20,final=Tmetal,final).

We can connect the temperature change of the water to energy. Recall from your textbook that the heat (energy) required to raise the temperature of 1 kg of water by 1°C is 1 Calorie ≈ 4200 J, so it takes about 4.2 J to raise 1 g of water by 1°C. How much did the temperature of the water in the styrofoam cup change when the block was added? How many Joules of heat energy did the water lose? What principle allows you to determine the change in the

energy of the block? By how much did its energy change? Did its temperature change by the same amount as the water?

Temperature and heat are related, but not the same! Each substance has a “specific heat,” the amount of energy needed to raise 1 g of the substance by 1°C. From the above discussion, the specific heat of water is CH2O ≈ 4.2 J/g-°C. (It can also be quoted in other units, for example, 1 Cal/kg-°C.) C does not depend on the mass, shape, or temperature of the material – it just depends on what the object is made of.

Use this definition and your measurements to figure out the specific heat, Cmetal, for your block. Report your result to one of the AI’s – we will compare the values measured by the different groups in all the labs. Does your metal or water have the higher specific heat?

Discuss and list some real-world situations when having a material with high specific heat or low specific heat can be important.

Part V: Electrical Energy and Power – a Watt is a Watt

The 100-Watt light bulb has been set up for safe immersion in water. (Don’t try this at home!)

1. First try plugging it into the Kill-A-Watt meter. Try the various buttons. How much electrical power is being used? We will try to measure this ourselves.

2. Put enough tap water in the plastic water jug to cover the bulb. 500 ml should do it. Figure out the mass of the water. (Why is it easy if you used 500 ml?) Put the probe of a digital thermometer in the water. It should not touch the wall of the jug or the bulb. Use a plastic ruler to stir the water gently. Measure the temperature of the water with the bulb off.

3. Turn on the bulb. Record the temperature at fixed time intervals, say every 30 seconds, for several minutes. Stir the water gently and continuously with the ruler. Plot the temperature vs. time.

4. We can connect the temperature change of the water to energy. Recall from your textbook that the heat (energy) required to raise the temperature of 1 kg of water by 1°C is 1 Calorie ≈ 4200 J, so it takes about 4.2 J to raise 1 g of water by 1°C. At what rate is energy being added to the water (J/s = Watts) while the bulb is on?

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Physics 115B Lab 2: Atoms, molecules, and measuring the

very small

In this lab we will make a series of measurements using simple equipment (rulers, beakers, lab balances) that will give us estimates of molecular-scale quantities such as the size of a molecule, the wavelength of light, and Avogadro’s number.

Learning goals

Understand the relative and absolute sizes of atoms, molecules, the wavelength of light, the average molecular spacing in a gas.

Apply mathematical and geometrical concepts to the physical world.

Understand some techniques for making realistic measurements of things that are invisibly small.

Warm-up 1: Guesses

Do this exercise individually, without consulting lab partners or other sources. You won’t be judged (or graded) on the accuracy of your results – we really mean guess. This is for later comparison with what you’ll measure in today’s lab.

1. In your notebook, rank in order of increasing size the following items:

The wavelength of visible light

The diameter of an atom

The length of a molecule. We’ll be using an oil (oleic acid) with chemical formula C18H34O2, that forms a long chain of atoms, so use that in your thinking. Here’s a model:

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(Wikipedia, Oleic Acid)

The average distance between molecules in air

2. Guess numbers for the sizes, indicating both a guess for the size of each item and the ratios between adjacent sizes in your list. (Example: you might guess that an atom is 25 times larger than the wavelength of light.)

Warm-up 2: Slick

You know that oil floats on water, and, at least since the Gulf oil spill, you know that oil on water forms a “slick” – a very thin layer or film. Under controlled circumstances, this film can be one molecule thick! (So an oil spill spreads very far indeed.) It isn’t easy to prove that the layer is one molecule thick, but, by assuming this, we can measure the size (it turns out to be roughly the length) of an oil molecule, something Lord Rayleigh did in 1890. Though there are some practical issues, the concept couldn’t be simpler.

Figure out how to determine the thickness t of the film formed by a volume of oil V by measuring the area of the film A.

We make the patch of oil visible by dusting a bit of the surface of the water with baby powder. The AIs will demonstrate this. They will also suggest some assumptions we are making and maybe test them.

Parts I, II, and III can be done in any order – the AIs will get your group started on one after the demo, then move on to the others. Don’t forget Part IV at the end.

Both Parts I and II will need you to measure the volume of a drop from an eyedropper. We will assume that all the droppers are the same (we tried a few, and they are at least similar), so your

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group need only measure one dropper for each part. We will not assume that a drop of oil-water mix (Part I) and a drop of turpentine (Part II) are the same size.1 Do this as part of Part I and II, as the graduated cylinders are available. For now, think how you would do it.

Using oil-water or turpentine and one of the 25-ml graduated cylinders, determine the volume of one drop from your eyedropper. Each group member should try this, using the same dropper. Record all the measurements for your group.

Part I: Measuring the size of a molecule with a ruler

The trick now is to make this quantitative, and what makes it hard is that V must be known and must be quite small to keep the film smaller than our pans. (Rayleigh used a very large pan; Ben Franklin, who started all this, publishing an account in 1774, used a pond.) As part of the Warm-up 2 demo, the AIs will make a 1:1000 mixture of oil:water. Oil and water, famously, do not mix, but we can do well enough by making a suspension of tiny oil droplets. When you put a drop of the mix in your pan of water, the water in the mix will just enter the water in the pan, leaving the oil droplets to join and form the film.

Measure the volume of a drop of the oil-water mix, as described in Warm-up 2.

Make several measurements of areas of films, each from a single drop, in the pans, recording the results. (You may only get to do one trial per pan. Let the AIs know if you need to reuse a pan – they can help dump it out and clean it for another trial.)

Estimate the size of a molecule of the oil, assuming that the film is one molecule thick. Remember that your drop was not all oil!

1 We used to assume this, but then we tried it – it isn’t true!

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Assuming that all the measurements you have made are accurate, discuss what effects could make your estimate of the molecular size too big? too small?

From the size of a molecule, estimate the number of oil molecules in a cubic centimeter of oil. This is called the number density, n. To do this, you need to make assumptions about the shape of an oil molecule. Assume that the molecules are cubes that are packed together. (This turns out to be a poor assumption – see the picture on page 1 – but it will do for now.) You can (and should) calculate the number of oil molecules in your film. It is surprising that you can measure such a number (even roughly) so simply!

For the next steps, we will need the density (mass per cubic centimeter) of the oil. The AI’s have some oil you can use to measure this. Give the oil back to them when you are done.

The oil we are using is oleic acid, the major component of olive oil. (Rayleigh and Franklin just used olive oil.) Oleic acid is described by the chemical formulas

C18H34O2 or CH3 (CH2)7 CH = CH (CH2)7 COOH .

The important constant Avogadro’s number, NA, is the number of atoms or molecules of a substance in a mole of that substance. The atomic or molecular weight is the mass in grams of a mole of atoms or molecules of a given type. The atomic weights of hydrogen, carbon, and oxygen are 1, 12, and 16, respectively, so, for example, a mole of carbon atoms has a mass of 12 g.

From this information, what is the molecular weight of oleic acid?

Using this molecular weight and your measurements, calculate an estimate of NA, Avogadro’s number.

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Discuss factors that could cause your estimate to vary from the accepted value, NA = 6.02×1023 molecules/mole.

Note from the right-hand chemical formula above and the picture on page 1 that oleic acid is a chain of carbon and oxygen about 20 atoms long. (The structure is a bit bent, but we’ll ignore that.)

From your measurements, what is the diameter of an atom?

Part II: The wavelength of light

As we will study later in the course, light is an electromagnetic wave. No matter is moving in this wave – what is oscillating as the wave propagates is the strength of electric and magnetic fields. Though this is very abstract, our usual picture of a wave applies, but with a different meaning of the graph.

We often ask questions like this, and it is worth noting what we are looking for.

We are not looking for “I could have forgotten a term in my equation.” or “I could have multiplied wrong.” or “I could have weighed the sample wrong.” Even though we call this exercise “understanding the errors,” that does not mean mistakes. We are looking for specific effects or inaccurate assumptions (explicit or hidden) that could have distorted a measurement or its interpretation. An example from this lab would be assumptions about the shape of an oil molecule. An example from last week would be the assumption that no heat was lost through the wall of a beaker. You should not only list such effects, but indicate, if possible, the direction they would change your result. For example: “Not taking into account the heat entering through the wall of the beaker during the measurement would make our measured specific heat too small.”

whole pattern moves

wavelength

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Instead of the graph representing, say, the height of the water surface in an ocean wave, it can be the strength of the electric field. A stationary observer would measure the electric field getting stronger and weaker as the wave went by, just as a person in the ocean would bob up and down as a water wave passed.

Like any sine wave, this wave has a wavelength, the distance between two adjacent crests or troughs, for which we use the Greek letter lambda, . We can ask, If light is a wave, what is its wavelength? You may know that the wavelength of light depends on its color. Since we just want an order-of-magnitude estimate, this won’t matter to us (though it will make the patterns we observe quite beautiful.)

Dip a wire-loop in the bubble solution. Go ahead – blow a bubble or two! Look specifically at the reflections of the room lights, noting that the reflections are pretty colors, not the white of the lights themselves. We can make the pattern of colors more regular. Dip the loop again. This time hold it so the loop is vertical and look at the reflections in the flat bubble spanning the loop. Note that there are horizontal bands of alternating green and red. The pattern moves as the water drains downward, making the film thinner. Eventually, there is no reflection at all from the top, even though the bubble hasn’t popped yet (though it is about to.) Each member of the group should make and observe this pattern.

As we’ll learn when we study waves later in the course, the bands of color appear when the film is about a wavelength thick, via a process called interference. In fact, for a color with wavelength , when the film is ¼×, ¾×, 5/4× thick you see a bright reflection of that color. Because gravity drains the water downward, the bubble in the loop is thinner at the top and thicker at the bottom, and there are horizontal bands of constant thickness. This means the green and red reflections alternate as each color

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goes through the multiples of its . As the water drains, the pattern shifts downward. Eventually, the top of the bubble is less than ¼-wavelength thick and there is no color that reflects, that is, no reflection at all. For our purposes, we just need the fact that when there are a few bands present, the film is about a wavelength thick.

To make this quantitative, we will use a variant of the same trick used in Part I. In this part, we use turpentine. We use turpentine because, for some reason, it forms films that show interference bands. This tells us the film is about a wavelength thick, even if we don’t know why that happens. Note that, unlike the oil we use in Part I, the turpentine evaporates quite rapidly. This means that there is some uncertainty about the volume of turpentine remaining. On the plus side, it means we can try several drops in succession in the same pan without dumping it and restarting.

