1st Science Physics Laboratory Manual

PHYC 10120 Physics in Medicine

2014-2015

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Contents

Introduction 4

Laboratory Schedule 5

Experiments: Springs 7

Investigation into the Behaviour of Gases and a

Determination of Absolute Zero

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Conservation of Momentum 25

Newton’s Second Law 33

An Investigation of Rotational Motion 43

Graphing 53

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Introduction Physics is an experimental science. The theory that is presented in lectures has its origins in, and is validated by, experimental measurement.

The practical aspect of Physics is an integral part of the subject. The laboratory practicals take place throughout the semester in parallel to the lectures. They serve a number of purposes:

• an opportunity, as a scientist, to test theories by conducting meaningful

scientific experiments; • a means to enrich and deepen understanding of physical concepts

presented in lectures; • an opportunity to develop experimental techniques, in particular skills

of data analysis, the understanding of experimental uncertainty, and the development of graphical visualisation of data.

Based on these skills, you are expected to present experimental results in a logical fashion (graphically and in calculations), to use units correctly and consistently, and to plot graphs with appropriate axis labels and scales. You will have to draw clear conclusions (as briefly as possible) from the experimental investigations, on what are the major findings of each experiment and whetever or not they are consistent with your predictions. You should also demonstrate an appreciation of the concept of experimental uncertainty and estimate its impact on the final result.

Some of the experiments in the manual may appear similar to those at school, but the emphasis and expectations are likely to be different. Do not treat this manual as a ‘cooking recipe’ where you follow a prescription. Instead, understand what it is you are doing, why you are asked to plot certain quantities, and how experimental uncertainties affect your results. It is more important to understand and show your understanding in the write-ups than it is to rush through each experiment ticking the boxes. This manual includes blanks for entering most of your observations. Additional space is included at the end of each experiment for other relevant information. All data, observations and conclusions should be entered in this manual. Graphs may be produced by hand or electronically (details of a simple computer package are provided) and should be secured to this manual.

There will be six 2-hour practical laboratories in this module evaluated by continual assessment. Note that each laboratory is worth 5% so each laboratory session makes a significant contribution to your final mark for the module. Consequently, attendance and application during the laboratories are of the utmost importance. At the end of each laboratory session, your demonstrator will collect your work and mark it. Remember,

If you do not turn up, you will get zero for that laboratory.

You are encouraged to prepare for your lab in advance and bring

completed pre-lab assignments to the lab

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Laboratory Schedule

Your physics laboratories will be on Tuesday 11-13, the weeks depending on your timetable. Please consult the notice boards, Blackboard, or contact the lab manager, Thomas O’Reilly (Room Science East 1.41) to see which of the experiments you will be performing each week. This information is also summarized below.

Semester Room Dates Week Science East 143 Science East 144 Science East 145

15-19 Sept 2 Springs: 1

Springs: 2

22-26 Sept 3 Springs: 3

Springs: 4

29 Sept-3 Oct 4 Gas: 2

Momentum: 1

6-10 Oct 5 Gas: 4

Momentum: 3

13-17 Oct 6 Gas: 1

Momentum: 2

20=24 Oct 7 Gas: 3

Momentum: 4

10-14 Nov 10 Rotation: 1

Newton 2: 2

17-21 Nov 11 Rotation: 3

Newton 2: 4

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Name:___________________________ Student No: ________________ Partner:____________________Date:___________Demonstrator:________________

Springs What should I expect in this experiment? You will investigate how springs stretch when different objects are attached to them and learn about graphing scientific data. Introduction: Plotting Scientific Data

In many scientific disciplines, and particularly in physics, you will often come across plots similar to those shown here. Note some common features:

• Horizontal and vertical axes; • Axes have labels and units; • Axes have a scale; • Points with a short horizontal

and/or vertical line through them; • A curve or line superimposed.

Why do we make such plots? ______________________________________________________________________

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What features of the graphs do you think are important and why? ______________________________________________________________________

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The equation of a straight line is often written as y = m x + c.

y tells you how far up the point is, whilst x is how far along the x-axis the point is. m is the slope (gradient) of the line, it tells you how steep the line is and is calculated by dividing the change in y by the change is x, for the same part of the line. A large value of m, indicates a steep line with a large change in y for a given change in x. c is the intercept of the line and is the point where the line crosses the y-axis, at x = 0.

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Apparatus In this experiment you will use 2 different springs, a retort stand, a ruler, a mass hanger and a number of different masses. Preparation Set up the experiment as indicated in the photographs. Use one of the two springs (call this one spring 1). Determine the initial length of the spring, be careful to pick two reference points that define the length of the spring before you attach any mass or the mass hanger to the spring. Record the value of the length of the spring below.

