Name_____________________________ Date_________________

5

LAB 4: Work and Energy

OBJECTIVES

To understand the concept of work in physics as an extension of the intuitive understanding of effort.

To explore kinetic energy, gravitational potential energy, and conservation of mechanical energy.

INTRODUCTION

While Newton’s laws provide the basic framework for attacking any problem in physics, the calculations they use are sometimes more complicated than are necessary to find a solution. One of the first shortcuts which physics uses is the concept of work and energy.

We intuitively think of “work” being synonymous with “effort”. This, however, comes closer to merely meaning force than anything else. When a physicist says that work has been done, then, he means something else: namely, that force has been exerted, and that what it has been exerted on has moved in the direction of the force.

If gravity pulls on your textbook sitting on your desk, it does no work. If, however, you knock your book off the desk, gravity works as the object accelerates to the floor.

Interestingly enough, when an object is moving opposite to the force, the force does negative work. When you pick your textbook off the floor, you do work in lifting it, but gravity does negative work as it continues to pull down as the book goes up.

Energy is a trickier concept. We see energy in various forms all around: we turn on lights, hear sounds, burn fuel in fireplaces, and avoid getting run over by rapidly moving cars. However, quantifying energy is not nearly so intuitive. As it turns out, however, the work done on an object can be represented by the following equation:

This quantity is defined to be the kinetic energy (energy of motion) of the object or particle. The difference between initial and final kinetic energy is the work done on the object.

One consequence of this is that all energy – not merely kinetic, but also chemical, electromagnetic, nuclear, and even potential energy – is measured in Joules, just like work.

One of the most powerful ideas in all of physics is the law of conservation of energy. In this lab, you will be working with only one form of this law: the conservation of mechanical energy in situations where there is no friction. (Friction steals energy from systems and dissipates it in another form, usually heat.)

INVESTIGATION 1: THE CONCEPTS OF PHYSICAL WORK AND POWER

You will need the following materials for this investigation:

force probe

ruler

protractor

PASCO motion track and cart

materials to incline track

two ½ kg masses

motion sensor

Activity 1.1: Work and Power: Motion In Line and out of Line with Force

In this activity you will measure the force needed to pull a cart up an inclined plane using a force probe. You will examine two situations. First, you will exert a force parallel to the surface of the track, and then you will exert a force at an angle to the track. You will then be able to see how to calculate the work when the force and displacement are not in the same direction in such a way that the result makes physical sense.

Note: the hook on the force probe can be removed so that it can easily be inserted through the hole at one end of the cart. You can then screw the hook back on the force probe. The following experiment works best when you place the motion sensor at the bottom of the incline and record the velocity of the cart as you push it.

Set up the cart and track as shown in the diagram below. Place the two weights in the top of the cart. Support one end of the track so that it is inclined to about 15°-20° or a little steeper.

Open Data Studio. If the Force Probe data appears but says “Force, push positive,” click Setup and check the box next to “Force, pull positive” under the force sensor. Then, if there is not a graph already displayed, open a graph for force data. Make sure you also have a graph of velocity so you can watch it as you move the cart.

Hook the force sensor onto the cart and take force data while pulling the cart up the track. The cart’s velocity should be as constant as you can make it.

Use the Fit or Statistics tool to find the mean force exerted:

Mean force pulling parallel to track: _____________ N

Next you will pull the cart at an angle to the track surface.

Prediction 1-1: The force you will be exerting on the cart will be inclined 45° to the track. Predict what the force sensor will read as you pull the cart. (It may be helpful to draw a free-body diagram.)

Predicted force: __________ N

Test your prediction with the cart and inclined track, using the protractor to keep the force probe at the correct angle. Pull the cart slowly and steadily, and make sure that the wheels do not leave the track surface. Analyze the data as you did before, comparing a period of time when the average velocity was equal to that from the first trial.

Mean force pulling at 45° angle to track: ________ N

Question 1-1: It is only the component of the total force which is parallel to the track which we use in calculating work. Find a formula for determining this component for any angle at which one might pull the cart. Express your answer in terms of , the angle between the track and the force vector. Refer to Prediction 1-1, especially if you drew a free-body diagram.

