Energy and Momentum

Submit all results in the manner established for previous experiments.

Experiment 15: The Quantity F * `ds is conserved, sort of.
Experiment 16: The total of the potential and kinetic energies is constant, sort of, as a car rolls down an incline; this might be so for a ball rolling down a smooth incline or a grooved track.
Experiment 17: The rate at which a human heart beats, in excess of rest rate, is proportional to the rate which the attached physiological mechanism produces useful work (sort of).
Experiment 19: When two objects moving along a common straight line collide and maintain motion along the same line as before collision, the total of their momenta immediately after collision is equal to the total immediately before collision.
Experiment 20: When two objects moving in the same direction collide, the total of their momenta immediately after collision is equal to the total immediately before collision, whether or not their motion after collision is along the same line as their original velocities.

Experiment 15. The quantity [ sum (F * `ds) ] is conserved, sort of.

We test whether the area under the F * x graph for a rubber band or a spring is equal to the f * `ds product of an object accelerated by the stretched elastic object, where f is the constant frictional force against which the object slides during and after being accelerated. If so we have evidence that the force * displacement total is in some sense conserved.

See CD EPS01 for Lab Kit Experiment 15.

Using a calibrated rubber-band balance or some other accurate means of measuring force, we obtain a force vs. stretch profile for a rubber band used to propel your curved-end ramp, here called a 'rail', along a smooth floor or table. We then compare the force * distance total for the rubber band with the force * distance produce for the 'rail', where the force is that exerted by the rail against the frictional resistance of the surface over which it travels and distance is the distance it travels before coming to rest.

We first obtain data from which we determine the force vs. stretch profile for the rubber band, convert this data to a force vs. distance table, and sketch a graph of force vs. distance for the rubber band.

Set up the apparatus as shown on the video clip, using a heavy rubber band.
Be sure that your rubber band balance will give a reading for pullbacks up to at least 8 cm. If not, use the medium-weight rubber band to propel the shelf-standard balance.
Using a calibrated rubber band balance, as shown in the video clip, pull the rail back 1 cm, 2 cm, 3 cm, . . ., until the end of the rubber band on the balance is beyond the end of the scale. Record the reading on the balance at every distance.
Using the calibration program, find the force in Newtons corresponding to each pullback distance. Place your force vs. distance information into a table.
Sketch a graph of your force vs. distance information.

Next determine the force the rail exerts against friction as it slides across the surface.

Using an appropriate calibrated rubber band balance, as demonstrated on the video clip determine the force required to slide the rail that constant velocity across the floor.
Why should we expect that the rail will exert this force as it slides across the floor?

For various pullbacks observe the distance the rail slides across the floor.

Pull the rail back 8 cm and release it. If the rail does not move in a straight line, or if the motion of the rail is interrupted or otherwise interfered with, try again.
Note the total distance traveled by the rail from the position at which it was released to the position at which it stops.
Repeat twice more and record your results.
Now estimate how far the rail should slide with a 4 cm pullback.
Repeat, for the 4 cm pullback, the procedure used for the 8 cm pullback.
How did your estimate differ from your observation? What were you thinking when you made your estimate? Was your thinking correct, or should be modified? If you believe your thinking was correct, explained how your observation supports it. If you believe your thinking was incorrect, speculate on what might have been wrong, and on how you might modify your thinking.
Now repeat the procedure for pullbacks of 1 cm, 2 cm, 3 cm, 5 cm, 6 cm, 7 cm, 9 cm and 10 cm (the last two if your rubber band balance allows you to go this far).
Arrange your data in a table.

For each pullback distance determine the force * distance total for the rubber band and for the rail as it slides across the floor.

For each pullback distance you have the distance the rail slides. Multiply this distance, in cm, by the number of Newtons of force exerted by the rail as it slides across the floor.
For each cm of pullback, determine the maximum and minimum forces exerted by the rubber band. Average these forces and multiply the average (in Newtons) by the 1 cm distance through which this approximate average force is applied.
You will obtain a force * distance product for each centimeter of pullback.
For each pullback position, the force * distance total is the total of all force * distance products up to and including that position. Find the force * distance total for each position.

For each pullback position, compare the force * distance product for the sliding rail with the force * distance total.