Think about how precisely you can determine the length of the spring. Repeat the procedure for spring 2. Calculate the initial length of spring 1. ______________________________________________________________________

Calculate the initial length of spring 2. ______________________________________________________________________

Procedure Attach spring 1 again and then attach the mass hanger and 20g onto the spring. Measure the new length of the spring, using the same two reference points that you used to determine the initial spring length. Length of springs with mass hanger attached (+ 20 g)

Spring 1 Spring 2 Position of the top of spring

Position of the bottom of spring New length of spring

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Add another 10 gram disk to the mass hanger, determine the new length of the spring. Complete the table below. Measurements for spring 1

Object Added Total mass added to the mass hanger

(g)

New reference position (cm)

New spring length (cm)

Disk 1 30 Disk 2 40 Disk 3 Disk 4 Disk 5

Add more mass disks, one-by-one, to the mass hanger and record the new lengths in the table above. In the column labelled ‘total mass added to the mass hanger’, calculate the total mass added due to the disks. Carry out the same procedure for spring 2 and complete the table below. Use the same number of weights as for spring 1. Measurements for spring 2

Object Added Total mass added to the mass hanger

(g)

New reference position (cm)

New spring length (cm)

Disk 1 30 Disk 2

Graphical Analysis Add the data you have gathered for both springs to the graph below. You should plot the length of the spring in centimetres on the vertical y-axis and the total mass added to the hanger in grams on the horizontal axis. Start the y-axis at -10 cm and have the x-axis run from -40 to 100 g. Choose a scale that is simple to read and expands the data so it is spread across the page. Label your axes and include other features of the graph that you consider important.

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Graph representing the length of the two springs for different masses added to the hanger.

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In everyday language we may use the word ‘slope’ to describe a property of a hill. For example, we may say that a hill has a steep or gentle slope. Generally, the slope tells you how much the value on the vertical axis changes for a given change on the x-axis. Algebraically a straight line can be described by y = mx + c where x and y refer to any data on the x and y axes respectively, m is the slope of the line (change in y/ change in x), and c is the intercept (where the line crosses the y-axis at x=0). Suppose the data should be consistent with straight lines. Superimpose the two best straight lines, one for each spring, that you can draw on the data points. Examine the graphs you have drawn and describe the ‘steepness’ of the slopes for spring 1 and spring 2.

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Work out the slopes of the two lines.

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______________________________________________________________________ What are the two intercepts? ______________________________________________________________________

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______________________________________________________________________ How can you use the slopes of the two lines to compare the stiffness of the springs? ______________________________________________________________________

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In this experiment the points at which the straight lines cross the y-axis where x = 0 g (intercepts) correspond to physical quantities that you have determined in the experiment. How well do the graphical and measured values compare with each other? ______________________________________________________________________

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Look at the graphs on page 7, the horizontal and vertical lines through the data points represent the experimental uncertainty, or precision of measurement in many cases. In this experiment the uncertainty on the masses is insignificant, whilst those on the length measurements are related to how accurately the lengths of the springs can be determined. How large a vertical line would you consider reasonable for your length measurements? ______________________________________________________________________

What factors might explain any disagreement between the graphically determined intercepts and the measurements of the corresponding physical quantity? ______________________________________________________________________

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The behaviour of the springs in this experiment is said to be “linear”. Looking at your

graphs does this make sense?

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Name:___________________________ Student No: ________________ Partner:____________________Date:___________Demonstrator:________________

Investigation into the Behaviour of Gases

and a Determination of Absolute Zero What should I expect in this experiment? You will investigate gases behaviour, by varying their pressure, temperature and volume. You will also determine a value of absolute zero. Pre-lab assignment If quantity A is inversely proportional to quantity B, what two quantities could you plot to get a straight line graph? Define zero on the centigrade temperature scale. Introduction: The gas laws are a macroscopic description of the behaviour of gases in terms of temperature, T, pressure, P, and volume, V. They are a reflection of the microscopic behaviour of the individual atoms and molecules that make up the gas which move randomly and collide many billions of times each second. Their speed is related to the temperature of the gas through the average value of the kinetic energy. An increase in temperature causes the atoms and molecules to move faster in the gas. This idea is encapsulated in the equation:        1/2  𝑚  𝑣! = 3/2 k T. The rate at which the atoms hit the walls of the container is related to the pressure of the gas. The volume of the gas is limited by the container it is in. Using the above description to explain the behaviour of gases, state whether each of P,V,T goes up (↑), down (↓) or stays the same (0) for the following cases.

