1 Set up the apparatus as shown in Fig.10.5. In order to create an elastic collision, put a piece of Velcro fastener on the front side of cart #1 and a piece of the opposite type of Velcro on the back of cart #2. Attach a piece of thread to cart #1. The thread should be about 10 cm longer than the distance from the cart to the floor. Connect the Smart Pulley to your computer, and make sure the proper connections have been made before going on.
Cart #2
Cart #1
Velcro
Eraser
Thread
Universal Clamp
Figure 10.5: Equipment setup for inelastic collision
2 Insert the Smart Pulley software disk into your computer disk drive and start up the computer.
3 Determine the mass of each of your lab carts. Record these as m_{1} and m_{2}, respectively, in Table 10.2. Fasten the thread running from cart #1 to the mass hanger by wrapping 45 turns of thread around the notched area of the hanger. The purpose of the mass hanger is to keep a small amount of tension on the thread to turn the pulley as the cart moves.
4 Move cart #1 until the mass hanger almost touches the floor. With the cart motionless and the holes in the pulley positioned so that the LED on the photogate is off, select <M> on the main menu.
5 Give cart #1 a push so that it collides with cart #2 near the center of the table. Continue timing until the thread runs out, then press <RETURN> to halt the timing process.
6 When the computer finishes processing the times, move to the graphing function. Select <A> to tell the computer that you are using the Smart Pulley to monitor a linear motion. Next choose <V> which will give you a velocitytime graph.
7 When finished looking at the graph, press <RETURN>. Choose <T> to display a table of the velocitytime data. Copy the data Table 1 (you need only be concerned with the data for timing intervals before the collision and 5 intervals after the collision).
8 Change the relationship between m_{1} and m_{2} by adding mass to either of the two carts. Repeat steps 37 five times with different mass combinations.
Part B) Elastic Collision
1 Set up the apparatus as shown in Fig.10.6. In order to create an elastic collision, attach a piece of thread to cart #1. The thread should be about 10 cm longer than the distance from the cart to the floor. Connect the Smart Pulley to your computer, and make sure the proper connections have been made before going on.
2 Insert the Smart Pulley software disk into your disk drive and start up the computer.
3 Determine the mass of each of your lab carts. Record these as m_{1} and m_{2}, respectively, in Table 10.2. Fasten the thread running from cart #1 to the mass hanger by wrapping 45 turns of thread around the notched area of the hanger. The purpose of the mass hanger is to keep a small amount of tension on the thread to turn the pulley as the cart moves.
4 Move cart #1 until the mass hanger almost touches the floor. With the cart motionless and the holes in the pulley positioned so that the LED on the photogate is off, select on the main menu.
5 Give cart #1 a push so that it collides with cart #2 near the center of the table. Continue timing until the thread runs out, then press to halt the timing process.
Cart #2
Cart #1
Eraser
Thread
Universal Clamp
6 When the computer finishes processing the times, move to the graphing function. Select <A> to tell the computer that you are using the Smart Pulley to monitor a linear motion. Next choose <V> which will give you a velocitytime graph.
7 When finished looking at the graph, press <RETURN>. Choose <T> to display a table of the velocitytime data. Copy the data Table 10.1 (you need only be concerned with the data for timing intervals before the collision and 5 intervals after the collision).
8 Change the relationship between m_{1} and m_{2} by adding mass to either of the two carts. Repeat steps 1115 five times with different mass combinations.
4. DISCUSSIONS AND CONCLUSIONS
For Inelastic Collision:

Calculate the momentum of cart #1 immediately before the collision, and the momentum of the combined masses immediately after the collision for each of your trials.

Determine the percentage difference between the pre and postcollision momentum for each trial.
For Elastic Collision:

Calculate the momentum of cart #1 immediately before the collision, and the momentum of the combined masses immediately after the collision for each of your trial.

Determine the percentage difference between the pre and postcollision momentum and energy for each trial.
5. QUESTIONS

List the sources of error effecting the values of initial and final velocities.

What is the percentage error of the value of v_{2f }?

In the elastic collision was K_{i}=K_{f }? If not, what are the possible reasons for this difference? Can you justify this difference by the errors due to the instrument?

Are v_{i} and v_{f} equal in magnitude and direction in elastic collision ? Is the momentum conserved? Is the energy conserved ?

Compare the values of K_{i} and K_{f} and explain the difference ?

How did the inelastic collision effect the total momentum and total kinetic energy in the system ?