To make our measurements of the wavelength of light, we will drop one drop of turpentine into a tray of water. To know the film thickness is about a wavelength, we need to see the colored reflection bands. We have found that in the black pans we are using, it is easy to see both the bands and the edge of the slick without using any baby powder.

Measure the volume of a drop of turpentine, as described in Warm-up 2.

Make observations of what happens when you put a drop of turpentine on the water. Make sketches of what you see or describe it in words. We have always seen one oddity: the first drop spreads to cover the whole tray, but subsequent drops make near-circular films of reasonable size. Measure the diameters of all films after the first. Note roughly how long after the drop you see the colors appear, when they vanish, and when the film itself vanishes. We found that you can do five or six drops in the pan before it gets too messy.

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Estimate the wavelength of light using your observations and measurements.

Part III: Weighing air

It seems odd, but it turns out to be easy to weigh air. Your equipment: a flask with a valve, a balance, and a vacuum pump. The pump can suck just about all the air out of the flask in about a minute. (The flasks have a protective coating and are glued inside the boxes for safety.) Note the valve with the black handle on the flask. It is open when the handle points straight out from the stem.

Figure out how to find the mass of air in the flask and do it – record the result. Pretty easy. Do it again (have each group member try it) to see if you get the same result. What would happen if you tried to weigh helium this way?

Use your result to find the density (g/cm3) of air. What do you need to do this? Figure this out with the tools at hand.

Air is mostly nitrogen molecules, N2. (For simplicity, we’ll assume it is all nitrogen. The oxygen molecules have about the same mass, so this is a fine approximation.) The molecular weight of nitrogen is 28, so one mole of N2 has a mass of 28 g. In Part I, you determine the number of molecules in a mole, called Avogadro’s number, from measurements you made. Don’t worry if you haven’t done Part I yet – here we’ll use the actual value, NA = 6.02×1023 molecules/mole.

From your measured density of air, the molecular weight of N2, and the value of NA, find the number density, that is, the number of molecules per cm3 of air.

Now figure out the average distance between air molecules. This will take some thinking and discussion in your group. Indicate your reasoning in your notebook.

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Part IV: Summary

Remake the same list from Warm-up 1, using the results of this lab, including the sizes and ratios from your measurements.

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Physics 115B Lab 3: Motion

Learning goals

Perform an experiment that demonstrates the predictive power of Newton’s laws.

Apply the laws of motion in two dimensions.

Determine and observe the constraints placed on spinning objects by conservation laws (energy and angular momentum.)

Warm-up 1

As director of research for a toy manufacturer, you regularly receive proposals for new toys. Indicate which of these seem possible for an inexpensive toy with no internal power source (battery, etc.):

a) A plastic top that spins on a plastic base for weeks.

b) A simple top that, while spinning, flips over so that its larger, heavier end is on top.

c) A specially weighted wheel that rolls back and forth on a V-shaped ramp for weeks.

d) A specially weighted wheel that rolls back and forth on a V-shaped ramp, reaching a higher point on the ramp on each pass.

e) The situation in c) and d) if you know there are magnets in the wheel and the base.

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Part I: Catch the ball

On the table is a track on which a cart can roll with very little friction. The top section of the cart is a launcher that shoots a ball straight up. Ask an AI to show you how to activate the launcher. Never stand with your face directly above the launcher – it can fire without warning!!! There is also an adjustable spring that can give the cart a reproducible velocity along the track.

First: read through all of Part I so you’ll understand the overall goal of these steps.

Determine how high the ball goes when launched with the cart at rest. This takes at least two people: one to trigger the launcher and hold the meter stick, and one or more to watch how high the ball goes. Do quite a few trials with each member of your group launching and watching the height. Everyone should have a record of all the launches. Look for reproducibility in two forms: precision, how close in height are the launches? are there outliers, occasional misfires that go to a quite different height? Also, be sure to decide what you should define as the initial height of the ball.

At this point, ask the AI to take the ball away. Ask your AI to demonstrate the cart-launching spring and how to adjust it.

You will use a light gate (an LED light source and light sensor that straddle the track and a timer) to measure the cart’s speed just before it launches the ball. You shouldn’t have to do anything to the timer other than turning it on (power switch on the back) and hitting the reset button on the front. Figure out how this system works. (Try passing your hand between the LED and the sensor. What happens?) How can you use this setup to measure the speed of the cart?

Note the setting of the cart-launching spring (see the scale on the side of the launcher) and record it in your notebook.

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Without launching the ball, launch the cart several times, again looking for reproducibility in both forms.

Knowing the velocity of the cart and the height the ball reaches, compute the location where the ball will land (see formula below). Place the cup at that location, making sure the bumper is between the ball-launch point and the cup.

Call the AI over. He’ll give the ball back and observe.

Launch the cart with the ball-launcher armed. If the ball is caught in the cup, congratulations! Try it again. If not, check the calculations and the velocity and try again.

Now remove the cup, move the bumper to the far end of the track. Discuss where the ball (launched straight up from the cart) will land – ahead of the cart? behind the cart? in the cart? What if the cart is launched faster? Slower? Write your predictions in your notebook. Now try a few more launches. Vary the cart speed by just giving it a push with your hand and recording the velocity without adjusting the launcher. What do you observe? (No numbers needed, just watch.) Why does it make sense?

Useful formula:

The time it takes an object to fall a height h starting at rest is

g

ht fall

2 ,

where g is the gravitational acceleration, g = 9.8 m/s2 = 980 cm/s2. (See pp. 11-12 of Chapter 3 in your textbook. If you know how to derive this result, show your lab partners, but for this lab we will take it as given.) Figure out how to use this formula in your calculation of where to put the cup. Show your reasoning and all calculations in your notebook.

The Big Picture: this is the essence of Newtonian determinism, discussed in lecture. Given the initial position and velocity of an

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object and knowledge of the forces, the object’s future position is completely predictable. Where in your calculation did you express the knowledge of the forces? The understanding of situations like those in your last measurements (without cup or bumper) were essential to Galileo’s realization that the Earth could, in fact, be moving.

Part II: Spinning toys

This lab will be our only look at things that spin. We’ll mostly just ask that you observe, but we do want you to realize that conservation of energy and, as we will see in Part III, a new quantity called angular momentum, are universal.

The toys in this section fall into three categories.

Category 1: spin the “Top secret top,” the little silver top on the round black stand. Start the “Space wheel,” the little satellite thingy, by putting it on its rails (the two vertical transparent plates on a black stand) near one end and letting go from rest. Just start each one once and let them do their thing while you go on to the other toys.

Category 2: Before the first spins, guess how long each toy will spin. Take turns spinning the “Quark top” and the “Euler disk” on their mirrors. (The mirrors are just smooth surfaces.) Time how long each spins. To set the Euler disk going, you place on edge at a slight angle (tilted on the edge that that is rounder; the sharper edge tends to scratch the surface) on its mirror (the slightly concave one). If you give it a good spin it will spin like a coin. Record in your notebook interesting things you observe with the Euler disk as it comes to rest.

Category 3: There are three similar tops, two green wooden ones and one orange/green plastic one, that do something odd. (Have an AI show you how to work the button on the

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plastic top.) Why is this flipping behavior unexpected? What must be true about the motion for this to be consistent with conservation of energy? Though the sippy birds are not spinning, their iconic behavior is odd. What might provide the energy needed for them to keep going despite friction?

Now look back at the Category 1 toys. Can they actually be frictionless? (Hint: no.) What if we told you (truthfully) that they’d keep going for days or weeks? What would you suspect was going on? (Hint: aren’t the bases a bit heavy?) Try starting the satellite thingy at a point about halfway to one end.

Compare your understanding with your responses to the Warm-up.

Part III: Angular momentum (and more toys)

If you are uncomfortable trying this, do it very slowly and/or ask an AI to help you. You must be careful – if you spin too fast you can hurt yourself by falling off or someone else by hitting them, especially when you get dizzy! Sit on the comfy chair. Sit upright (don’t lean in any direction), with you arms extended horizontally to the sides. Push yourself with your feet, or have a lab partner push your arm so you turn slowly. When you are turning steadily, bring you hands in to your chest. What happens to your rate of spin? Put your arms back out. What happens? Everyone should try this – if you are nervous, do it in the reverse order, starting with your arms in. This will be quite tame. You can try this with the dumbbells in your hands. If you do, start with arms in.

Sit on the stool again. Have a partner spin up the bicycle wheel as fast as possible (note that this one is meant to turn only in one direction). Have him/her hand the wheel to you

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with the axle oriented vertically. With the stool at rest and with you holding the axle firmly at both ends, turn the wheel over, so that the axle that was on the bottom is now on the top. What happens?

Stand on the floor (no chair this time) and spin the wheel with the axle horizontal. Try to turn the wheel as if you were steering it for a left turn, keeping it upright. Hold the axles tightly, but let your arms respond to what the wheel does.

Find an AI and describe your observations to him. We will try to extract the properties of a new quantity called angular momentum from your observations.

Find the colored plastic oblongs. Try spinning them on a table, first counter-clockwise then clockwise. Why is this behavior odd? Does it violate the law of conservation of angular momentum? Why or why not?

Just for fun: turn over the pair of connected bottles to see the tornado. Spin the water-filled gyroscope.

Find the book held closed by the rubber band. Try flipping it in the air about each of its three perpendicular axes. What do you observe? This odd behavior is universal and well understood from Newton’s Laws.

Part IV: Air

An AI will do a demonstration in which the big vacuum cleaner can support a ball on its air jet. What must the direction of the net force of the air on the ball be?

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Physics 115B Lab 4: Radioactivity

In this lab, we will study radioactivity. Chapter 4 of your textbook is a good introduction to the physics and the terminology, but also to the effects of radiation. You probably know a bit, and you will learn more in this course, about the technological, political, and health impacts of radioactivity. From a scientific and philosophical point of view, however, radioactivity and radiation are a direct, observable connection to the atomic and sub-atomic world.

Though some of the sources of radioactivity we will use are created for scientific purposes, several are from the grocery store or the hardware store and are found in almost every home. We will get the radioactive atoms for one part of the lab from the air. None of the sources used in this lab are dangerous. Still, we will apply sensible rules that should always be applied to radioactive sources.

1. Minimize contact. Hold the needle sources by the stopper only. Keep them in their tubes when not in use. Hold the plastic disk sources by the edge. Don’t touch the actual source in the smoke detector. Decide what you are going to do with a source before you pick it up, and then do it.