P V T

A sealed pot containing water vapour is placed on a hot oven hob.

Blocking the air outlet, a bicycle pump is depressed.

A balloon full of air is squashed.

A balloon heats up in the sun.

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From the arguments above you can see that at constant temperature, decreasing the volume in which you contain a gas should increase the pressure by a proportional amount. Thus VP 1∝ or introducing k, a constant of proportionality, V

kP = or

kPV = . This is Boyle’s Law. Similarly you can see that at constant volume, increasing the temperature should increase the pressure by a proportional amount. So TP ∝ or introducing κ , a

constant of proportionality, TP κ= or κ=TP . This is Gay Lussac’s Law.

These laws can be combined into the Ideal Gas Law that relates all three quantities by

nRTPV = where n is the number of moles of gas and R is the ideal gas constant. In these laws the temperature is measured in Kelvin. Experiment 1: To test the validity of Boyle’s Law

The equipment for testing Boyle’s Law consists of a horizontally mounted tube with a graduated scale (0 to 300 cm3) along one side. The tube is sealed by a standard gas tap (yellow colour) next to which is mounted a standard pressure gauge. The effective length of the air column inside the tube is controlled by altering the position of an inner rubber stopper moved via a screw thread. The volume of the air column can be read of the scale and the corresponding pressure taken from the pressure gauge. Procedure

• Ensure that the yellow gas tap is in its open position (i.e. parallel to the tube). • Move the rubber stopper (located inside the tube) to at least 200 cm3 on the

graded scale by unscrewing the thread knob (anticlockwise movement). • Close the yellow gas tap by turning it 90o clockwise compared to the “open”

position. • Observe that initially the pressure reading is at approximately 1.0x105 Pa which

is atmospheric pressure; in the next steps you will increase the pressure of the air column inside the tube from approx. 1.0x105 Pa to the maximum pressure written on the pressure gauge.

• Start compressing the air inside the tube by screwing in the thread knob (clockwise direction).

• At 10 different positions of the screw, take readings of the air pressure and the corresponding air volume. For safety reasons do not exceed the maximum pressure on the pressure gauge scale! On the volume scale, each volume value is read by matching the position of the left extremity of the rubber stopper to the graded scale.

• Tabulate your results below. • Once you finished taking readings, depressurize the tube to atmospheric

pressure by turning the gas tap 90o anticlockwise.

Air Pressure (Pa) Volume of air (m3) Air Pressure (Pa) Volume of air

(m3)

Compare Boyle’s Law: V

kP = to the straight line formula y=mx+c.

What quantities should you plot on the x-axis and y–axis in order to get a linear relationship according to Boyle’s Law? Explain and tabulate these values below. ______________________________________________________________________

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x-axis= y-axis=

Use the straight-line equation to work out what the slope of your graph should be equal to.

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Similarly, what should the intercept of the graph equal?

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Make a plot of those variables that ought to give a straight line, if Boyle’s Law is correct. Use either the graph paper below or JagFit (see back of manual) and attach your printed graph below.

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Comment on the linearity of your plot. Have you shown Boyle’s Law to be true? Does how precisely you can determine the pressures and volumes influence your answers? ______________________________________________________________________

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What is the value for the slope determined from the graph? What is the value for the intercept determined from the graph? Use the Ideal Gas Law, at a temperature of 20oC, to figure out how many moles of air are trapped in the column. (The universal gas constant R=8.3145 J/mol K)

Number of moles of gas =

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Experiment 2: To test the validity of Gay Lussac’s Law

The equipment for testing Gay Lussac’s Law consists of an enclosed can surrounded by a heating element. The volume of gas in the can is constant. A thermocouple measures the temperature of the gas and a pressure transducer measures the pressure.

Procedure Plug in the transformer to activate the temperature and pressure sensors. Switch on the multimeters labelled pressure and temperature, by turning the dial to the V scale. Connect the power supply to the apparatus and switch on. At the start your gas should be at room temperature (≈ 200 mV) and pressure (≈ 100 mV). You need to divide by a factor of 10 to convert temperature in mV to centigrade. You can directly use the mV unit of pressure as Nm-2 or Pa. During the course of this experiment the gas contained within the apparatus will be heated up to a final temperature of ≈75-100oC. Immediately switch off the power supply. While the gas is heating, take temperature and pressure readings at regular intervals (say every 5oC) and record them in the table below.