How did the elastic collision effect the total momentum and total kinetic energy in the system ?

What effect did friction have on the momentum and kinetic energy of the system for inelastic and elastic collision ?

What are other possible contributors to the variation of the experimental data from the expected (theoretical) values for inelastic and elastic collision ?

Two clay balls of equal mass and speed strike each other headon, stick together, and come to rest. Kinetic energy is certainly not conserved. What happened to it? Is momentum conserved ?

Consider a one dimensional elastic collision between a given incoming body A and a body B initially at rest. How would you choose the mass of B, in comparison to the mass of A, in order that B should recoil with (a) the greatest speed, (b) the greatest momentum and (c) the greatest kinetic energy ?
Table 10.1: Velocities of Carts 
Trial 1
Before After

Trial 2
Before After

Trial 3
Before After

Trial 4
Before After

Trial 5
Before After






Table 10.2: Momenta and Energy of Carts 
Trial

M_{1}

M_{2}

Inıtial
Velocity
 Initial
Momentum

Initial
Energy

Final
Velocity

Final
Momentum

Final Energy










GENERAL PHYSICS
PART A: MECHANICS
EXPERIMENT – 11
1. PURPOSE
The purpose of this experiment is to show the conservation of mechanical energies of rolling object on an inclined plane.
2. THEORY
An object at the top of an inclined plane has gravitational potential energy (GPE). Moving down the inclined, at accelerates, and this GPE is converted into kinetic energy (KE). If friction is small, the conversion is nearly complete. However, if object is a rolling ball, the kinetic energy shows up in two forms: transnational kinetic energy and rotational kinetic energy. Both types of energy must be considered in determining whether mechanical energy is conserved.
Fig. 11.1 shows a disk of mass m and radius r at the top of inclined plane. The axis of the disk is parallel to the edge of the plane so that the disk, when released, rolls straight down the plane. If the frictional force is great enough, there is no sliding and the disk rolls without slipping. The thickness of the disk is not important here, except that for a given material and radius, the total mass of the disk depends upon its thickness.
r
h
Figure 11.1: A disk rolling down an inclined plane. Its potential energy at the top is
transformed into translational and rotational kinetic energy at the bottom.
At the top of the plane the disk has a potential energy relative to its position at the bottom. Here h is the vertical distance through which the center of mass moves from the top to the bottom of the plane. If the disk rolls down to the bottom of the plane without slipping, the n all of the initial potential energy is completely transformed into kinetic energies of rotation and translation at the bottom. Because the disk rolls without slipping, we can neglect energy loss due to friction. Then we may extend the idea of conservation of mechanical energy to include rotational as well as translational kinetic energy. Consequently,
∆PE + ∆ KE_{trans} + ∆KE_{rot} = 0 (11.1)
here KE_{trans} is the kinetic energy due to translation of the center of mass of the disk and KE_{rot} is the kinetic energy of rotation about its center of mass. The sum of these two terms is the total kinetic energy of the rolling disk:
KE_{tot } = KE_{trans} + KE_{rot} = mv^{2} + Iw^{2} (11.2)
We see that the magnitude of the decrease in potential energy is equal to the gain in total kinetic energy, or
mgh = mv^{2} + Iw^{2} (11.3)
where v and w are the linear and angular speeds of the disk when it reaches the bottom of the plane and I is the moment of inertia of the disk about its center of mass. Since the center of mass is moving steadily in a straight line, a point on the rim must be in rotation about it at the same angular velocity. Thus, the angular velocity and the linear speed are related through v = rw.
3. EXPERIMENTAL PROCEDURE

Set up the apparatus as shown in Fig.11.2. Connect the Smart Pulley photogate to your computer, and start up the computer.
Ramp
Ball
Smart Pulley Photogate
Figure 11.2: Equipment Setup

Move the ball slowly through the photogate, using the meter stick as shown in Fig. 11.3. Determine the point at which the ball first triggers the photogate timerthis is the first point at which the LED turns ON and mark it with a pencil on the side of the channel. Then determine the point at which the ball last triggers the timer (the LED turns OFF), and mark this point. Measure the distance between these marks and record this distance as d. Determine the midpoint of this interval, and mark it in pencil on the channel.
LED comes ON
LED comes OFF
Mark with a pencil on side of channel
Meter stick
LED comes ON
LED comes OFF
Mark with a pencil on side of channel
Meter stick
Figure 11.3: Measuring d.