2. Keep track. If your group is using one or more sources, each member should know where each source is. Never put the source in your pocket or somewhere hidden where it can’t be seen. Return the source to an AI when done.

3. No eating (or food) or drinking in the lab. Wash your hands before leaving the lab.

4. When in doubt, ask an AI.

To start this lab, look at the Learning Goals and the questions in Part III; use them to guide your investigations in Parts I and II.

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Learning goals

Perform measurements of things you cannot see.

Distinguish different types of radiation and different types of radioactive materials by measuring their penetration power, half-life, and tracks in the cloud chamber.

State evidence that radiation is made of particles.

Explain how half-life is a property of exponential decay.

Measure radiation from everyday objects.

Part I: Geiger-Mueller counter

The Geiger-Mueller counter is a very sensitive device that can detect the passage of individual particles (beta and gamma and maybe alpha rays) from radioactive decays. Because it can detect individual decays, it allows you to count them, hence “counter.” Your AI can describe how it works. Our G-M counters (GM-45s) are connected to computers, so the results can be stored and plotted. The computers are trained to make the traditional “clicks” that let you get a sense of how many decays are detected without plotting. We will find it most useful to measure the rate of detected decays. The computer does this by counting for a minute, plotting the result, then doing the same thing for the next minute, etc. The displayed rate is the counts per minute, or CPM.

The instructors will demonstrate how to use the GM-45 counters. They will also introduce some of the concepts we need to study radioactivity.

Note that the counting is a statistical process. Governed directly by quantum mechanics – it is fundamentally random. Even with measurements of perfect precision, we do not expect to get the same number of counts in each minute! This means that we must do some averaging, either numerically or by eye. It also means that we must be patient and let the average unfold. During the longer measurements, you can work on Part II.

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First make sure the computer is operating and recording data. The first thing to measure is the room background, the rate of counts with no source present. This may be partly an occasional misfire of the GM-45, but it is mostly radioactivity in the materials used to make the device itself and in the air and other surroundings (including you). Record this number – you will want to subtract it from all measurements of other sources.

The AIs will give each group a source. Write down what your source is. Place it gently on the GM-45 (the mesh is fragile). Record the rate. Try some other sources.

Use the ringstand and clamp to hold the Co-60 or Cs-137 source at various heights above the GM-45, say 4, 8, 16,… cm. Before you take the measurements, what do you expect to see? (We are looking for something more specific that “It will decrease.”) Plot the count rate vs. distance. Discuss the result.

We can learn a bit about the radiation from the source by trying to block it. There are various sheets of materials from paper to metal in various thicknesses. Carefully remove the plastic cover from the smoke detector and the metal cover from the source itself, then put it face-down on the GM-45, as the AI did in the demo. Try the various materials between your source and the GM-45. Start with one, then 2, then more sheets of paper, then go to the heavier materials. What do you conclude about the source?

Try this shielding test with another source if there is time.

In a radioactive material, individual nuclei decay to other nuclei, emitting particles that we call radiation (the old name was rays). We now know what these rays are: alpha rays are helium nuclei (these are made of two protons and two neutrons, an especially stable configuration that can be emitted as a unit from a larger, less stable nucleus); beta rays are electrons, and gamma

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rays are photons, the quantum-mechanical particles of light, but with energies much higher than visible light. X-rays are just less-energetic gamma rays.

Note how the particle model is “obvious” here: we count them. (We’ll address this again in a later lab.) The particles have specific properties, notably mass and electric charge. The most intuitive property, the particles’ sizes, turns out to be unimportant here. They are all so small compared to the size of atoms that none of our measurements reveal any information about particle size.

If the radioactive nuclei are decaying, should they eventually be gone entirely? Yes! For the sources we’ve been using so far, this takes years (Co-60), or tens of years (Cs-137, Sr-90), or hundreds of years (Am-241), or even billions of years (K-40), so you haven’t seen it in your measurements.

When radioactive nuclei decay, they do so in an odd way. Each type of unstable nucleus has its own half-life: the time it takes for half of the individual nuclei of that type in a sample to decay. Example: the half-life of Americum-241 is 432 years. The odd thing is that after 432 years, when half the original Americium is gone, it is not the case that the other half is about to kick. In fact, half of the remainder will live another 432 years, exactly as if the whole surviving half had just been born. Your text calls this “dying, but not aging.” Mathematically, this is exponential decay, which, along with exponential growth is observed in many natural phenomena (see Chapter 5 in your text).

The problem with doing experiments with radioactive nuclei with a half-life short enough to measure in a 3-hour lab is that if you buy some on Monday, it’s mostly gone by Tuesday. Instead, we need something that is made continuously that we can harvest and observe. Fortunately (in a sense, see below for the downside), uranium and thorium in the Earth’s crust is slowly decaying, and one of the resulting nuclei, Radon-222 (Rn-222), is a chemically inert but radioactive gas. It has a long enough half-life (about 4

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days) that it can seep out of the ground. When it seeps into a basement, it can collect there, and, if levels are high, can become a health problem. The nuclei that Rn-222 decays to (see Figure 1) are metals. They stick to dust particles in the air, and we can collect and concentrate them onto a paper filter by using the yellow air samplers that have been running in the adjacent classroom. Note from the figure that lead-210 (210Pb) has a long half-life. It is thus a bottleneck, and what we will see is the decays that precede this. We will be most sensitive to the beta-decays of lead-214 and bismuth-214. Because we are seeing a chain of linked decays, we will not observe a single half-life. For example, for awhile, as long as there is 218Po (polonium-218), it will replenish the lead-214, and likewise, the decay of the lead-214 replenishes the bismuth-214. Still, the nuclei will decay in tens of minutes, and we can find when the number of decaying nuclei has dropped to about half of its initial value.

Note that we will be measuring the rate of decays (counts per minute), while our discussion of half-life above was in terms of surviving radioactive particles. Based on your understanding of half-life, predict the shape of the measured CPM vs. time you will see if you observe a sample with a single type of radioactive nucleus for several half-lives. Make a sketch, labeling the location of one half-life.

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Figure 1. The radioactive decay sequence of uranium-238. The portions relevant to this lab are highlighted. The half-life of each nucleus is in its box (note the different units!), with the type of decay (, ) between it and the next box. (Note: a + is a positive electron: a bit of anti-matter!)

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When you’ve read the above description, make another room background measurement by leaving your GM-45 with no source on it for a few minutes. Then place an unused paper filter on the GM-45 to see if it changes the count rate.

Go with an AI to get a sample from the air samplers. Choose one and shut it off, noting the time and the time on the blackboard indicating when the sampling started. Carefully remove the paper ring holding the filter paper and take the filter back and place it on your GM-45, noting the time again. (Note that holding the filter is no more dangerous than cleaning the filter in a clothes drier, which is doing the same thing that the air sampler did.)

You will leave this running for much of the rest of the lab. This would be a good time to do Parts II and III.

When you have enough data: does the shape of the graph of CPM vs. time match your prediction? Use the graph made by the computer to estimate the half-life of this decay chain. (Call over the AI to help you print copies of the graph for your notebook.) Look back at Fig. 1 and think about whether your result for the half-life makes sense.

Part II: Cloud chamber – seeing the rays

Set up:

In its operating state, the chamber should have isopropyl alcohol soaked all the way up the blue filter paper on the side, with a couple mm of liquid on the bottom. If the whole thing is dry, add about 30 ml of isopropyl.

Make sure there is ice in the water cooling bath (a tub or Styrofoam cooler). Make sure the rubber hoses from the pump (small square black plastic thing) to the chamber and from the chamber to the bath are connected and the pump is submerged in the ice water. Plug in the pump and make sure the water is circulating.

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After the water has circulated for a few minutes, plug in the chamber. This does several things. It turns on a Peltier refrigerator in the base, cooling the bottom of the chamber. It applies a voltage to the yellow wire, which, if connected to the metal needle through the stopper, is intended to help clear away old tracks. It also turns on lights inside to illuminate the tracks.

If starting from scratch, it takes about 20 minutes for the chamber to show tracks. Check periodically that there is still ice in the bath.

How it works:

The alcohol wicks up the sides of the chamber and evaporates near the top, forming a clear vapor filling the chamber. The vapor near the bottom is cooled by the fridge in the base. In fact, it supercools, falling below the temperature at which it would normally be liquid. This state is unstable, needing only a triggering event to cause condensation. When ionizing radiation, that is, a charged particle such as a beta or alpha ray (but not a gamma ray), passes through the vapor, it can remove electrons from the atoms, leaving a trail of positive ions along the path of the ray. The ions trigger the condensation of the supercooled alcohol vapor along the path of the ray, and the resulting tiny droplets form a visible track. Neutral particles such as gamma rays do not make tracks. However, a gamma ray can strike an electron in an alcohol molecule, making the electron travel through the vapor. The electron (now identical to a beta ray) then leaves a track.

When the chamber is working, you will start seeing little blobs of fog near the bottom. You may also see the occasional longer tracks. These are due to traces of radioactivity in the lab and also perhaps to cosmic rays, which are due to high energy radiation from space hitting the atmosphere. Watch them for awhile, noting where in the chamber the tracks form.

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Have the AI bring you a radioactive source. The “thoria” and Po-210 sources produce alphas, the Sr-90 produces betas. These two needle sources can replace the stopper at the top of the chamber. Record notes on what you see in your notebook. Some sketches would be appropriate. The Co-60 and Cs-137 produce betas and gammas. These sources have to stay outside the chamber. Explore moving these sources around until you see the best tracks. Try blocking the radiation with the same materials used in Part I or other things lying around the lab. Again, record some notes on what you observe.

Part III: Some conclusions

Discuss these questions within your group and briefly note your conclusions in your notebook.

How might you distinguish between alpha, beta, and gamma radiation with measurements, like those you’ve made today and others?

What evidence have you seen today that “radiation” is particles?

List at least two ways to protect yourself from a source of radiation. (Let’s say it was small enough to carry, and you had to move it.)

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Physics 115B Lab 5: Waves

Learning goals

Describe two kinds of wave motion (transverse and longitudinal).

Define and measure the speed, frequency, wavelength, and amplitude of waves.

Recognize and create a standing wave (say in a string).

Give examples of waves in everyday life and describe how you experience their speed, frequency, and amplitude.

Part I: Introduction to waves

The instructors will demonstrate some of the basic properties of waves using a long stretched spring. Points to note:

These are transverse waves: the motion of any piece of the spring is perpendicular to the direction the wave itself is traveling. The maximum transverse displacement of a point on the spring as the wave passes is called the amplitude of the wave.

The wave is a disturbance of any shape; it doesn’t have to be a sine wave.

The wave reflects off the ends of the spring, even if they are held in place.