Temperature (oC)

Pressure of gas (Nm-2)

Temperature (oC)

Pressure of gas (Nm-2)

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Gay Lussac’s Law statesP =κT and by our earlier arguments heating the gas makes the atoms move faster which increases the pressure. We need to think a little about our scales and in particular what ‘zero’ means. Zero pressure would mean no atoms hitting the sides of the vessel and by the same token, zero temperature would mean the atoms have no thermal energy and don’t move. This is known as absolute zero. The zero on the centigrade scale is the point at which water changes to ice and is clearly nothing to do with absolute zero. So if you are measuring everything in centigrade you must change Gay Lussac’s Law to read P =κ (T −Tzero ) where T is the temperature measured in Centigrade and Tzero is absolute zero on the centigrade scale. Compare P =κ (T −Tzero ) to the straight line formula y=mx+c. What should you plot on the x-axis and y–axis in order to get a linear relationship according to Gay Lussac’s Law? ______________________________________________________________________

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______________________________________________________________________ According to the equation of the straight line, what should the slope of your graph be equal to? Similarly, what should the intercept of the graph equal? Use the equation to determine, how you can work out Tzero? ______________________________________________________________________

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Make a plot of P versus T. Use either the graph paper below or JagFit (see back of manual) and attach your printed graph below.

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Comment on the linearity of your plot. Have you shown Gay Lussac’s Law to be true? ______________________________________________________________________

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From your graph, what is the value for the slope? Similarly, what is the value for the intercept? From these values, calculate an experimental value for absolute zero temperature. Comment on the precision with which you have been able to find absolute zero

Absolute zero =

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Name:___________________________ Student No: ________________ Partner:____________________Date:___________Demonstrator:________________

Conservation of Momentum What should I expect in this experiment? You will investigate whether or not momentum is conserved in collisions. Pre-lab assignment A car of mass 1000kg hits a stationary car of the same mass at a velocity of 30 km/h and they move off together. What is the velocity of the two cars after the collision? Introduction

Momentum is a vector and is defined as mass multiplied by velocity: vmp = The principle of the conservation of momentum states that in any interaction the total momentum before is equal to the total momentum after. Momentum is conserved in both collisions (think of two snooker balls colliding) and explosions (the vector sum of the momenta of all the debris will sum to zero). In this experiment you will take data before and after collisions, and see whether or not the experimental data confirms or rejects the theory of conservation of momentum. First though, let’s try an example of the principle in action. Suppose your 50 kg friend faces you, at rest, wearing roller-blades and you throw a 5 kg bowling ball at 1 ms-1 to them which they catch. How fast will they move backwards?

What is the momentum of the bowling ball before the collision?

What is the momentum of the skater before the collision?

What is the total momentum before the collision?

What is the total momentum after the collision?

What is the total mass after the collision?

What is the velocity of the combined ball + skater after the collision?

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Apparatus The apparatus you will use is shown here and consists of two carts that can travel along a low friction track and an Explorer GLX data logger with two photogates to time the motion of the carts. Each cart has a mass of 0.5kg which can be adjusted by the addition of steel blocks each of mass 0.5kg. To make sure gravity isn’t assisting the carts moving in one direction, ensure the track is completely level!

(Use the height adjustment and a spirit level and don’t move the track once everything is correctly aligned.) Setting up the GLX

• Connect the GLX to its AC adapter and press the power( ) button. • Place photogate 1 at the 40cm mark on the track and Photogate 2 at the 80cm

mark. • Connect the photogates to the digital adapter and connect it to the GLX. • A screen will appear, select “Photogate timing” and enter the flag length

(0.0250m) and photogate spacing (0.4000m) by pressing to edit and again to confirm. Then press the home key.

• Press F2 to navigate to the table screen and then press twice and select “Time between Gates”.

Investigation 1: Test the conservation of momentum in a head-on collision between carts of equal mass where one cart is at rest. You are going to fire cart ‘A’ at a stationary cart ‘B’ and observe the collision. Small springs on the edge of cart ‘A’ allow it to be launched from the end of the track. After an initial acceleration due to the spring it travels at constant velocity. First of all you must estimate the velocity of cart ‘A’. Use the wooden block to depress the spring to ‘fire’ the cart. Explain how you can measure the velocity of cart ‘A’ using ruler and stopwatch. ______________________________________________________________________

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Use the GLX to make three independent estimates of the velocity of cart ‘A’. Fill in the table below using the information you have and fill in the table below.

• Attach the flags to the carts, you will only be using cart A for this part of the experiment.

• Compress the spring of cart A and place at the 0cm mark on track.

• On the GLX, Press the key to begin recording data and press again once the cart has completed a run.

• Repeat the above steps until you have 3 competed runs. Reset the GLX once the data is collected.

Make an estimate of the uncertainty on your estimate of the velocity. Think about whether distance or time has the most significant uncertainty.