Make a table of the force * distance product for the sliding rail and the force * distance total vs. pullback position.
Make a graph of the force * distance product for the sliding rail vs. the force * distance total.
How well do your results support the hypothesis that the force * distance total for the rubber band is equal to the force * distance product for the sliding rail?
How well do your results support the hypothesis that the force * distance total for the rubber band is proportional to the force * distance product for the sliding rail?

As demonstrated on the video clip, quickly stretch then unstretch the rubber band as it is held against your upper lip.

Does the rubber band gain or lose thermal energy as it is stretched?
Does the rubber band gain or lose thermal energy as it recoils?
Just after the rubber band is stretched, if it is left in the stretched position does it gain or lose thermal energy?

Analyze the energy situation for this experiment.

When we slowly stretch a rubber band we must expend a force * distance total. We call this total the work done to stretch the rubber band. Alternatively, we call this total the energy expended to stretch the rubber band.
When the rubber band recoils, if it is at the same temperature as when it was stretched it will exert the same forces through the same distances, though in reverse order. Thus the rubber band will expend the same energy as that expended to stretch it.
We therefore say that the stretched rubber band has a potential energy, which is equal to the energy it will expend as it recoils. If it exerts the same forces when recoiling as when being stretched, then this potential energy would be equal to the work done to stretch to rubber band.
This potential energy will then be converted into two forms of energy: the kinetic energy, or energy of motion, of the rail, into the energy necessary exert the force necessary to overcome the frictional resistance as the rail slides across the surface.
In the end, the kinetic energy will be used up to overcome frictional resistance, until the rail comes to rest. So ultimately the potential energy stored in the rubber band will be converted to the energy required to slide across the floor.

The energy required to slide across the floor is equal to the work done by the rail as it slides across the floor. For each pullback, this work is equal to the force * distance product you recently calculated.
You probably observed that the force * distance products (the work\energy expended) for the rail sliding across the floor were in fact less than the corresponding force * distance totals (the work\energy for the rubber band) for the stretching the rubber band.

In the above analysis, we assumed that the forces exerted by the rubber band on recoil were the same as those exerted when the rubber band was being stretched.
Assuming that a rubber band gets stiffer and therefore exerts more force when it cools, and less force when it gets warmer, what do you conclude about the assumption that the forces exerted on recoil are the same as those exerted on stretching?

Is this assumption valid? How could you easily test it?

Given that before being released the rubber band is left to rest for at least a brief time, during which time it exchanges thermal energy with the surrounding air, how does this modify your conclusions?

Explain the meaning of the following statement: When you do the work to stretch the rubber band, this work goes into the potential energy of the rubber band and into thermal energy, some of which is dissipated into the surrounding air. The potential energy available to propel the rail across the floor is therefore less than the work done to stretch the rubber band.
Explain why we expect that the force * distance product for the sliding rail will be less than the total force * distance associated with stretching the rubber band.

Experiment 16. The total of the potential and kinetic energies is constant, sort of, as a car rolls down an incline; this might or might not be so for a ball rolling down a smooth incline or a grooved track.

By accelerating a friction car for different distances down various inclines we obtain data that allow us to compare velocities attained with altitude differences, testing the hypothesis that the product `dW = weight * `dy of the weight of the cart and its vertical displacement is equal to the KE attained.

See CD EPS01 for Lab Kit Experiment 16.

In this experiment we test the hypothesis that, for a friction car rolling on a constant incline, the increase in the kinetic energy of the car is very nearly equal to the decrease in its potential energy, and that the difference is within experimental error equal to the work done against friction. This is, however, not the case for a ball which rolls without slipping down either an incline or a grooved track; the difference is due to the rotational kinetic energy of the ball as rolls.

Begin by setting up the wooden incline and determining the slope at which the frictional force and the component of the gravitational force directed down the plane are in equilibrium. From this determine the frictional force for small slopes.

Adjust the incline, as was done in previous experiments, until the friction car when given a small initial velocity tends to roll at a constant velocity down the incline.
Measure the slope accurately, using a level and the beam from the shelf-standard balance.
From the weight of the car determine the frictional force being exerted on the car as it moves down the incline.

For several different inclines, obtain the data you will need to determine the change in the potential energy mgh of the car and its change in kinetic energy .5 m v^2.