A sinusoidal wave has a frequency and period (in time) and a wavelength (in space), as well as an amplitude. The frequency and period are related by f = 1/T.

The speed a wave propagates along the spring is a different quantity from the speed of any bit of the spring. The speed the wave propagates is approximately independent of its

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shape and amplitude, and, if sinusoidal, its frequency and wavelength. (This speed depends on the properties of the spring: its mass and the tension.) This speed-independent-of-frequency is not true for some other types of wave. It is true for sound waves.

For a sine wave, the velocity of the wave (along the spring), and its frequency and wavelength are related by v (m/s) = (m) f (Hz). Units: 1 Hz = 1 Hertz = 1 cycle/s = 1 s−1.

When sinusoidal waves are traveling back and forth along the spring, they can form standing waves. A standing wave is a fixed pattern that does not move along the spring. It just oscillates in place. Though it is not moving along the spring, it retains the relation v = l×f, with v the speed of traveling waves along the same spring.

For a spring of a given length, standing waves can have only certain wavelengths. This and the constant wave speed mean that only certain frequencies make standing waves.

Note that these properties are the basis for the sounds formed by musical instruments of all types: string, wind, percussion.

Part II: Wavy toys

Play with the long spring. Feel a short pulse traveling back and forth along the spring. Make standing waves of several wavelengths. What must you do to the frequency with which you shake to go from a standing wave with one hump to two? Sketch the waves you make.

Play with a slinky. Lay one on a table or the floor and have a person hold each end. Make transverse waves. Make a longitudinal wave (a wave where pieces of the spring move parallel to the direction of motion of the wave itself. (Sound is a longitudinal wave.) Though it has little to do with

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waves, make the slinky walk down a stairway of potato-clock boxes.

Play with the colorful percussion pipes. Find a way to get a good tone by striking the pipe on anything non-breakable. (For purposes of this lab, we will consider all parts of an instructor to be breakable.) The waves in this case are sound waves. Compare the tones of the different pipes. Sketch some standing waves that can be formed in the tubes. (Assume that the open end of a tube is a node.) What is the difference in the tone when one end of a tube is capped? (Assume that the capped end of a tube is an antinode – where the standing wave is a maximum.) Sketch some standing waves possible in a tube capped at one end. From you sketches, how should the tone change when you cap one end?

Play with the little motors with the red strings. To use one: 1) Find one with both wires attached from motor to battery holder. 2) Put a battery in the holder – the motor will start; there is no switch. 3) Hold the free end of the string and let the motor hang.

Note the distinction: with the long spring, you have to shake it at just the right frequency to get a standing wave, and yet the percussion pipes and stringy motors seem to make a standing wave easily. Discuss this in your group and write your conclusions in your notebook.

Play with the other toys.

Part III: Speed of sound

The apparatus for this part is the long vertical tube. There is a can attached by a hose that contains a reservoir of water. By sliding the can up and down, the water level in the tube can be raised and lowered. Try it and see. At the top of the tube is a speaker. If one puts an alternating electrical current through the

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wires connected to the speaker, it makes the speaker cone vibrate with the frequency of the current. The cone pushes the air, making sound. (This is, of course, the same thing that happens in your iPod earbuds.) For our speaker, the alternating current is provided by a signal generator (the box with knobs). Turn it on. Note how the frequency (tone) and amplitude (volume) are adjusted by their respective knobs. (There is no need to touch the other knobs.)

What is less well known about a speaker is that it works backward as well: if you shake the speaker cone, the speaker produces an alternating current of that frequency in the wires connected to it. (That is, a speaker is also a microphone.) The amplitude of this current is measured by the attached meter.

Back to the tube. The water represents a fixed wall to the sound in the tube, where the amplitude of the air displacement must be zero (a node). The speaker is the other end of the tube. Less obviously, the speaker end is also a node. (Recall when we shook the long spring in Part I: the shaken end was approximately a node.) Your measurement will not depend on this fact, but see if you can verify it with your data.

Sketch some standing waves that have a node at both ends.

Raise the water can all the way and let the tube fill with water as high as it goes. Set the signal generator frequency to a convenient frequency (1000 Hz = 1 kHz = 1000 cycles/s works fine). Adjust the volume to get an audible but not-too-large sound. Lower the water can all the way in one go and let the water level in the tube drop steadily. Note the approximate heights where the sound gets louder. These are air columns whose length is right for a standing sound wave of frequency 1000 Hz.

With the water can still at its lowest setting, adjust the volume so the meter reads a pretty high value, say about 20. Raise the can to sweep the water height past each of the maxima found in the first sweep. Adjust it up and down

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around each maximum, using the meter to determine a more precise value for each height. (One person moving the can, one watching the meter, and one reading the height works well.) Note that the meter only goes up a bit – your ears are very sensitive!

Use the positions of the maxima to make scale drawings (that is, the length of the air column is proportional to the length in your drawing) of the standing waves at each volume maximum. (Note that the wavelength of 1000-Hz sound waves is a fixed value.)

From your measurements and drawings, what is the wavelength of the 1000-Hz sound waves?

Use this result to determine the speed of sound in air. From this speed, devise an easy-to-remember method for determining the distance to lightning by comparing the times of the flash and the boom (or check the method you have.)

Part IV: Seeing sound waves

Start with the pair of tuning forks, each mounted on a wooden box with an open end. The rest of the kit is a rubber mallet and a blob of clay. Start with no clay on either fork.

Take turns hitting one of the forks with the mallet until you can get a nice loud tone from it. Try the other fork. Note that they have the same frequency (that is, the same pitch). Now move the boxes so that their open ends face each other, a small distance apart. Strike one with the rubber mallet. Listen to the nice tone. Grab the fork you hit to stop the sound. The sound continues. Grab the other fork. What happens?

The other fork resonates. This is a word whose technical meaning matches its everyday usage. Technically, this resonance occurs only when the two frequencies are the same. This is a much

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better way of showing that the two forks’ frequencies are identical than just listening to them.

Hit both forks in succession. It should sound about the same as one alone. Now change the frequency of one of the forks by adding most of the blob of clay to it, near the top of one tine. Hit the fork with the clay – the resonance vanishes even if you can’t hear that the pitch has changed.

You will now hit both forks in succession. You might expect, with two frequencies very close together, to hear a nasty, dissonant sound. But this isn’t what happens.

Try it! Do it a few times, using a smaller blob of clay each time. Oddly, the smaller the blob (and thus the closer the frequencies are to each other), the more noticeable the effect. What you hear are beats: a surprisingly pure tone that wavers in volume. With a small blob, the beat period can be longer than a second.

For the rest of this part, have an instructor show your group the computer-based sound analyzer. The microphone turns the displacements of the air in sound into alternating currents in the wires. The program displays this current as a function of time in the top plot. Sound of a single frequency (say, from a tuning fork) appears as a pure sine wave. The bottom plot decomposes the sound into sine waves of all frequencies (a mathematical theorem states that any function can be decomposed this way) and plots the amplitude of each frequency. This is called the spectrum of the sound. Think of it as plotting the volume at each frequency. (Actually, what is plotted is the energy at each frequency, which is proportional to the square of the amplitude.)

We will generally just leave the program running (“On” in the upper-left corner). There is no need to change any of the settings – please don’t.

Try a tuning fork. Try the electronic keyboard (actually a sound synthesizer that simulates various instruments.) Set it

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to simulate a flute, which gives a pretty clean sine wave. Try some notes. Pick a note (if you have a musician in your group, write down which note it is.) Hold it down. Measure the period of the wave using the top plot. (For this one measurement, you can freeze the upper plot and get a readable horizontal scale in milliseconds (ms) by clicking “Capture scope” in the box at the lower-right.) Turn this period into a frequency (in Hz) and relate it to the peak frequency on the bottom plot.

How do notes that differ by one octave differ in the plots?

Try any two notes, a few keys apart, at the same time (not so easy on a real flute!)

Try any musical instruments you brought. Try singing or whistling a clear tone. (If you don’t have a large selection of musical instruments, use the electronic keyboard. If you hit the wrong button and get cheesy music, stop it by pressing the same button again – please.)

A flute gives a very “pure” (or “thin”) tone. A violin or trumpet gives a “rich” tone. How do the spectra of the flute and trumpet differ, even when playing the same note? (You can switch the instrument while holding down a key and watching the spectrum.) What makes the tone rich?

Back to beats: Set the keyboard back to flute. Hold down two keys next to each other at the high (right) end of the keyboard. How does what you see in the top plot relate to what you heard with the tuning forks?

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Notes on Beats Beats seem very mysterious: how can two nearby frequencies sound like a changing

volume? In fact, just a little math (one step, in fact), and there is no mystery here at all.

We start with two sine waves with slightly different frequencies, f1 and f2. Can you see that a sine wave with frequency f has the form sin(2f t), where t is the time? (Think: if the frequency is f, then the period is T = 1/f. Then, after one period, sin(2f t) = sin(2) = 0, and the function is back where it started.) So our two sound waves are

S1 = sin(2f1 t) and S2 = sin(2f2 t).

When two waves are present at the same place at the same time, the rule is that they just add. So what you hear is

S = S1 + S2 = sin(2f1 t) + sin(2f2 t).

Now, for our one step of math. There is a trig identity that does exactly what we need here:

2

cos2

sin2sinsinbaba

ba .

In our case, a = 2f1 t and b = 2f2 t, so

tff

tff

S2

2cos2

2sin2 2121 ,

or

t

ftfS diff

avg 22cos2sin2 ,

where favg is the average of the two frequencies and fdiff is the difference between the two frequencies. For the case of beats, since the two frequencies are close together,

favg ≈ f1 ≈ f2 and fdiff << favg.

So, what does this sound like? We can plot this for f1 = 20 Hz and f2 =18 Hz, where favg = 19 Hz and fdiff / 2 = 1 Hz. The plot of t192sin (which looks about the same as

sin(220 t) or sin(218 t)) is

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The plot of 2cos(2t) is

The formula says that sound you hear is the product of these two graphs. At each time, take the value in the top plot times the one in the bottom plot. Because the second plot changes slowly, its effect is to make the amplitude of the top plot go from large to small to large – the result looks like this

Look at the first two plots till this makes sense!

So how does this sound? It sounds like a pitch about equal to the original pitches, but varying up and down in volume (amplitude) with a frequency that is equal to fdiff , the difference between the two original frequencies. (Count how many cycles of volume there are in the 2 sec shown in the plot.) This gives the impression of a pure tone that is wavering rapidly in volume. The closer together the original frequencies are, the slower the wavering, because fdiff is smaller.