Cart ‘A’ Distance (m) Time (s) Velocity (m/s)

Trial 1

Trial 2

Trial 3

What is your best estimate of the velocity of cart A before the collision? Why is this the best estimate? How large do you think the uncertainty is? ______________________________________________________________________

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Place cart ‘B’ at rest on the track and fire cart ‘A’ toward it. After the collision, how could you work out the velocity of each cart? ______________________________________________________________________

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The GLX must now be configured for collision timing. This allows each photogate to measure velocity using the length of the flag and the time it spent in the gate. Measurements of the velocity of each cart are taken after the collision. If the total momentum of the carts is equal to the momentum of cart A before the collision, momentum is conserved. Fill in the table below for the velocities of each cart after the collision. Repeat the measurements three times.

• Move the first photogate to the 50cm mark and the second to the 90cm mark.

• Place cart B on the track so that the rear of the cart is at the 45cm mark with the flag attached to the front.

• Place cart A at the start of the track. • Switch on the GLX. • Make sure that the photogates are connected to the digital adapter and

connect the adapter to the GLX. • A menu should appear, select ‘Collision Timer’. • Ensure that ‘Velocity 1’ and ‘Velocity 2’ are set to visible. • Navigate to the home screen and select ‘Tables’. • Press the select key and set ‘Velocity 1’ to be displayed in column 1 and

‘Velocity 2’ to be visible in column 2.

• Press the button to begin recording data. Take great care to ensure that you do not accidentally trigger the photogates as this will affect your results.

Distance (m) Time (s) Velocity (m/s)

Trial 1 Cart A

Cart B

Trial 2 Cart A

Cart B

Trial 3 Cart A

Cart B

What is your best estimate of the velocity of cart A after the collision? Comment if necessary. ______________________________________________________________________

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What is your best estimate of the velocity of cart B after the collision? Comment if necessary. ______________________________________________________________________

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Analysis What is the total momentum of the carts before the collision? What is the total momentum of the carts after the collision? (Don’t forget the units) Do your results confirm or reject the hypothesis that in collisions momentum is conserved. ______________________________________________________________________

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Investigation 2: Test the conservation of momentum in a head-on collision between carts of unequal mass where one cart is at rest. Change the mass of the stationary cart by adding a series of steel bars each of 0.5 kg. Measure the velocities of each cart after the collision and fill in the table below. For the heavier weights (1.5kg and 2.0kg), move the second photogate closer to the front of cart B. Do not forget to include units in your table.

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Mass Cart

B

Velocity / Momentum / Total momentum

0.5kg Cart A

Cart B

1.0kg Cart A

Cart B

1.5kg Cart A

Cart B

2.0kg Cart A

Cart B

Comment on the numbers in the final column. What should they be if the conservation of momentum holds? ______________________________________________________________________

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______________________________________________________________________ Plotting the GLX data: Use the arrow keys to highlight your data file in the RAM memory and press “F4 files”. Select the “copy file” option and then choose the external USB as the destination for the file, and press “F1 OK”. Transfer the USB data stick to the lab computer and create a graph.

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Draw a graph of the velocity of cart B after the collision against the mass of cart B. Use either the graph paper below or JagFit (see back of manual) and attach your printed graph below.

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Suppose you were sitting in a car at rest when another car hit you. With reference to the graph and your data, explain whether you would prefer being in a more or less massive car? ______________________________________________________________________

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Suppose you are driving a car and around a bend see a stationary car which you can not avoid. With reference to the graph and your data, would you hope to hit a more or less massive stationary car? ______________________________________________________________________

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Comment on your graph. How accurately do you think you could determine the velocities of the carts? Is this accuracy the same for all of your measurements?

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Name:___________________________ Student No: ________________ Partner:____________________Date:___________Demonstrator:________________

Newton’s Second Law. What should I expect in this experiment? This experiment has two parts. In the first part you will apply a fixed force, vary the mass and note how the acceleration changes. In the second part you will measure the acceleration due to the force of gravity. Pre-lab assignment Acting under constant acceleration, a cart starts from rest and travels 1m west in 5s. What are the acceleration and average velocity of the cart? Introduction

Newton’s second law states amF = , a force causes an acceleration and the size of the acceleration is directly proportional to the size of the force. Furthermore, the constant of proportionality is mass. Apparatus The apparatus used is shown here and consists of a cart that can travel along a low friction track. The cart has a mass of 0.5kg which can be adjusted by the addition of steel blocks each of mass 0.5kg. String, a pulley and additional masses allow forces to be applied to the carts. A PASCO Xplorer GLX, two photogates, a PASCO angle indicator and PASport didgital adapter are also required. Take care to ensure that the track is completely level before starting the experiments.