Set the incline so that the car requires about 2 seconds to coast the maximum possible distance on the incline, starting from rest.
As accurately as possible obtain the information you will need to determine the final velocity obtained by the car.
Obtain the information you will need to determine the change in the potential energy of the car.
Repeat for inclines where the potential energy change is approximately double that of the first run, then triple that of the first run and quadruple that of the first run.

For each incline determine from your data the change in potential energy, the change in kinetic energy, and the work done against friction, and determine whether the hypothesis that energy is conserved is supported by your results.

Make a table showing potential energy change, kinetic energy change and work done against friction for each slope.
Do your data support the contention that the potential energy loss is gained in the form of kinetic energy and work done against friction?
Do your data support the contention that the sum of the potential energy change of the car, its kinetic energy change and the work done by the cart against friction is zero?
Do your data support the contention that the sum of the potential energy of the car, its kinetic energy and the work it does against friction is constant?

Repeat the entire experiment for a steel ball rolling down the curved-end ramp.

Set up the ramp near the edge of the table with the curved end horizontal on the table, so that you can determine the final velocity of the ball by the horizontal distance it travels as a projectile.
Roll the ball down the ramp, from rest, and determine its final velocity by its projectile motion.
Use this velocity to determine the kinetic energy gained by the ball. You need not know the mass of the ball; simply denote the mass as m. Your kinetic energy will then be expressed as a multiple of m.

Measure the change in the altitude of the ball from the instant it is released to the instant it leaves the end of the ramp, and use this change in altitude to determine the change in the potential energy of the ball. Simply denote the mass of the ball as m. Your potential energy will then be expressed as a multiple of m.

Compare your expressions for potential and kinetic energies. Are they equal and opposite?

Assuming that the force exerted against friction is approximately .02 m g, where g is the acceleration of gravity, can we conclude that the sum of the change in potential energy, the change in kinetic energy, and the work done against friction is zero?
Are your results consistent with the claim that you are missing a significant amount of the kinetic energy of the ball?

The ball is in fact rotating as well as moving through space. A rotating ball, even if it is not moving through space, consists of moving particles each of which has a mass and therefore a kinetic energy. So the rotation of the ball accounts for the 'missing' kinetic energy.
From your results how much kinetic energy do you think is present in the rotation of the ball?

Analysis of Errors

Estimate the uncertainties in your data, and use these estimates to obtain ranges for the actual potential and kinetic energies.
Assuming that the force exerted against friction might actually be between .01 m g and .03 m g, determine the range for the actual work done by the ball as it rolls down the incline.
Since for a ball which rolls without slipping, the rotational KE is 2/7 of the translational KE, we expect that the 'missing' kinetic energy should be 2/7 of the observed translation kinetic energy, or approximately .3 of the translation KE.
Are there energies within the ranges you have obtained such that the 'missing' kinetic energy is less between .25 and .35 of the observed kinetic energy?

Experiment 17. The rate at which a human heart beats, in excess of rest rate, is proportional to the rate which the attached physiological mechanism produces useful work (sort of).

We observe the heart rate of an experimental subject vs. the sustained rate at which work is done by the subject to test the hypothesis that heart rate in excess of resting rate is proportional to the rate at which work is done.

This experiment does not use the lab kit.

Follow the directions for the Pulse Rate Experiments.

For each situation, construct an accurate graph of pulse rate vs. the rate at which energy is expended.

Answer the following questions:

How well does the experiment validate the claim that pulse rate is linearly dependent on the amount of energy expended?
How well does the experiment validate the claim that pulse rate is linearly dependent on the rate at which energy is being expended?
How well does the experiment validate the claim that pulse rate is linearly dependent on the length of time during which energy is being expended?

Experiment 19. When two objects moving along a common straight line collide and maintain motion along the same line as before collision, the total of their momenta immediately after collision is equal to the total immediately before collision.

Note: This experiment has been revised. Click here for the revised version.

By setting up head-on collisions of various spherical objects and allowing them immediately after collision to fall a known distance under the influence of gravity, we can from the horizontal ranges of their falls determine their velocities immediately after collision. If one object is stationary prior to collision, and if the velocity of the other immediately before collision is determined, we can then compare total momentum before collision to total momentum after. This comparison constitutes a test of the Law of Conservation of Momentum for two objects.