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Physics 115B Lab 6: Light as a Wave

Learning goals

Turn intuitive sense of “light rays” into an ability to explain optical effects such as reflection and refraction and optical devices such as parabolic mirrors, periscopes, and optical fibers.

Explain how phase + superposition gives interference. Demonstrate the geometrical construction for 2-slit interference and explain the dependence on wavelength and slit separation.

Know properties that tell us that light is a transverse wave.

Laser safety

We will be using only laser pointers, which are reasonably safe. However, as with our radioactive sources, you should use good, safe practices.

Do not shine a laser pointer at yourself, at anyone else, or around the room.

Realize that there can be reflections that are nearly as bright as the laser itself – watch out for them and warn others.

In many of our applications, the laser is held horizontally. Much of the original and reflected light will be in a horizontal plane. Keep your eyes out of this plane!

Part I: Reflection

The instructors will demonstrate some of the features of optics using reflection with a pair of curved (parabolic) mirrors. As you may know, such a mirror focuses light, that is, it brings rays coming in parallel to its axis to the same

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point, no matter where they hit the mirror. It also does the reverse: takes all rays from that point that hit the mirror and directs them in a beam parallel to the axis. In this demonstration, the light is infrared, that is, below red in frequency. It is not visible, but it carries energy and is felt as heat. After watching the demo, make a sketch that shows how the rays go from heater to match.

Play with the periscope and make a sketch of how its flat mirrors direct the light.

Reflection has a simple rule (“angle of incidence equals angle of reflection”), but it can still lead to complicated situations. Take a look at the black mirrored bowl on the table. Standing back a bit, shine a laser pointer on the pig. Now go ahead and grab the pig.

Part II: Reflection and refraction: Optics

Drawings like you made in Part I are called ray tracing. In this part, you will use the beam from a green laser pointer to make its own trace by directing the beam at a grazing angle along the white paper on top of the box. We will use the plastic optical elements (mirrors, prisms, and lenses) to study reflection and refraction. This means you will have to direct the ray grazing the paper onto or into these elements as they sit on the paper. This takes some practice and some patience – ask an instructor for help if it gets too frustrating. Take turns with each member of your group pointing the laser for at least one element, and everyone making suggestions and sketches. In the pictures for each part, the arrows indicate that you should move the laser or the element, keeping the other fixed.

Flat mirror. Sketch the reflection at at least two orientations of the mirror. Do you see the simple rule referred to in Part I?

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Concave mirror. Sketch rays with the laser in different positions (keeping the beam parallel.) Do you see the behavior described in Part I? Locate the focus of the mirror.

Refraction. Use the rectangular prism. (A pretty oblique angle of incidence, as shown, helps.) Sketch what you see. It’s intricate! Just one setup here, but show all the important rays.

Converging lens. Sketch rays with the laser in different positions (keeping the beam parallel.) Locate the focus of the lens.

Diverging lens. Sketch rays with the laser in different positions (keeping the beam parallel.)

Semicircular prism. Here we are looking for a different effect. What happens when the beam is near one edge of the prism (still perpendicular to the flat face)? Make a sketch.

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This is called total internal reflection – it is the basis for fiber optics and much of modern communication (see next item).

The wavy rod is our optical fiber. Direct the laser beam into it at one end. (You are no longer trying to have the beam graze the paper. You can see the beam inside the rod.) Sketch what happens.

Part III: Polarization

Like a wave on a shaken string (but unlike sound), light is a transverse wave. This means there are two independent directions of oscillation. (For the string this would correspond to shaking vertically and shaking horizontally.) This direction is called the polarization. A polarizer blocks one polarization and lets the other through. The string analogy would be a picket fence that lets vertical shaking through but blocks horizontal shaking.

Look at the setup on the overhead projector. It has two polarizers; the bottom one rotates. The light from the projector itself is unpolarized – it is a random, changing mixture of both polarizations. If such light passes through a polarizer, it emerges with a single polarization determined by the orientation of the polarizer. Turn on the projector and focus the circular patch on the wall. Rotate the bottom polarizer and describe what happens. Discuss and write down a good explanation. (Here’s one: “Polarized light only goes through a polarizer if the light and polarizer have the same orientation.” Does that explain your observations?) Include sketches with your explanation.

Use the small polarizers to view reflections of one of the ceiling lights (fluorescents work best) from the plastic bag or the tray of water on the nearby table. Rotate the polarizer. What do you observe? Look at the light directly with the polarizer and rotate it. Is the light directly from the lamp

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polarized? What do you conclude? This effect is why sunglasses are often polarized “to reduce glare”. If you have your sunglasses with you, try the same thing with them. Try looking through them with a polarizer.

Some materials are optically active – they affect the polarization of light. You can see this by placing such a material between crossed polarizers that otherwise pass no light at all. Though explaining this is beyond our scope, it is striking and useful, so have a look – no writing required. Hold up and look at the thick plate of glass with the crumpled cellophane (and the weird drawing of dancing alligators – no one knows what’s with that) in the room lights. Now set the polarizers on the overhead projector to block all light. Put the alligator plate between the polarizers. Rotate the bottom polarizer. Cross the polarizers again. Put the tuning-forky plastic thing between the crossed polarizers. With it still between the polarizers, squeeze the forks together. Stress makes the plastic optically active. Viewing models made of plastic with polarizers allows engineers to study the stresses in designs of buildings and machines.

Part IV: Interference and Diffraction

The property of adding constructively or destructively, a property known as interference, is fundamental to waves of all types.

Diffraction is a related effect, where a wave passing through a narrow aperture (or past an edge) spreads out on the far side. You saw an example of this in your last homework, where you played with a simulation of water waves passing through a gap in a barrier. This same simulation is running on the computer in the lab so you can refresh your memory. We need one result from this: even though the section of wave that goes through the gap is traveling only in one direction (to the right), if the gap is narrow,

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the waves on the far side spread out so that the crests are circles: the wave on the far side is propagating outward in all directions.

Consider what this implies when the beam of one of our laser pointers shines on an opaque card with a single tall-but-narrow vertical slit in it. (The width of the beam is much bigger than that of the slit.) Diffraction makes the beam fan-out horizontally on the far side of the card. If there was a screen some distance away on the far side of the card, you would see a very dim horizontal streak of light on it. (Dim because only a little light gets through the slit and it is spread out.) There is little spreading vertically because the slit is tall, so the streak has a vertical spread that is about what you get for the original laser beam at that distance.

This is just some background information for the case we are really interested in: what if the laser beam illuminates two (or many) parallel slits, very close together?

Shine a green laser through the “500-line-per-mm” diffraction grating onto the wall. (The big book with a rubber band is a convenient mount for the slide. Try to touch only the frames of the slides.) “Line” means “slit,” so this grating has 500 slits in each millimeter. Be sure to realize that the dots you see are not just the projection of the light through a single slit – recall that each slit alone would give a dim streak. Each dot is the result of interference of the light coming through all the slits hit by the beam.

Two-slit interference

Our goal is first to understand this conceptually and quantitatively, and then to use it to make some precise measurements. Since we will need a more detailed understanding than the one established in lecture so far, we will develop it in some detail here.

We will consider two slits a distance s apart. A top view of the setup is shown in the figure below. There is a screen a distance D

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from the slits, with D >> s. The slits are so narrow that each alone makes a streak all the way across the screen horizontally (from top to bottom in the figure). The figure shows the ray from each slit that reaches a point on the screen a distance y from the center.

Since both slits see the same light wave incident from the left,

the two waves leaving the slits start out in phase with each other, that is, the crests (maxima) occur at the same time in both slits. The key to the interference of the two waves at the screen is that they travel different distances to the screen. (Can you see in the figure that the ray from the top slit to the screen is shorter than the one from the bottom slit to the same point on the screen?)

What matters: how long the extra distance traveled by the bottom wave is compared to the wavelength of the light, . If this extra distance is /2 (or 3/2, 5/2,…), the waves arrive at position y on the screen out of phase, with a crest (maximum) of one arriving at the same time as a trough (minimum) of the other and vice-versa. This means that the two waves destructively interfere (that is, they cancel) and the screen is dark at y. If this extra distance is 0 or (or 2, 3,…), the waves arrive at position y on the screen in phase. They then constructively interfere, and the screen is bright – these are the dots you observed.

D

s

y

Incident

light

Screen

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To do this quantitatively, we need to determine this extra distance. We reproduce the figure, with a line making an isosceles triangle with the top ray and most of the bottom one.

Since it is an isosceles triangle, the two long sides are equal, and the extra distance in the bottom ray is the short segment marked x. When y = 0 (at the center of the screen), x = 0, the two rays have the same length, the two waves are in phase at the screen, and there is a bright dot.

On the table at the center of the lab, there is a tape representation of this figure. Using meter sticks for the rays and rulers as needed, find the location of the next bright dot (that is, the first one away from the center) assuming a wavelength = 5 cm. When your group has this set-up and agrees it is correct, call over and show it to an instructor. Where would the next dot (on the same side of the center as your first one) be?

With this conceptual picture and sketches in your notebook (using the meter-stick setup if you want), answer these questions: What happens to the pattern of dots if the spacing of the slits is reduced? What happens to the pattern of dots if light of a longer wavelength is used?

D

s

y

Incident

light

Screen

x

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Back at your laser setup, mount the 1000-line-per-mm grating next to the 500-line-per-mm grating so you can switch the beam back and forth between them. What happens to the dots? Is this consistent with your prediction?

Point the red and green lasers at the 500 grating simultaneously. (This takes some fussing.) Adjust the pointing of one laser so that a red and a green dot line up. How does the spacing of the red and the green dots compare? Is this consistent with your prediction? Look at the room lights through the 500 grating (you should look through it at an angle to the lights.) Explain in your notebook why it makes a rainbow. Note that this is a different mechanism than in a prism or a rainbow in the sky.

Look at the room lights reflected in the CD and DVD. The rainbow indicates that the narrowly spaced tracks are acting as a diffraction grating. Reflect the green laser off the CD onto the wall. (This is best done with the CD flat on the table and pretty close to the wall.) Slide the DVD in front of the CD to compare the pattern of dots. Which has the smaller track spacing, thus holding more information?

Shine the green laser on different regions of the slide marked “Unit cell”. Instead of long, narrow, essentially one-dimensional slits, these patterns are two-dimensional arrays with various symmetries, meant to evoke the patterns of atoms in crystals. In fact, x-ray diffraction is used to infer the structure of atoms in crystals from the pattern of dots the diffracted x-rays make. Such information was crucial to the determination of the structure of DNA.