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Configuring the Xplorer GLX In order to conduct the experiment in an accurate fashion, the PASCO Xplorer GLX must be configured to give you data about the velocity of the cart at two points along the track and how long it takes the cart to travel between the gates.

• Connect the AC adapter to the wall and plug it into the GLX. • Insert the connector for the first photogate into the port marked ‘1’ on the digital

adapter and insert the second into the port marked ‘2’. • Connect the digital adapter to the GLX. A menu should pop up. • Place the photogates at the 50 and 80 centimetre marks on the track. • Use the arrow keys to navigate to ‘Photogate Timing’ and press the button

to select it. • Using the navigation and select keys, set Photogate spacing to 0.3000m and

ensure that only ‘Velocity In Gate’ and ‘Time Between Gates’ are set to visible. • Press the Home key to navigate to the home screen. • Press F1 or navigate to the Graph section. Once on the Graph screen press F4

to open the graph menu. Navigate to ‘Two Measurements’ and press select. • Press the select key and navigate to the ‘Y’ which should be visible on the right

hand side of the graph. Press select and choose ‘Time Between Gates’. Time between gates will now be displayed to the right of the graph.

• You can now record data by pressing to begin and stop recording data. • Velocity will be displayed as a line and the time between gates will be displayed

as both a single point on the graph and as a numerical value along the right Y axis of the graph.

• To read the data from your completed run accurately, press F3 and select ‘Smart Tool’. Use the navigation buttons to see the precise values for the points on the graph.

Plotting the GLX data: Use the arrow keys to highlight your data file in the RAM memory and press “F4 files”. Select the “copy file” option and then choose the external USB as the destination for the file, and press “F1 OK”.

Transfer the USB data stick to the lab computer and create a graph.

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Investigation 1: To check that force is proportional to acceleration and show the constant of proportionality to be mass. To calculate the acceleration due to gravity.

The apparatus should be set up as in the picture. Attach one end of the string to the cart, pass it over the pulley, and add a 0.012kg mass to the hook.

The weight of the hanging mass is a force, F, that acts on the cart. The whole system (both the hanging mass mh and the cart mcart) are accelerated. So long as you don’t change the hanging mass, F will remain constant. You can then change the mass of the system, M, by adding mass to the cart, noting the change in acceleration, and testing the relationship F=ma. If F=ma and you apply a constant force F, what do you expect will happen to the acceleration as you increase the mass? ______________________________________________________________________

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______________________________________________________________________ In the configuration you are using, the GLX will give you data regarding the carts velocity at two points along the track and the time elapsed travelling between the gates which are a known distance apart from each other. You can use the equation

𝑎 =   !!!!

, to formulate the average acceleration of the cart where v is final velocity,

u is initial velocity and t represents time. If the cart starts from rest write down an expression for ‘a’ . Place different masses on the cart. Using the GLX, fill in the table.

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mh (kg)

mcart (kg)

M= mh+mcart

(kg)

u

(m/s)

v

(m/s)

t

(s)

a (m/s2)

M.a (N)

1/a (m-1s2)

0.012 0.5

0.012 1.0

0.012 1.5

From the numbers in the table what do you conclude about Newton’s second law? ______________________________________________________________________

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Analysis and determination of the acceleration due to gravity. If Newton is correct, ammF carth ⋅+= )( You have varied mcart and noted how the acceleration changed so let’s rearrange the formula so that it is in the familiar linear form y=mx+c. Then you can graph your data points and work out a slope and intercept which you can interpret in a physical fashion.

Fm

mFa

ammF hcartcarth +=⇒⋅+=

11)(

So if Newton is correct and you plot mcart on the x-axis and 1/a on the y-axis you should get a straight line. In terms of the algebraic quantities above, what should the slope of the graph be equal to? In terms of the algebraic quantities above, what should the intercept of the graph equal? Plot the graph and see if Newton is right. Do you get a straight line? What value do you get for the slope? What value do you get for the intercept?

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Use JagFit to create your graph (see back of manual) and attach your printed graph below.

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Now here comes the power of having plotted your results like this. Although we haven’t bothered working out the force we applied using the hanging weight, a comparison of the measured slope and intercept with the predicted values will let you work out F. From the measurement of the slope, the constant force applied can be calculated to be From the intercept of the graph you can calculate F in a different fashion. What value do you get? You’ve shown that F is proportional to a and the constant of proportionality is mass. If the force is the gravitational force, it will produce an acceleration due to gravity (usually written g instead of a) so once again a (gravitational) force F is proportional to acceleration, g. But what is the constant of proportionality? Are you surprised that it is the same mass m? (By the way, a good answer to this question gets you a Nobel prize.)