Formula: Ftrack = _____________

INVESTIGATION 2: WORK DONE BY CONSTANT AND NONCONSTANT FORCES

Few forces in nature are constant. Most vary according to the distance over which they are acting. In this investigation you will measure the work done by a constant force, namely gravity, and a spring force varying with the distance the spring is stretched from the spring’s point of equilibrium.

You will need the following equipment:

motion detector

force sensor

400 g mass

index card and masking tape

PASCO cart and track

spring

force accessory bracket (black S-shaped metal object)

Activity 2.1: Work done by a constant lifting force

In this activity you will measure the work done when you lift an object from the floor through a measured distance. You will use the force probe to measure the force and the motion detector to measure distance.

1. The motion detector should be on the floor, pointing upward.

1. Open the experiment file Work in Lifting 1 from the Physics 131 folder.

1. Tape the index card to the bottom of the 400 g mass as shown. (This is so that the motion detector can see the system more easily. You will not be moving the mass fast enough for drag to build up and introduce error in this experiment.)

1. Zero the force probe with the hook pointing vertically downward. (Push the “Zero” button on it.)

1. Hang the 400 g mass from the probe’s hook. The index card must be very close to horizontal for the motion detector data to be accurate.

Prediction 2-1: You will be lifting the weight at constant velocity. Predict the upward force you will have to exert to obtain a constant upward velocity.

1. Hold the force probe and hanging weight at least 20 cm above the detector. Begin graphing and lift the whole system at a slow, constant speed through a distance of about 1 m. Check the position vs. time graph to make sure you have clean data.

1. When you get a graph in which the mass was moving at a reasonably constant velocity, print a graph of force and velocity vs. time to turn in at the end of lab.

Question 2-1: Did the force needed to move the mass depend on how high it was off the floor, or was it reasonably constant?

There should be a force vs. position graph minimized behind the position-time graph you have been working with (if you cannot find it, Window > Force vs Position should bring it forward). Print out a copy of this graph also.

Use the Statistics tool to find the average force during the period the mass was being lifted. Record this force and the distance below. (You can use the Smart Tool to find the distance.)

Average Force: ____________ N Distance lifted: ____________m

Question 2-2: Did this average force agree with your prediction? If not, why not?

Calculate the work done in lifting the mass. Show your calculation below:

Work: _______________ J

Look at your second graph geometrically. Take any rectangle on it: its width is measured in meters, while its height has units of force. Therefore, the area of the rectangle has units of N•m, which simplifies to joules. Find the area of the region under your data between start and finish using Data Studio’s area function. To use the area function, select “area” in the pulldown menu for Statistics ().

area: ___________________ J

Does this equal your calculation of work from step 10? Should it?

Activity 2.2: Work Done by a Varying Spring Force

In this activity you will measure the work done when you stretch a string through a measured distance. First you will collect data on the force exerted by a spring as it moves from being stretched back to its equilibrium position. From the force vs. position graph you obtain you will be able to calculate the work done in the experiment as you did in the last activity.

1. Set up the track, motion detector, force probe, and spring as shown in the diagram.

1.

To mount the force probe, you will use the Force Accessory Bracket. It should be mounted to the metal track using the two bolts on the side. The nuts fit into the slot on the side of the metal track, as shown in the diagram below. Unscrew one of the accessory bolts on the bracket and put it through the hole on the Force Sensor. Use this screw to mount the Force Sensor to the bracket so that the hook on the Force Sensor sticks out towards the track.

(See diagram on the next page.)

Track

Slot

Use these two bolts to attach bracket to metal track

Use this bolt to attach the force probe to the accessory bracket

After:

Continue to use the Work in Lifting activity file.

When the spring is at equilibrium, record the position of the cart.

Zero the force probe with the spring hanging loosely from its hook.

Holding the cart with your hand (but letting the motion detector see only the cart!) begin graphing and slowly move the cart toward the motion detector, stretching the spring until you have displaced the cart at least one meter.

When you have a good force-position graph, print out a copy to turn in at the end of lab.

Question 2-1: Compare this graph to the force-position graph from Activity 2.1. Apply your fit tools to this graph to determine how the spring’s force varies with distance. Describe the relationship (e.g. linear, parabolic, etc.):

Question 2-2: Can you use the first definition you were given for work, , to calculate the work in this experiment? Why or why not?