To make this quantitative, we need a way to determine x analytically. Though it isn’t absolutely necessary, making an approximation (a very good approximation) makes this a lot easier. Look back at the isosceles triangle in the most recent figure. Take the short side as the base. The height of the triangle is roughly D

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and the base is roughly s. (These aren’t the approximation we mentioned – that’s coming later. For this step, rough numbers are all we need. If you wish, consider the triangle to the center of the screen, y = 0.)

In your notebook, sketch an isosceles triangle with a base s and height D. Blow the dust off your trigonometry and find an expression for the base angle (the ones shown with little arcs in the figure) in terms of s and D. Consider your laser setup with the 500-line-per-mm grating. What is s? For now, take D, the distance to the “screen,” to be about 1 meter. Use your expression and your calculator to find the base angle in degrees.

You should have found that the base angle is very close to a right angle: when the screen is much farther away than the separation of the slits (a factor of about a million for your laser setup), the isosceles triangle is, bizarrely, also very close to a right triangle, with angles very close to 90-90-0! The two rays are, to very good accuracy, parallel. We redraw our figure once again below. Since on the page we can’t draw it D ~ 106×s, we can’t draw it with the rays about parallel and also meeting at the screen, so we keep the essential geometrical features and let the top ray point funny. We have made one more approximation: measuring y from the bottom slit rather than the middle. Why is this also a very good approximation? (Hint: how far apart were the dots from the grating?)

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Using this new figure and identifying similar triangles, find

an expression for x in terms of s, D, and y. If y measures the separation between the center dot and the next one, find an expression for the wavelength of the light in terms of s, D, and y.

Convince yourself that this two-slit result works for the grating (with many slits a distance s apart). Hint: start by considering two slits near the center of the grating. If the path difference to a point on the screen is one wavelength, the light from those two slits constructively interferes. For the same point on the screen, what is the path difference for one of the slits and, say, the 10th slit to the left? How does light from these two slits interfere at that same point on the screen?

Mount the 500-line-per-mm grating in its book. Measure D for this setup. For the green and the red lasers, measure y. From these measurements, find the wavelengths of green and red light. Give your answer in nanometers (1 nm = 10−9 m).

You estimated the wavelength of light in Lab 2. Look back at your result in your notebook. Here you measured it quite precisely (using a ruler again!) In your new measurement, which do you think has a bigger impact on the accuracy: the

D

s

y

x

12

approximations we made to draw the last figure and derive our expression for , or your ability to read the ruler?

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Physics 115B Lab 7: Measurement and Uncertainty

Learning goals

Know that all measurements have uncertainty and that a measurement reported without its uncertainty is of limited value.

Be able to interpret measurement and uncertainty expressed as m ± s, where s is the standard deviation.

Estimate the uncertainty of a single measurement in two ways.

Estimate the uncertainty of the average of several or many measurements from the uncertainty in a single measurement.

Estimate the uncertainty of a measurement that involves counting.

News

A recent New York Times/CBS News poll of likely Republican primary voters indicated than Herman Cain had the support of 25% of the voters, Mitt Romney had 21%, and other candidates much less. A question to think about (we’ll return to this at the end of the lab): Did this poll indicate that Cain was ahead? (Note that in two of the last three presidential elections, the margin of victory was less than this.)

Uncertainty

The uncertainty in a measurement expresses the range of true values of the measured quantity consistent with the measurement. Of course, we think the true value of a quantity is a single (perhaps unknown) number. The range comes from the fact that measurements themselves are not perfect. Uncertainties can be

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categorized as random or systematic. An example of a random uncertainty is trying to measure a length to better than half a millimeter with one of our rulers. If you repeatedly line up the ruler and read it again, you may randomly read numbers that differ by a few tenths of a mm. A systematic uncertainty would result from, for example, the scale on the ruler being printed slightly too small. (It, too, cannot be perfect.) This would make all measured lengths too large. In general, random uncertainties cause a spread in the data around the true value; systematic uncertainties cause a shift in the data.

The goals of any measurement should include quantifying the uncertainty and, if possible, reducing it. There is a lot of mathematics in quantifying uncertainty, but in our lab we will introduce and use only general rules of thumb. An understanding of these rules of thumb will serve you well in interpreting political polls, making medical decisions, and in Las Vegas.

By the way, scientists use the terms “uncertainty” and “error” interchangeably. Since most people use “error” to mean “mistake”, we will try to stick to using “uncertainty.” Go ahead and correct us if we say “error” (unless we are telling you that you made a mistake!)

To study measurement, we need measurements to study. You will measure your reaction time in several ways. This is quick and easy, but will give you lots of data to interpret and several complete measurements to compare.

Part I: Dropping the ruler – uncertainty in a single measurement

Our first method is indirect but simple. Your lab partner will hold a ruler dangling from one end. You will hold your fingers around the ruler at the bottom, lined up with the zero mark. Your partner will drop the ruler, and you will close your fingers, catching it, as soon as you see it drop. You will read the scale (in

3

cm) where you are pinching it. Clearly, the longer you took to respond, the farther the ruler fell. We’ll interpret this as a time later. For now, let’s consider the distance the ruler falls, the “reaction distance,” as the measurement of interest.

Practice this a few times until both you and your partner are comfortable with the procedure. Work on some aspects of technique: Take care to start with your fingers aligned at zero. Try to read the ruler the same way each time. If you read the ruler at the top of your finger, align at zero the same way. The dropper should try not to signal the drop is coming by moving. Try to minimize measurement uncertainty – a little care now will make the interpretation easier later.

Open the Excel spreadsheet called react.xls. You will enter your data in the columns labeled “Distance (cm)”. (Don’t change the others.) Put your name at the top of one of the columns.

Do 16 drops and write the results in your notebook. You should record every drop, no matter the outcome. Eliminating data is sometimes necessary, but is always suspect. It would be easy to bias the results by ignoring long distances because “I wasn’t ready.” Do not just eliminate points that are “not typical” – real measurements have real fluctuations. On the other hand, you will find that a 2-cm distance is not likely to be an actual reaction. If you must eliminate a data point, justify it in your notebook in detail and take a new point to replace it. Enter your 16 measurements in your column in Excel. Note that it plots them. Is there any trend? If so, can you think of a reason?

Switch roles with your partner and repeat the measurements. (If you are in a group of three, each person should drop and catch at least once.)

Each of your 16 measurements has various contributions to its uncertainty. One approach to quantifying the uncertainties is to estimate as many of these as can be identified by considering

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properties of the measurement itself. We start with random uncertainties.

Do another drop/catch or two. Look carefully and estimate the uncertainty in your measurement of where you caught the ruler, that is, how precisely (in cm) did you determine the location of each catch? Look carefully and estimate the precision with which you lined up with zero to start. The uncertainties in both the start and the catch contribute to the precision of each of your drop distances – add the two together as an estimate of their combined effect.1 Each of you should make her/his own estimates.

Open the program histogram.exe. You will use this to plot your distance measurements in a different way and to extract more information. Go back to your Excel spreadsheet, highlight your 16 measurements with the cursor and copy them to the column on the left side of the histogram window. (You can ignore the error message that sometimes appears.) Click “Plot Histogram” at the lower-right. If necessary, make some adjustments to the plot: Set “X-Axis Scale” to manual. Set “Xmin” to 0 and “Xmax” to a round number that encompasses all the data in your group (say, 30 or 40). Set “Bin Type” to “Select Bin Width”. The bins are the segments along the x-axis into which the measurements are grouped. Try several settings for “Bin Width” (1, 2, …) and see what happens. A bin width of 1 is probably a good setting for this set of data

You have probably seen a histogram before – they are commonly used to display test scores. If you are not familiar with them, take a look at this one. The x-axis is your measured quantity. The y-axis is labeled “Frequency”. The histogrammer looks at each data point (drop distance, test score,…) in the data

1 Technically, this is not correct. For this case, one should square each term, add the squares, then

take the square root of the total. If you try it, you’ll see that the result is a somewhat smaller uncertainty. Just adding is good enough for our purposes.

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column, finds what bin it belongs in, and adds one to that bin. “Frequency” is how often a measurement in that bin happened. Try changing “Bin Width” again. Make sure you understand what a histogram is. Set the Bin Width back to 1 when you are done.

Print a copy of the histogram of your data. (Make sure the printer is on, then File > Print > Print Chart.)

Each member of the group should print his/her plot using the same x-axis scale and bin width. To remove the previous data set, highlight the data in the column, then Edit Table > Cut.

The histogrammer also gives two numbers summarizing the data you entered in the column. The first is the average (also called the mean), which we will call a. The second is the standard deviation, which we will call s. Both of these quantities refer to your data, and both have the same units as your data, in this case cm. You already understand the average. It is a good way to summarize your measurements in a single number. This seems right intuitively. The standard deviation is a measure of the spread of your measurements. It is thus related to the uncertainty of any single one of your measurements. It has a mathematical definition, but we are more interested in a more practical, working definition, which we will now develop.

On your printed histogram, put an easy-to-see dot with x = a (the average of your measurements) and y about halfway between the top and bottom of the plot. Now put little tick marks at the same height, but with x = a + s and x = a − s. Connect the three marks with a horizontal line. (This is a standard way to indicate the mean and standard deviation graphically. It is called an “error bar”, but really should be called an “uncertainty bar”.)

Count how many of your 16 measurements are between the tickmarks. What fraction of the measurements is this? Compare with your lab partner(s). Even for histograms that

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look very different from each other, this fraction is usually about 0.68. Write your and your partners’ average, standard deviation, and fraction of data between a − s and a + s in your notebook. Our working definition of standard deviation of a set of measurements is that the interval from a − s to a + s contains about ⅔ of the measurements. It is a measure of the spread of the measurements.

You know that we often take several measurements of the same quantity because the average is somehow more precise. (We’ll see how precise in a minute.) We now see another reason: this is the second way of estimating the random uncertainty in a single measurement – looking at the spread of values. Note another important interpretation of standard deviation: a “typical” measurement will be about a standard deviation from the average (that is, about ⅔ of the measurements will be closer and ⅓ farther away).

Compare the standard deviation of your measurements to the uncertainty you estimated from reading the ruler (start point and catch). Is ruler-reading the largest source of variation in your measurements? Do you think your actual reaction time is the same for each drop? If not, would a perfect single measurement be a perfect prediction of your reaction time for the next drop?

Our two methods for estimating the uncertainty of a single measurement are thus 1) studying the measurement itself to determine individual contributions to the uncertainty and 2) looking at the spread (standard deviation) of a bunch of measurements. They are useful in different ways. The first method can be used when only a single measurement can be done. It also can give you the sources of the uncertainty and let you work to reduce the largest sources. However, the true uncertainty can only be larger than the combination of all the contributions you can think of. The second method, finding the standard deviation, covers everything, but doesn’t explain anything.