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______________________________________________________________________ So since that force was just caused by the mass mh falling under gravity, F= mhg, you can calculate the acceleration due to gravity to be Comment on how this compares to the accepted value for gravity of about 9.8 m/s2. How does the accuracy of your measurements affect this comparison? ______________________________________________________________________

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Investigation 2: Acceleration down an inclined plane.

The apparatus should be set up as in the picture. Raise one end of the track using the wooden block in place of the elevator. The carts can be released from rest at the top of the track.

A cart on a slope will experience the force of gravity which causes the cart to accelerate and roll down the incline. The acceleration due to gravity is as shown in the schematic. The component of this acceleration parallel to the inclined surface is, g.sinθ and this is the net acceleration of the cart (when friction is neglected). Note the acceleration depends on the angle of the incline.

Vary the angle of the incline (repeat for at least four different angles) and measure the

acceleration of the cart using a = v−ut . You will need to determine the angle of the

incline. Summarise your results in this table.

Angle (radians)

Sin (Angle) u (m/s) v (m/s) Time (t) Acceleration (ms-2)

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Since the acceleration θsinga = , there is a linear dependence on the sine of the angle. Make a graph with sin θ on the x-axis and a on the y-axis. What value do you expect (theoretically) for the slope? What value do you expect (theoretically) for the intercept? What value do you obtain (experimentally) for the slope? What value do you measure for the intercept? Comment on your results. How could you improve your measurements of g? ______________________________________________________________________

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______________________________________________________________________ Use JagFit to create your graph (see back of manual) and attach your printed graph below.

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Name:___________________________ Student No: ________________ Partner:____________________Date:___________Demonstrator:________________

An Investigation of Rotational Motion What should I expect in this experiment? This experiment introduces you to some key concepts concerning rotational motion. These are: torque (τ), angular acceleration (α), angular velocity (ω), angular displacement (θ) and moment of inertia (I). They are the rotational analogues of force (F), acceleration (a), velocity (v), displacement (s) and mass (m), respectively. Pre-lab assignment What is the angular velocity of a spinning disk that completes 2 full revolutions in 10 seconds? Introduction: The equations of motion with constant acceleration are similar whether for linear or rotational motion:

Linear Rotational

12

12

ttssvaverage −

−=

12

12

ttaverage −

−=

θθω

Eq.1

221 vvvaverage

+= 2

21 ωωω

+=average

Eq.2

atvv += 12 tαωω += 12 Eq.3

s2 = v1t + at2

2 θ2 =ω1t +αt2

2

Eq.4

Furthermore, just as a force is proportional to acceleration through the relationship F=ma, a net torque changes the state of a body’s (rotational) motion by causing an angular acceleration.

ατ I= (Eq.5)

The body’s moment of inertia is a measure of resistance to this change in rotational motion, just as mass is a measure of a body’s resistance to change in linear motion. The equation ατ I= is the rotational equivalent of Newton’s 2nd law maF = .

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You will use two pieces of apparatus to investigate these equations. The first lets you apply and calculate torque, measure angular acceleration and determine an unknown moment of inertia, I of a pair of cylindrical weights located at the ends of a bar. The second apparatus lets you investigate how I depends on the distribution of mass about the axis of rotation and lets you determine the value of I, already measured in the first part, by a second method. You can then compare the results you obtained from the two methods. Investigation 1:To measure the moment of inertia, I, from the torque and angular acceleration.

Place cylindrical masses on the bar at their furthest position from the axis of rotation. The bar is attached to an axle which is free to rotate. The masses attached to the line wound around this axle are allowed to fall, causing a torque about the axle

The value of the torque caused by the falling masses is Fr=τ where F is the weight of the mass attached to the string and r is the radius of the axle to which it is attached. Calculate the value of τ.

Mass, m attached to string (kg)

Force, F = m . g (N)

Radius, r of axle (m)

Torque, τ = F × r (Nm)

Next you have to calculate the angular velocity, ω, and the angular acceleration, α. Distinguish between these two underlined terms. ______________________________________________________________________

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Wind the string attached to the mass around the axle until the mass is close to the pulley. Release it and measure the time for the bar to perform the first complete rotation, the second complete rotation, the third, fourth and fifth rotation. Estimate your experimental uncertainties.