Mark a starting point for the ball 30 cm up the ramp from your midpoint. Measure h_{1}, the vertical distance from the surface of the lab table to this point. Also measure h_{2}, the distance from the lab table to your midpoint (see Fig.11.4). Record these values in Table 11.1. The difference between these two heights determines the gravitational potential energy gained by the ball as it rolls from the starting point to the photogate. Record the difference in heights as h. Also record the distance along the ramp (30 cm) as d.
h_{1}
Starting Point
h=h_{1}h_{2}
d
h_{2}
Mark a starting point for the ball 30 cm up the ramp from your midpoint. Measure h_{1}, the vertical distance from the surface of the lab table to this point. Also measure h_{2}, the distance from the lab table to your midpoint (see Fig.11.4). Record these values in Table 11.1. The difference between these two heights determines the gravitational potential energy gained by the ball as it rolls from the starting point to the photogate. Record the difference in heights as h. Also record the distance along the ramp (30 cm) as d.
h_{1}
Starting Point
h=h_{1}h_{2}
d
h_{2}
Figure 11.4: Measuring d and h.

Select option <G>, the gate function, from the Main Menu on the computer. Set the ball at starting point. Hold it at this position using a ruler or block of wood. Make sure the computer is not actively timing and is ready to take a new time. Release the ball so that it moves among the ramp and through the photogate. Record the time as Time 1 in Table 11.2.

Repeat your time measurement carefully 5 times, recording the time obtained in each trial.

Move the ramp to a different angle, and repeat steps 35. Do this for at least 3 angles.
r
R
W / 2
Finally, measure the mass of your ball, m, the diameter of the ball, D, and the inside width of the ramp channel, W. Record these values on your data sheet, along with R, the radius of the ball (D/2).
4. DISCUSSIONS AND CONCLUSIONS

Calculate the average time obtained for each trial. Use your average time to determine the velocity of the ball as it passed through the photogate (Final Velocity=d/t).

For each trial, calculate the gravitational potential energy (GPE=mgh) that the ball had at the beginning. Also calculate the final kinetic energy, KE. Compare these two values.

You should have found a discrepancy between initial and final energies. For a rolling ball, the rotational kinetic energy, RKE=½I^{2}, where I is the moment of inertia of the ball and is its rotational velocity. For the sphere, I=2/5mR^{2}. To solve for the RKE one must convert the transitional velocity, v, into rotational velocity, , using the relationship v=R. Calculate the RKE and add it to the KE to verify conservation of energy.

If your figures are still not quite in agreement, a final tuneup is in order. The ball doesn’t actually rotate on its full circumference. In figure 11.5, notice that the effective radius, r, is related to the overall radius, R, and the width of the channel, W, by the relationship: r^{2}=R^{2}4(W/2)^{2}.

Recalculate the rotational energy, given that relationship between the linear velocity and the angular velocity is actually v=r. At this point you have considered all of the mechanical energy forms possible, with the exception of the very small amount of energy that might have gone into heat due to friction.
5. QUESTIONS

Why do car owners go to the trouble to balance automobile tires? What happens when car wheels are unbalanced.

Can a diver pull into a tuck and rotate while diving if he leaves the diving board with no angular velocity. Why?

What would happens to the planets if the gravitational force had a tangential component as well as radial component

Explain how ice skaters can quickly go from a slow to a fast spin and vice versa. What happens to their angular velocity and their moment of inertia Is an external torque required.

A ball rolls across the floor. Is it possible for its translational and rotational kinetic energies to be the same?

Two spheres of equal mass are released from rest at the top of an inclined plane. One sphere is solid and uniform density. The other sphere is a shell of uniform density. (a) Which sphere reaches the bottom of the plane first? (b) Which sphere will have the greatest translational kinetic energy at the bottom?

About what axis would a uniform cube have its minimum rotational inertia?

A solid wooden sphere rolls two different inclined planes of the same height but with different angles of inclination. Will it reach the bottom with the same speed in each case? Will it take longer to roll down one incline than the other? If your answer is yes to either question, explain why.

Data from accelerating ball:
∆d = _______________
m = ________________
D = ________________
W = _______________
 Table 11.1: Measuring ∆h 
Trial

h_{1}

h_{2}

∆h

1
2
3
4
5
6





Table 11. 2: Measuring of trials

Quantity

Trial 1

Trial 2

Trial 3

Trial 4

Trial 5

d
Time 1
Time 2
Time 3
Time 4
Time 5
Average
Time
Final
Velocity
GPE
KE