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However it is estimated, we append the uncertainty s to a single measurement h by writing h ± s. Also useful is the fractional uncertainty (or relative uncertainty or precision) s/a, often given in percent, where we can use the average or a single measurement in the denominator.

Using the average and standard deviation of your measurements, what is the fractional uncertainty, in percent, of your individual measurements of the reaction distance?

All of our discussion so far has involved random uncertainties. As an example of a systematic uncertainty, consider the effect of starting the drops with the zero of the ruler well below your eye level. It is likely that this angle of view would make you always hold your fingers too low (try it). This would affect all the measurements, and thus the average, by the same amount. This would not be reflected in the spread of the measurements. Though we won’t do it here, the best way to deal with a systematic effect is to measure it and correct for it. If you cannot measure it, you can include a systematic uncertainty with your result.

Part II: Reaction time – “propagation of error,” uncertainty in an average, significance

We already know the relation between reaction distance and reaction time – it is the freefall time we used in our ball-launch in Lab 3,

g

ht fall

2 .

If we take h as our measured reaction distance, tfall is our reaction time.

Take a look at your spreadsheet again. It automatically applied this formula to each of your measurements. Click on one of the entries under “Reaction time (s)” and look at the formula it uses. Can you see that this takes your reaction distance in cm and

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gives the reaction time in seconds? If you are not familiar with Excel, ask your partner or an AI for a brief explanation. Hit “Enter” before clicking on anything or typing to keep the formula unchanged.

Take your column of reaction times and enter it in histogram.exe. Choose a reasonable Xmax and Bin width (they can and should be less than 1). Ask an AI if you aren’t sure. Make sure that you and your partner use the same settings. Print copies of your histogram for you and your partners. Each one in your group should do this, so you both/all have everyone’s plots. Tape them in your lab book, one above the other, so the ends of the x-axes line up to allow comparison.

What is the fractional uncertainty of your reaction time measurements? (Give a single number, in percent.) Compare this to the fractional uncertainty in the reaction distance measurement you already calculated.

Did you note that the precision of the reaction time measurements was much better than the precision of the reaction distance, even though the latter was the only input to the former? This is a (particularly favorable) example of error propagation – finding the impact of the uncertainty in a measurement on conclusions drawn mathematically from that measurement. In this case, the square root in the formula means that a largish fractional change in h makes a smaller fractional change in t.

OK, we now know how to estimate the uncertainty in one measurement of reaction time. But, we have 16 measurements and took the average. We know that should be better than one measurement, but how much? It turns out that if you have n measurements with about the same uncertainty, the uncertainty in the average is the uncertainty in one measurement divided by n , that is,

n

ssav .

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This works for any random uncertainty, hence here it reduces the impact of the ruler reading and even the actual variations in reaction time. This is the reason we average! (Note that averaging does not reduce a systematic uncertainty.) If we have n measurements m ± s, we can present our result as

avsa ,

where a is the average, and sav is given in the previous formula. This sav has the same property as our standard deviation of a single measurement: if we do a bunch of experiments, each with the same number of measurements, we expect that about ⅔ of the experiments will give results (each of which is an average!) within sav of the average of all the measurements of all the experiments.

Write down your result for your average reaction time, including its uncertainty, and those of your lab partners in your notebook. Draw error bars showing avsa on each of the plots.

With your results in this form, we can finally ask and answer meaningful questions. For example: is your reaction time different from your lab partner’s? Your averages are almost certainly different. The question is: are they significantly different? By this we mean: is the difference larger than might be expected from the uncertainties alone?

Look at this graphically in your notebook by extending the ends of the error bar you added to your histogram vertically to reach the error bar on the histogram of your. Do the error bars overlap?

More rigorously, we write our difference in our now-standard form

diffsd , where sdiff is the standard deviation, that is, the

uncertainty, of the difference. This clearly depends on the uncertainties in the two measurements that make up the difference. Here we will use the real version mentioned in Footnote 1 in Part I. For a difference d = x – y, the standard deviation is

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22yxdiff sss , where sx and sy are the standard deviations of x

and y. (If x or y is itself an average, the corresponding standard deviation is calculated from our relation for sav. Note also that if sx ≈ sy, ssdiff 2 , where s is either sx or sy.)

Write the difference between your measured reaction time and your partner’s and its uncertainty in this form.

We assess significance by comparing the difference to its standard deviation. Centuries of experimentation have taught us to be careful, especially when the stakes are high. Do you think a one-standard-deviation difference is proof that the difference is real? Why or why not?

How many standard deviations is the difference between your reaction time and your partner’s?

Physicists consider a 3-standard-deviation difference to be “evidence for” an effect and a 5-standard-deviation difference to be the “discovery” of an effect. ( diffsd 5 contains 99.99994%

of the data if the uncertainties are truly random!)

By these standards, how do you interpret the difference between your reaction time and your partner’s? Is it likely to be a real effect?

Part III: Timer

We can, of course, measure time directly. Here we will redo the reaction-time measurement with different apparatus. We are using the timers you used on the light-gates in Lab 3. Note that one tap-key starts the timer and the other stops it. The simplest way to run this is for the starter to hit her key hard enough to make a sharp sound. The stopper reacts to the sound. Some technique: can you see why the stopper should not see the starter’s hand or arm?

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After a bit of practice, each lab group member should do 16 trials. You shouldn’t use Excel for this, just enter your column of 16 times directly into histogram.exe. Print your plot for your notebook.

Compare the fractional uncertainty from this method to that of the ruler drop.

How significant (that is, how many standard deviations) is the difference between your reaction times from the two methods?

Part IV: Uncertainty in counting

There should be no measurement uncertainty in counting objects. If there are 10 potato clocks and you count 11, that’s just a mistake. However, when counting occurrences of some event, randomness can enter in any of a number of ways. You’ve seen this at its most fundamental when you counted radioactive decays. You also know that each side of a die has a 6

1 probability of

occurring, but you don’t expect exactly two ones in 12 rolls.

A very useful rule of thumb for the random variation in the counting of such occurrences is that the standard deviation is approximately the square root of the number of occurrences. For example, if you roll 100 dice, you expect, on average, 7.166

100

ones. Using our rule of thumb, that would be

1.47.167.167.16 .

This uncertainty has our usual meaning: about ⅔ of rolls of 100 dice should have 13-20 ones. (You can’t have 12.6 ones in one roll.)

We actually have 100 dice. Each group should roll all the dice a few times and record the number of ones in each roll on the blackboard. The dice have holes in them (a long story), but that doesn’t affect their behavior, and you can clearly tell which side is a one.

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When all the rolls of all the groups are done and on the board, does “13-20” roughly describe the spread in values? Find the fraction of rolls with 13-20 ones.

You may have a sense that if you roll more dice at once, the number of ones will get closer to 6

1 of the number of dice. Is

this the case? How far from 17 ones will a typical roll of 100 dice be? How far from 167 ones will a typical roll of 1000 dice be?

Now imagine betting on rolls of dice. Let’s say you are $2 behind after 10 rolls. Can you expect to be closer to breaking even after 100 rolls? (Thinking that you will be closer because “it will average out” is called the Gambler’s Fallacy.)

It is odd to think that randomness has predictable features. As we discussed in Lab 4, radioactive decay is a fundamentally random process. We ran into this as soon as we tried to measure something as simple as the “room background”: the number of counts in each minute with no source on the Geiger counter. Look back at the plots from Lab 4 in your notebook to remind yourself of how this looked.

Open the file G47_10-19-2010.bmp (on the Desktop of the computer). This is from an overnight half-life measurement we did after last-year’s Lab 4. Recall that each point is the result of counting for one minute. Look at the fluctuations in the room background before the source was put on and after it had decayed away. Now open the file G47_10-19-2010.txt, which has that same data in numerical form.

In your notebook, plot the room background, the first 11 data points. Your x-axis can be minutes, with the first point at 1,… All the data is clustered around the average, so you need not start your y-axis at zero.

Calculate the average room background (in CPM) for the first 11 points. Draw a horizontal line at the average through your

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plot. From our counting-measurement rule of thumb, what do you expect the standard deviation of a single minute of counting to be? Draw horizontal lines at a ± s on your plot. What fraction of the points is within one expected standard deviation of the average?

It not particularly impressive that about ⅔ of the points are within 1 standard deviation of the average – this is our working definition of standard deviation. What is impressive is that you were able to predict the standard deviation from just the average! This is the magic of the statistics of counting and doesn’t apply to measurements in general.

Test your understanding with this example from recent news:

The October 25 New York Times/CBS News poll interviewed 1475 voters, chosen randomly. Of that group, 455 respondents identified themselves as likely Republican primary voters. Among this group of 455, support for the various candidates was

Herman Cain 25%

Mitt Romney 21%

Newt Gingrich 10%

Ron Paul 8%

Rick Perry 6%

Others <2% each

Was the support for Cain significantly different from the support for Romney among these voters on October 25? Answer this by calculating the significance as defined above and interpreting it. Hint: you must work with actual counts, not percent.

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Physics 115B Lab 8: The Quantum World

Part I: Electrons as particles and waves

This part of the lab uses two similar devices. Each is a glass bulb with a filament inside, like a light bulb, but in this case the electric current in the filament heats it enough to allow electrons to escape from it. A second power supply (the first supplied the filament heating current) applies a voltage between the filament and a nearby piece of metal, making an electric field that applies a force on the electrons. Each electron accelerates, and the piece of metal is designed to let most of the electrons miss it. This produces a beam of electrons that can then be directed by subsequent electric and magnetic fields.

The bulb of the first device is almost at vacuum, but a small amount of gas is left inside. When electrons hit the gas molecules, the molecules can gain energy, then release it as light (a nice example of quantum mechanics). We use this to make the beam visible. You will need to dim the room lights and maybe use the black cloth to see the beam.

Turn on the “Leybold” power supply. This box supplies both the filament heating current and the electron accelerating voltage. The filament current is not adjustable – it turns on when you flip the main switch. You should see the filament glow – let it warm up for a minute or so. The upper-right knob controls the accelerating voltage. Turn it up slowly – the voltage is shown on the meter on the bulb stand. Watch for the beam as the voltage passes 100-150 V or so. Set the accelerating voltage to about 150-250 volts. Hold a permanent magnet near the beam (avoid banging the glass) and watch what happens. Flip the magnet end-for-end. Let everyone in your group try this. Sketch what you saw.

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The other power supply provides current to the coils outside the bulb. This configuration makes a quite uniform magnetic field throughout the bulb. Turn on the supply and turn the current up slowly. Stop at about 0.5 A. Sketch what happens to the beam. If you know a little about magnetic fields, explain what is happening to your lab partners. Vary the strength of the magnetic field and the electron-acceleration voltage a bit and describe what happens.