Number of

Rotations θ (rad) Time (s)

0 0 0 1 2 3 4 5

Since you have the angular distance travelled in a given time you can use Eq. 4 to find the angular acceleration α. Explain how you can do this and find a value for α. ______________________________________________________________________

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α =

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Using Eq.1, calculate the average velocity, averageω , during each rotation. If the acceleration is constant, this will be the instantaneous velocity at a time, tmid, exactly half-way between the start and the end of that rotation. Can you see this? Explain why it is so. ______________________________________________________________________

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Enter the values in the table below and make a plot of averageω on the y-axis against tmid on the x-axis.

Number of

Rotation averageω (rad / s) tmid (s)

1

2

3

4

5

Since this graph gives the instantaneous velocity at a given time, Eq. 3 can be used to find the angular acceleration, α. What value do you get for α?

Now you know τ and α, so work out I from Eq. 5.

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Use JagFit (see back of manual) to create your graph of averageω on the y-axis against tmid on the x-axis and attach your printed graph below.

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Investigation 2: To determine how the moment of inertia, I, depends on the distribution of mass in a body. Take the metal bar on the bench and roll it between your hands. Now hold it in the middle and rotate it about its centre so that the ends are moving most. Which is easier? Which way does the bar have a higher value of I?

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Attach the weights and the bar to the rotational apparatus known as a torsion axle. This consists of a vertical axle connected to a spring which opposes any departure from the angle of rotational equilibrium.

Note: The apparatus is very delicate. Please take care not to strain it.

When you rotate the bar (always taking care to compress the spring, but not to over-compress it) the spring causes a torque about the axis of rotation which acts to restore the bar to the equilibrium angle. Usually the bar overshoots, causing an oscillation to occur. This is exactly analogous to the way a mass on a linear spring undergoes simple harmonic motion. The period of the oscillations, T, is determined by the restoring torque in the spring, D, and the moment of inertia, I, of the object rotating, in this case the bar and cylindrical weights. They are related by:

DIT π2=

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The value of D is written on each torsion axle. Note it here.

To investigate the influence of mass distribution vary the position of the cylinders along the torsion bar, measure the period of oscillation, T, and use the equation above to calculate the moment of interia, I for the combined system of cylinders plus rod. (To improve the precision with which you measure T take the average over 10 oscillations.)

r – Position of the cylinders along rod (m)

T – Period of oscillation (s)

I for the combined system of cylinder +

bar (kgm2)

I for the cylinders (kgm2)

0.05

0.10

0.15

0.20

0.25

Just as you can simply add two masses together to get the total mass (e.g. if the mass of a cylinder is 0.24kg and the mass of a bar is 0.2kg, then the mass of cylinder plus bar is 0.44kg) you can also add moments of inertia together. Given that the moment of inertia of the bar is 0.00414 kgm-2, will let you fill in the fourth column in the table above.

Plot the value of Icylinder against the position r.

Plot the value of Icylinder against the position r2.

What do you conclude? ______________________________________________________________________

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Use JagFit to create a graph of Icylinder against the position r and attach your printed graph below.

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Use JagFit to create a graph of Icylinder against the position r2.and attach your printed graph below.

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The last measurement in the table above, with r=0.25m, returns the bar + cylinders system to the orientation used in your first investigation. You have thus measured the moment of inertia of the bar + cylinder system in two independent ways. Write down the two answers that you got.

I from investigation 1:

I from investigation 2:

How do the two values compare? Which is more precise?

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Using JagFit In the examples above we have somewhat causally referred to the ‘best fit’ through the data. What we mean by this, is the theoretical curve which comes closest to the data points having due regard for the experimental uncertainties. This is more or less what you tried to do by eye, but how could you tell that you indeed did have the best fit and what method did you use to work out statistical uncertainties on the slope and intercept? The theoretical curve which comes closest to the data points having due regard for the experimental uncertainties can be defined more rigorously1 and the mathematical definition in the footnote allows you to calculate explicitly what the best fit would be for a given data set and theoretical model. However, the mathematics is tricky and tedious, as is drawing plots by hand and for that reason.... We can use a computer to speed up the plotting of experimental data and to improve the precision of parameter estimation.

In the laboratories a plotting programme called Jagfit is installed on the computers. Jagfit is freely available for download from this address: http://www.southalabama.edu/physics/software/software.htm

Double-click on the JagFit icon to start the program. The working of JagFit is fairly intuitive. Enter your data in the columns on the left.

• Under Graph, select the columns to graph, and the name for the axes. • Under Error Method, you can include uncertainties on the points. • Under Tools, you can fit the data using a function as defined under

Fitting_Function. Normally you will just perform a linear fit. Three of the experiments in this manual was developed by the Physics Education Group, CASTeL, Dublin City University

1 If you want to know more about this equation, why it works, or how to solve it, ask your demonstrator or read about ‘least square fitting’ in a text book on data analysis or statistics.

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