THREE-CAR COASTER PROCEDURE

1. Keep the same track configuration as shown in Figures 4 and 5. Make certain the photogate is directly over the top of the loop. Hold a car at the top of the loop and adjust the photogate up or down so the photogate flag will block the gate. Note that the flag must be in a particular side of the car in order to pass through the gate.

2. Connect 3 Mini Cars together as shown in Figure 7.

3. Place a flag in each car as shown in Figure 8.

4. Set the Smart Timer on Time: Fence Mode to measure the speed of each cart at the top of the loop.

5. Press the Button #3 on the Smart Timer to ready the timer. Place the 3-car coaster at the top of the hill on the left and release it from rest.

6. After the coaster passes through the loop, press Button #3 on the Smart Timer to stop timing. The first time displayed will be the time between blocks on the first car’s photogate flag. See Figure 9. Then step through the subsequent times by repeatedly pressing Button #2.

Figure 9: Photogate Timing

Figure 9: Photogate Timing

The second and third times are when the second car’s photogate flag blocked the photogate. Take the difference between the second and third times to find the time for the passing of the second car’s flag. Similarly, the fourth and fifth times correspond to the blocking by the third car’s flag. Calculate the speed of each car using the block-block distance of the flag (1 cm):

QUESTIONS

1. Which car is going the fastest at the top of the loop? Considering energy conservation, how can each car have a different speed at the top of the loop?

2. Which car experiences the least normal force at the top of the loop?

3. How is the experience of the riders in the first car of a coaster different from the experience of the riders in the last car of a coaster?

1. Configure the two tracks as shown in Figures 10 and 11. Attach a photogate at the end of the two tracks as shown in Figure 11. Also put catchers on the end of each track to keep the cars from going off the end of the tracks. Put a photogate flag in each of the cars, in the sides nearest to the other car so both flags will block the photogate.

2. If the cars are started from rest on the left end of each track at the same time, predict which car will reach the right end first. Try it to test your prediction.

3. Prediction: Which car will have the greater speed at the right end of the track?

4. Set the Smart Timer for Speed: Collision Mode. Press the Button #3 on the Smart Timer to ready the timer. Place the two cars on the left end of the track and release them from rest.

5. After the cars pass through the photogate, press Button #3 on the Smart Timer to stop timing. The speeds of the cars will be displayed.

QUESTIONS

1. Which car has the greater speed at the right end of the track? How does energy conservation explain the result?

2. Which car reaches the end of the track first? Why?

QUESTIONS

1. How does increasing the mass of the car change the total energy?

2. How does increasing the mass of the car change the speed of the car at the bottom?

3. Does the car lose a greater percentage of its energy when it has the extra mass or not?

1. Configure the track as shown in Figures 2 and 3. Attach photogates at the top of the hill and on the straight portion at the bottom. Also put the catcher on the end of the straight part to keep the car from going off the end of the track.

2. Place the Mini Car at the top of the hill on the left. Mark on the white board where you start the car. Measure the initial height of the car: Measure from the table to the center of the car.

3. Place the car at the top of the small hill in the center and measure the height of the car.

4. Place the car at the bottom on the flat part of the track and measure the height of the car from the table.

5. Place the car at the top and release it from rest. Use the Smart Timer on Velocity: 2 Gate Mode to measure the speed of the cart at the top of the center hill and at the bottom.

6. Calculate the initial total energy of the car.

7. Calculate the total energy of the car at the top of the center hill.

8. How much energy is lost? Where does it go?

9. Calculate the percent of total energy lost.

10. Calculate the total energy of the car at the bottom. Calculate the percent of the total energy lost between the starting position on the left and the final position on the right.

QUESTIONS

1. Using the speed of the car at the top of the middle hill, calculate the normal force on the car. You will need to estimate the radius of a circle that matches the curvature of the hill by drawing the circle on the white board.

2. How fast would the car have to go to cause the normal force to be zero at the top of the hill? How high would the car have to start to make this happen?

1. Configure the track as shown in Figures 4 and 5. Attach a photogate at the top of the loop. Also put the catcher on the end of the track to keep the car from going off the end of the track.

2. Put a peg in the center of the loop. Place the Mini Car at the top of the loop. Mark on the position of the center of mass of the car on the white board. Measure from the center of the center peg to the center of mass of the car at the top of the loop (see Figure 6).

3. Measure the distance from the center of mass of the car at the top of the loop to the table.

4. Using Conservation of Energy, predict the minimum height from which the car can be released on the left end of the track so the car will just make it completely over the loop.

5. Draw a horizontal line from the top of the circle you drew for the loop to the left part of the track. Measure from this line to mark the starting position calculated in Part 4.

6. Place the center of mass of the car at the marked predicted position and release it from rest.

QUESTIONS

1. Does the car make it over? If not, why not? If so, does it just make it or did you start too high?

2. Once you have determined the release position where the car will make it over the loop, observe and mark the highest position reached on the right side of the track. In theory, where should this position be? How far above or below is this position from the horizontal line drawn in Part 5? Use the loss in height from the starting position to calculate the percent energy lost.