Note that the language we have been using treats the electrons as particles. (For example, we talk about “accelerating each electron.”)

An instructor will now show you the second device. (This is a bit fragile, and we only have one – please let an instructor turn the knobs.) It is almost identical to the first. In this case there is just vacuum in the bulb, so the beam is not visible. However, the end of the bulb is coated on the inside with a phosphor, which glows when the beam strikes it. (The combination of the two devices – an electron beam steered by electric and magnetic fields striking a phosphor-coated screen – is a good model of how old TV’s and computer monitors work.)

Try a magnet on this invisible beam to see that it behaves similarly to the other one. (Please don’t bang the bulb with it.)

The main difference between this device and the first one is that there is a target that the beam must go through before hitting the screen. The target is polycrystalline, that is, the material is a crystal with regularly-spaced atoms, but the target is made of many small crystals with random orientations.

Observe what happens when the accelerating voltage is increased. At sufficient electron momentum, a pattern appears. Sketch this pattern.

This pattern is a diffraction pattern, with the atoms on the crystal acting like the slits of the gratings you used in the last lab.

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In the last lab, what was the pattern of a 1-d array of slits? Of a regular 2-d array (the “unit cell”)? Can you explain why randomly-oriented chunks of a crystal, all with the same atomic spacing, would give the pattern you see? (Imagine a regular single crystal making a 2-d array of dots. Now take little sections of the crystal and rotate them randomly around the axis of the electron beam. Each piece still makes a dot at the center. What happens to the other dots?)

According to quantum mechanics, the “wavelength of a particle” (whatever that means!) is inversely related to the particle’s momentum: = h / p. Increasing the accelerating voltage increases the momentum of the electrons. If the pattern we see is a diffraction pattern, what should this do to the spacing of the pattern? Does it?

What does the fact that electrons diffract demonstrate?

Write down the value of the accelerating voltage (note that the display reads kilovolts) and the spacing of the pattern on the screen. Measure also the approximate distance from the crystal (piece of metal farthest into the bulb) and the glowing screen.

You worked out the geometry of interference from a diffraction grating in Lab 6. Recall our picture:

D

s

y

x

Incident

light

4

where s was the spacing between adjacent slits, D was the distance from the slits to the screen, and both rays actually converge at y on the screen.

What was the requirement on x for there to be a bright spot at y?

Look back in your notebook and remind yourself of your derivation for the distance y from the bright spot at the center to the next bright spot. You found:

22 yD

ys

.

In this case, the “slits” are the atoms of the crystal. We can estimate their separation from what we found in Lab 2. A typical result for the length of an oleic acid (olive oil) molecule was 1.6×10−7 cm = 1.6×10−9 m, and this was about 20 atomic spacings, making the atoms about 0.8×10−10 m apart. (In Lab 2, this was our estimate of the size of an atom, since they are essentially touching. This is true in a crystal as well.)

Now use the numbers you wrote down to calculate the wavelength of the electrons! (Which you have now measured!!)

There is a shortcut to knowing the momentum of our electrons: an electron accelerated through one volt gets, by definition, one electron volt (eV) of kinetic energy, and 1 eV = 1.6×10−19 J.1

From the accelerating voltage you recorded in your notebook, find the kinetic energy of the electrons in eV and in Joules. From the definitions of kinetic energy and momentum, derive a formula that gives the momentum of an electron in terms of its kinetic energy (in Joules) and its mass. The electron mass is 9.1×10−31 kg. Find the momentum of the electrons in the beam.

1 If you know some electrostatics, don’t be confused! A volt is a unit of electrical potential; an

electron volt is a unit of energy. A 1.5 volt flashlight battery does not contain 1.5 eV of energy.

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Now, use your measured wavelength of the electrons, your calculated electron momentum, and = h / p to determine Planck’s constant h. Be sure to give the units as well. (Precise measurements give h = 6.63×10−34 kg·m2/s.)

Congratulations – you have just measured Planck’s constant. Take a minute to think back to all the steps (in three labs) you carried out to do this!

Part II: Light is particles

In Lab 6, the diffraction of light proved that light was a wave. Here we will demonstrate that light is particles by observing them individually. The pipe has an LED at one end, connected to a box that can make it flash rapidly. Your instructor will show you the LED – you can see it flashing.

At the other end of the tube is a light detector called a photomultiplier tube (PMT). It is extremely sensitive, so we have to be a little careful. It works via the Photoelectric Effect, itself a quantum phenomenon, which you studied in lecture. Your instructor will describe how it works. Because the PMT is so sensitive, we have to protect it from seeing too much light. The whole setup is designed to keep the room light out. Between the LED and the PMT is a metal block with a 1-cm hole through it. The block has slots into which various filters can be rotated by a clever system of magnets.

By using the filters, we can reduce the intensity of the light in steps. In a classical electromagnetic wave (or any wave), the intensity (energy/sec) is proportional to the square of the amplitude. If we reduce the intensity by a factor of 100 with a filter, the amplitude of the wave drops by a factor of 10. We will just keep doing this and see what happens.

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Part III: Atomic spectra

In lecture and in lab, you have already looked at several light sources using our 500-line-per-mm diffraction gratings. Here we will make some quantitative measurements of the light emitted by hydrogen. The basic element is the same diffraction grating, but we will mount it in our “crossbow spectrometer” so we can make actual measurements.

Take a look at the setup with the ordinary light bulb in the black can. Look through the grating; note the rainbows off to each side.

Now go to your group’s station. The clunky box contains a hydrogen lamp. Note that the lamp only stays on while the button is held down. This is because the tubes have a limited lifespan, so only hold down the button when you are making observations. Set up the crossbow so that the front slit is right at the hydrogen tube. Make sure that you see the very narrow bright region in the front slit when you look through the grating.

Take turns holding the button and looking through the spectrometer. Note that instead of a rainbow, there are a few bright lines. (We generally call them spectral lines, even though the vertical extent is just due to the vertical extent of the lamp and front slit.)

We learned to use the grating quantitatively in Lab 6, but the setup here is a bit different. When we shined the laser through the grating, we viewed the resulting interference pattern (the dots) on a screen (the wall). The geometry was the one shown in the figure from Lab 6 reproduced in Part I of this writeup. (Have a look.) Now, you are looking through the grating.

What is the screen in this case? Hint: where does the light end up?

It turns out that we can still use the geometry shown in the figure, with D the distance from the grating to the meter stick, as if it

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were the screen. More importantly, the formula you derived in Lab 6, also shown in Part I of this writeup, works as if the meter stick were the screen, too.

Sketch and discuss with your group why this works. It has to do with where your eye tells your brain (i.e., you) where things are. (Hint: it has nothing to do with focusing by the eye and everything to do with where the rays appear to be coming from.) Don’t get bogged down on this – ask an instructor if needed.

One person should view the lines through the grating. She should direct her partners to record the observed places on the meter stick on both sides where the lines appear. (Use the pointers or little clamps to mark the spots if that helps.) All should view and agree on the locations. Write these down. Note that you can’t easily measure the distance from the center (which is behind the front slit). That’s OK. Taking the difference between the locations on the two sides and dividing by 2 gives the average separation of the two from the center – this average reduces random uncertainty a bit, but, more important, it removes most of the systematic uncertainty from alignment of the crossbow. Do this for as many of the lines as you can see (at least three, maybe four?) on both sides of the front slit. Take the difference over 2 to average the two sides for each line. Use our grating formula to find the corresponding wavelengths.

Recall our all-quantum picture for this:

1. Only certain energies are allowed for the electron as it “orbits” the nucleus – the energy is quantized.

2. When the electron goes from a higher energy level to a lower one, the energy is emitted as a single photon (particle of light).

3. This photon has an energy hfEEE mn , where n and m

are the initial and final energy levels and f is the frequency of the photon. The wavelength of the photon is, as usual,

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fc , so hcE . (The subscript in E is a Greek

gamma, often used for photons, as in “gamma ray”.)

It turns out that in the series of lines you are observing, the final state is always m = 2. The longest wavelength is the lowest frequency, and thus the lowest energy (initial level n = 3). In the next few steps, you will use your measurements to map out some of the energy levels of hydrogen.

Make a table of your observations. In the first column, list your observed wavelengths (in nm) from longest to shortest. Add columns for: the photon energy (in J, use in meters), the photon energy in electron volts (eV, see Part I), the initial energy level n (just assume that you are seeing all the lines going to m = 2 in increasing order of n).

Draw a vertical scale of energy in eV in your notebook. Start with m = 2 at the bottom (we can call it 0 eV for this purpose). Using your measured photon energies, add the energy levels m = 3, 4, 5,… for as many lines as you saw to your scale.

In the Bohr model for hydrogen, 20

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nEEn , with ,3,2,1n 2

Thus

22011

nmEEEE mn .

Calculate the value for E0 (in eV) using your first (longest) wavelength. (The accepted value is 13.6 eV.)

Check Bohr’s model by calculating E0 (in eV) using your next measured wavelength.

If there is time: How far below the m = 2 level is the ground state m = 1 (in eV)? What would the wavelength of a photon

2 Note our minus sign. We take E0 to be a positive number for convenience. The energy levels are all

negative – the electron would need additional kinetic energy to escape. This is the same convention we used for gravitational potential energy when discussing escape velocity.

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emitted by a 2 → 1 transition be? Why didn’t your see this line in the spectrometer?

Part IV: Superconductors

The instructors will give you a black disk made of a high-temperature superconductor. Superconductivity is a complicated quantum effect: when some materials are made very cold, their electrical resistance becomes exactly zero. In a high-temperature superconductor, this transition takes place above the temperature of liquid nitrogen, making it easy to demonstrate in the lab. It took humans 50 years to figure out superconductivity in its most basic form. High-temperature superconductors are still not understood. We will not try to explain them here!

One of the more striking consequences of superconductivity involves magnetism. You will also be given a small gold-colored cubic magnet. Some superconductors contain toxic metals, so don’t touch yours – use the tweezers to handle both the superconductor and the little magnet.

Test the magnet on ordinary metals. Put the black disk in a cut-down Styrofoam cup. Put the magnet on top of the disk. Is the black disk magnetic?

With the magnet sitting on the center of the disk, carefully add liquid nitrogen (use another cup) until the liquid just reaches the top of the disk. (In fact, just barely cover the disk with liquid, then let the liquid boil away to expose the top surface.) When the disk is cold, place the gold cube on the disk (use the tweezers.) What happens when the disk gets cold? Complete this sentence: If you can levitate something, then you are a _________.

Wash your hands before Part V.

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Part V: A Scientific Study of the Effect of Cryogenic Temperatures on Dairy Products