As your last activity should have shown, the area under the graph is equal to the work in the experiment. Use the area function to measure the work done:

area = _________________ J

This technique of measuring work (and external energy) by taking the area under the force curve tells us something more general about the relation of work/energy to force:

As you can see from this formulation, is a special case of the above equation, namely when force is constant.

INVESTIGATION 3: KINETIC ENERGY AND THE WORK-ENERGY PRINCIPLE

In systems where only one force is acting on an object to accelerate it, it is easy to observe that the object’s energy is increasing as it accelerates. As you have learned, kinetic energy (that is, energy of motion) is defined as having a value of

The units energy is measured in () are the same as for work. This may be confusing at first; however, the matter is cleared up when we realize that doing work on a system is precisely the same as adding energy to it. This is called the “Work-Energy Principle”: a system’s total energy increases or decreases by the amount of external work done on the system.

You will need the following materials for this investigation:

motion detector

force probe

spring

track

PASCO cart

two 500 g masses to put on cart

Activity 3.1: Kinetic Energy from a Spring

1. Set up the track, cart, motion detector, spring, and force probe as shown in the diagram that follows. Place both 500 g masses on the cart.

Open the experiment file Work-Energy 1 in the Physics 131 folder.

From the “Setup” menu, set the force probe and the motion sensor to take data at 50 Hz by pushing the (+) button. Otherwise, you will only get one or two data points.

The calculator should open when you open the experiment file; if it does not, you can open it with the “Calculate” button.

To enter an equation in the calculator, press “New”, enter the equation in the blank field, and press “Accept”. (We have entered a dummy equation for y in the following figure.)

You will then need to define values for all the variables in the equation:

If you wish to define a variable as an experimentally determined constant value (like a mass m), as for example we have done for L above, you must define that value below, in the “Experiment Constant” fields. A data measurement recorded by one of the sensors can also be selected. You can also use variables that you have defined (like “y”, above) in other formulas. They are also selected as a “Data Measurement”.

In our case, we want to input a formula for the Kinetic Energy. Our carts have m=0.497 kg, which you can use as an Experimental constant (add 1.0 kg for the two masses).

Note: When you click “Accept”, it may say “Define variable v”. If you then click on the double-arrow button, you can select “Data Measurement” and then “Velocity” from the menu it gives you. If “Velocity” does not appear as a choice, click on the “Setup” menu at the top of your window and check the “Velocity” box under the motion sensor parameters.

Zero the force probe with the spring hanging loosely.

Pull the cart along the track until the spring is stretched at least 75 cm from equilibrium.

Start taking data, then release the cart so that the spring pulls it back along the track. Catch the cart before it crashes into the force probe. Check a position vs. time graph for good data.

When you get a good graph of Kinetic Energy vs. time and Kinetic Energy vs. position, print out copies to turn in at the end of lab.

Note that the graph labeled Force vs. Position displays the force applied by the spring on the cart vs. position. Using Data Studio’s area function, as you did in the previous investigation, it is possible to find the work done by the spring force for the displacement of the cart between any two positions. If you also display a graph of the Kinetic Energy as a function of position, you can directly see the change in Kinetic Energy as the cart travels between two different positions.

If you have trouble making these two graphs, edit your Force or Kinetic Energy data first (as a function of time) so you can screen out the part of the data after the cart crashes. Then, you can plot your data versus position by clicking on the bottom axis label (“Time (s)”). A little menu will appear that will allow you to select “Position” instead. You can now use your Force vs. position and KE vs. position graphs to answer the following questions.

Find the change in kinetic energy of the cart between the release point (where EK = 0) and three different positions during the time the spring was pulling the cart back down the track. Find the work done by the spring during those same intervals. You can use the same starting point for each of the three intervals. From the Kinetic Energy graph, you can use the Smart Tool to read off the Kinetic energy at the end of each of the intervals, at the same position you ended your Force vs. position integral.

Position of cart (m)

Work done (J)

Change in kinetic energy (J)

Initial position:

Question 3-1: How well do your results for work done and change in kinetic energy agree? If one is less than the other, how can you explain it?