Note again that this experiment has been revised (see note in read just before the preceding paragraph). You need not be doing the version presented below unless you have been specifically instructed to do so. See CD EPS01 for Lab Kit Experiment 19.

In this experiment we will allow a ball to roll from rest down the curved-end incline and strike another ball head-on, after which both balls will fall as projectiles to the floor. We will obtain data to determine the velocities of the balls after impact and the velocity of the first ball before impact, from which we can make various tests of the conservation of momentum.

Begin by colliding two balls of equal mass.

Using a section of a straw, as in the video clip, set up one ball at the edge of a table just past the end of the ramp. Position the ball also that the collision will be head-on in both a horizontal and a vertical plane.
Position sheets of paper overlaid with carbon paper to detect the positions at which the balls strike the floor.

Release a ball of equal mass from the end of the ramp and allow it to collide with the 'target' ball, after which the two balls will fall to the floor and leave marks indicating the positions at which they struck.
Take any other data you will need to determine the velocities of the balls immediately after collision.

Using the same procedures as in previous experiments, determine the horizontal velocities of the falling balls.

Allow the ball rolling down the incline to fall freely without colliding with the second ball, and collect the data you will need to determine the velocity with which it left the ramp.
Determine the horizontal velocity of the ball as it falls.

Using m to stand for the mass of a ball, determine the momentum of the first ball before collision, and of each ball after collision.
Determine whether your data support the hypothesis that the total momentum of the two balls before the collision is equal to their total momentum after collision.

Now collide balls of unequal masses.

This time, set the smaller steel ball on the straw. Be sure that the balls will collide head-on in both horizontal and vertical planes. Obtain the same data as before.
From your data determine the initial and final velocities of the two balls.

Letting m1 stand for the mass of the larger ball and m2 for the mass of the smaller, write expressions for the total momentum of the two balls before collision and after collision.
Set the two expressions equal to obtain an equation expressing momentum conservation.
The resulting equation will have m1 and m2 as unknowns.

Using simple algebra, rearrange the equation to get only the ratio m2 / m1 on the left-hand side. The other side will reduce to a single number, which will be the ratio m2 / m1 of the mass of the smaller to the larger ball.
What do you get for the ratio?

Measure the diameters of the two balls. Is the ratio of the masses equal to the cube of the ratio of the diameters? If the balls are made of the same material, why would we expect that the ratios would behave in this manner?

Repeat the above procedure for the large steel ball and a small marble.

What do you get for the ratio of the masses?
Would you expect the ratio of masses to again be close to cube of the ratio of the diameters?

Using the program MOMSIM, analyze the first collision from a variety of reference frames.

Enter the velocities and the mass ratio, as requested, for the balls in the first collision.
If the collision is viewed from a vehicle which is moving smoothly along with the moving ball just before collision, and which continues moving smoothly at this velocity, then to an occupant of the vehicle it will appear that the this ball is initially standing still and that the second ball approaches and strikes the first. After collision the first ball will appear to move 'backwards'.

Using the simulation, give your vehicle a velocity equal to that of the moving ball just before collision, and observe the collision from this frame of reference. Describe what you see and why what you see makes good sense as a collision.
Now use the simulation to view the collision from a vehicle which is moving at half the before-collision velocity of the first ball. Describe what you see and why this collision makes good sense.
Repeat using a frame of reference of the second ball after collision.

Verify in detail that the conservation of momentum is validated in each of these frames of reference. Explain why the velocities are as indicated on the simulation, and calculate total momentum before and after collision for each frame of reference.

Analyze the first collision from the center-of-mass frame, using various coefficients of restitution.

The center of mass of the system at any given instant is the position relative to which the two balls would balance if their positions were frozen and they were placed on a beam rotating about the center of mass.
As the balls move toward or away from collision, their center of mass moves in almost every reference frame. The one frame in which the center of mass does not move is called the center-of-mass frame.
The total momentum in the center-of-mass frame is 0.
If the balls were to stick together after collision, they would be moving with the velocity of the center-of-mass frame. This velocity is easily found to be vCM = (m1 v1 + m2 v2) / (m1 + m2).
Determine vCM from the velocities of the balls before collision, and observe the collision from this frame.