INVESTIGATION 4: GRAVITATIONAL POTENTIAL ENERGY

Suppose that an object of mass m is lifted slowly through a distance y. To cause the object to move upward at a constant velocity, there must be an upward force equal to the downward force of gravity mg. The work done by this force in lifting the object does not actually cause any increase in the kinetic energy of the system; however, the Work-Energy Principle tells us that the total energy has gone up, so we define gravitational potential energy as equal to the work done by this upward force.

Gravitational potential energy is sometimes abbreviated GPE.

EGP is relative. We can set the “zero point” at any height we want, usually the lowest point that an object will reach in an experiment or scenario. It is usually not useful to talk about an object’s “absolute” gravitational potential energy, i.e. the kinetic energy it would acquire in free-fall all the way to the center of the Earth.

You will need the following materials for this investigation:

track

PASCO cart

protractor

block or other means to elevate one end of track

masking tape (to mark positions on the track)

You will be using Data Studio’s calculator more extensively in this experiment. This will allow you to use quantities that you are not directly measuring in your data analysis.

Activity 4.1: Gravitational Potential, Kinetic, and Mechanical Energy of a Cart Moving on an Inclined Track

1. Set up the ramp and motion detector as shown below. The ramp should be inclined at about 10° above the horizontal. Use trigonometry (skip the protractor) to measure the angle as precisely as possible – it doesn’t matter what it is as long as you know it well.

Make sure that the motion detector can see the cart all the way to the bottom of the ramp.

Open the experiment file Inclined Ramp 1 from the Physics 131 folder. Note the three empty equations in the experiment summary on the left: y, EK and EGP. You will need to fill these equations in.

The motion detector is measuring the distance s between it and the cart along the track. We are interested in the cart’s height y above our “zero point” at the bottom of the track. Find the height y as a function of s, the position measured by the motion sensor. (Hint: you must first decide where the cart will start on the track, i.e. determine the value of s0. Make sure you put the offset into the equation.) Draw a triangle!

y = ___________

Enter this equation into the Calculator, making sure that you put the angle in DEGREES (click the DEG button), if that is what you choose. Remember that distances close to the motion sensor are SMALLER than distances far away. Make sure your height has the behavior you want by sliding the cart up and down the track and plotting “y” vs. time. This will save you an immense headache later when your plots don’t make sense.

Predict from this value of y what EGP will be for any value of s. What sign do you use for g?

EGP = ____________

Enter this equation into the Calculator. Where is your potential energy zero? Again, move the cart up and down the track and look at a graph of EGP to see if it does what you think it should.

Enter the mass of the cart into the formula for kinetic energy on the screen.

Check the formulas for kinetic and potential energy on the calculator screen. Does the potential energy formula agree with the one you derived? Make sure that the formulas displayed yield answers with the correct units.

Prediction 4-1: Suppose that the cart is given a push up the track and released. It moves up, slows to a halt, and comes back down again. Predict, on the following three graphs, how a graph of kinetic energy versus position, gravitational potential energy versus position, and total mechanical energy versus position, would look. Plot the expected graphs for the same quantities vs. time.

time

time

time

Hold the cart at the bottom of the track and start the experiment. Keep your hand from getting between the cart and the detector. Give the cart a push up the ramp so that it goes up but does not come too close to the detector.

When you get good data for all of the energy plots, as a function of position and time, print out a copy of your graphs to turn in at the end of lab.

Question 4-2: How does the total mechanical energy change as the cart rolls up and down the track? Does this agree with your prediction? Explain.

Question 4-3: If there had been a significant amount of friction in the system, would you have expected mechanical energy to be conserved?

End-of-lab checklist:

Make sure you turn in:

Your lab manual sheets

Graphs from Activities 2.1, 2.2, 3.1, 4.1

Make sure you have answered all questions and have filled in all relevant data.

Please leave the Force Probes on the brackets when you are finished.

Modified from P. Laws, D. Sokoloff, R. Thornton

Supported by National Science Foundation

and the U.S. Dept. of Education (FIPSE), 1993-2000

and the University of Virginia

University of Notre Dame Physics Department

PHYS 131, Spring 2005

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