Now let the computer calculate and represent the actual velocities to be expected after collision, based on the velocities before collision. First run the simulation, then make this choice afterwards.
You will need to select a coefficient of restitution, which is the ratio of the magnitude of the relative velocity of the balls after collision to that before. Begin by selecting 0 and 1, and observe how the white dot, which is at the center of each circle before collision, tends to move differently after collision. These white dots represent the positions of the balls according to the computer's calculations, which are based on the initial velocities you provided..
See if you can find a coefficient of resitution which best models your observations by keeping the white dots as close to the positions of the balls as possible.
How closely were you able to model your observed velocities? How nearly 'true' then were your conclusions? How does this compare with the error in your experiment (the degree to which your results failed to verify momentum conservation)?

Now repeat the analysis for the second and third collisions, from various reference frames.

Analyze each collision from the point of view of the first ball before collision.
Analyze each collision from point of view of the second ball after collision.
Find the frame of reference in which the total momentum of the two balls is as close as possible to zero both before and after collision. Analyze each collision in this reference frame.
As before, attempt to model the collision in the center-of-mass frame.

Analysis of errors

Discuss possible sources of error in this experiment.
Estimate the possible error ranges in your data, and determine whether within the resulting ranges of observed momenta we can conclude that momentum is conserved.

Experiment 20. When two objects moving in the same direction collide, the total of their momenta immediately after collision is equal to the total immediately before collision, whether or not their motion after collision is along the same line as their original velocities.

Note: This experiment has been revised. Click here for the revised version.

Using glancing collisions of spherical objects, which as in Experiment 19 are permitted to freely fall a known distance after collision, we can determine the angle and magnitude of the velocity of each immediately after collision and test momentum conservation for two-dimensional collisions.

See CD EPS01 for Lab Kit Experiment 20.

The preceding experiment will now be modified by supporting a second ball on a cut straw just past the end of the ramp. The second ball is positioned so that the two balls will collide with their centers of mass at the same vertical position, but the collision is not head-on in the horizontal plane. The horizontal distance traveled after collision will therefore have a component in the direction of the ramp and a component perpendicular to this direction. We will test momentum conservation in both directions.

Begin by colliding two balls of equal mass.

As shown on the video clip, position the 'target' ball so that its center is at the same height as the center of the moving ball, but also so that the balls do not collide 'head-on' as viewed from above.
Define the direction of the x axis as that of the moving ball before collision, with the y axis perpendicular to this direction. Determine the distance traveled by each ball, falling as a projectile, in each direction.
From your data determine the x and the y velocities of each ball after collision.
Using m as the mass of a ball, write out expressions for the x and y momenta of each ball before and after collision.
You should have 4 expressions (and x and a y momentum for each ball) before collision and 4 expressions after collision. All but one of the four before-collision momentum components (namely the x momentum of the moving ball) will be 0, while none of the after-collision momenta will be zero.
Set the expression for the total before-collision and after-collision x momenta equal. Do the same for the y momenta.
Simplify your expressions and verify whether, within the limits of experimental error, total momentum is the same before collision as after collision.

Now collide to balls of unequal mass and from your observations determine the ratio of the masses.

Repeat the first 5 steps of the preceding procedure for the large and small steel balls, this time using m1 for the mass of the larger and m2 for the mass of the smaller ball.
Obtain two equations for m1 and m2 by setting the expressions for the total before-collision and after-collision x and y momenta equal.
Rearrange the equation for the y momenta so that the left-hand side is m2 / m1. The right-hand side will be a pure number, representing the ratio of the two masses.
Rearrange the equation for the x momenta so that the left-hand side is m2 / m1. The right-hand side will be a pure number, also representing the ratio of the two masses.
To what extent are your results consistent? That is, within the limits of experimental error are the two m2 / m1 ratios equal?

Model the collision from different reference frames using MOMSIM2.

Model the collision from the frame of reference of the first ball before collision and describe the collision from this frame.
Model the collision from the frame of reference of the first ball after collision and describe the collision from this frame.
Model the collision from the frame of reference of the second ball after collision and describe the collision from this frame.
Model the collision from the center-of-mass frame of reference and describe the collision from this frame.

View the simulation KINMASS0.

What is the coefficient of restitution for these collisions?
Do the collisions appear to be realistic?

Analysis of errors

Discuss possible sources of error in this experiment.
Estimate the possible error ranges in your data, and determine whether within the resulting ranges of observed momenta we can conclude that momentum is conserved.