INTRODUCTION

A car is started from rest on a variety of shapes of tracks (hills, valleys, loops, straight track) and the speeds of the car at various points along the track are measured using a photogate connected to a Smart Timer. The potential energy is calculated from the measured height and the kinetic energy is calculated from the speed. The total energy is calculated for two points on the track and compared.

The height from which the car must be released from rest to just make it over the loop can be predicted from conservation of energy and the centripetal acceleration. Then the prediction can be tested on the real roller coaster. Also, if the car is released from the top of the hill so it easily makes it over the top of the loop, the speed of the car can be measured at the top of the loop and the centripetal acceleration as well as the apparent weight (normal force) on the car can be calculated.

THEORY

The total energy (E) of the car is equal to its kinetic energy (K) and its potential energy (U).

E = K + U (1)

(2)

where m is the mass of the car and v is the speed of the car.

U=mgh (3)

where g is the acceleration due to gravity and h is the height of the car above the position where the potential energy is defined to be zero.

If friction can be ignored, the total energy of the car does not change. The Law of Conservation of Energy is stated as

E = constant

Figure 1: Step Configuration

1. Configure the track as shown in Figure 1. Attach a photogate to the straight portion at the bottom, positioned to measure the speed of the car just after it reaches the straight part (on the second peg from the left). Also put the catcher on the end of the straight part to keep the car from going off the end of the track.

2. Place the Mini Car at the top of the step on the left. Mark on the white board where you start the car. Measure the initial height of the car: Measure from the table to the center of mass of the car. Note that the center of mass of the car is approximately at the slot where the flag is inserted. The exact center of mass can be determined by balancing the car. Measure the car’s mass.

3. Place the car at the bottom on the flat part of the track and measure the height of the car from the table.

4. Place the car at the top and release it from rest. Use the photogate and Smart Timer (set on the Velocity: One Gate Mode) to measure the speed of the cart at the bottom of the step.

5. Calculate the initial total energy of the car.

6. Calculate the final total energy of the car.

7. How much energy is lost? Where does it go?

8. Calculate the percent of total energy lost.

9. Place the 50g mass on the car and repeat steps 2 through 8 above.

Setup Passport Sensor

1. Screw the Photogate to the frame of the Centripetal Force Apparatus.
2. Attach the entire Centripetal Force Apparatus as low as possible to the 90cm rod and base.
3. Attach the 45cm rod horizontally to the 90 cm rod with the multi-clamp.
4. Hang the Force Sensor from the horizontal rod.
5. Screw the Ball Bearing Swivel to the Force Sensor.
6. Thread the cable through the plastic pulley and attach the other end to the sliding post.
7. Plug the Photogate into the Photogate Port. Plug the Photogate Port into a PASPORT Interface Plug the Force Sensor into a PASPORT Interface.
8. Making sure it is off, connect the power supply to the Centripetal Force Apparatus with banana plugs.
9. Level the base.

Figure 1: Centripetal Force Apparatus

Software Setup
1. Open any of the files CentripetalForce_A (PP).ds, CentripetalForce_B (PP).ds, and CentripetalForce_C (PP).ds.
2. Select the “Setup” button in DataStudio. Click the Change Value button. Enter the value of 0.3142 for the arc length (in meters). This corresponds to a 0.050 m radius.

EXPERIMENT 3 – FORCE VS. RADIUS (Mass and Velocity Must be Constant)

1. Science Workshop Interface users, open the file “CentripetalForce_C.ds.” PASPORT Interface users, open the file “CentripetalForce_C (PP).ds.”
2. Make sure the power supply is off when you begin.
3. Keep the 30g mass attached to the string and the rotating platform as it was in Experiment 1. It should be able to slide freely back and forth in the slot. If it does not, adjust the nuts and washers accordingly.
4. Now, adjust the height of the force sensor so that the sliding mass maintains an approximately 0.050 m radius. Turn on the power supply and adjust the voltage from 0 to 5 V. Observe the vertical section of cable. Turn the power supply down to 0 V. If the vertical section of cable is not completely vertical, adjust the horizontal rod. Pull the mass to tighten the cable to determine the actual radius. Record the value of the radius in the Force v. Radius data table. Remember to adjust the fixed mass to match the radius of the free mass.
5. Select the “Setup” button in DataStudio. For ScienceWorkshop Users: Double click the Smart Pulley Icon. In the Sensor Properties Dialog Box select the “Constant” tab. Highlight “Spoke Arc Length.” For PASPORT Users: If necessary, scroll until the Smart Pulley (Linear) Icon is visible. Click the “Change Value” button.
6. Enter the value of the spoke arc length using the following equation: spoke arc length = 2 x ? x radius.
7. For ScienceWorkshop Users: Select “OK” and minimize the Sensor Properties Dialog Box to return to the previous graphs in DataStudio. For PASPORT Users: Select “OK” and minimize the “Experiment Setup” window to return to the previous graphs in DataStudio.
8. From the Experiment Menu, select “Monitor Data.”
9. Slowly increase the voltage until the velocity maintains a constant value; for example, 2.0 m/s.
10. Press the Stop button.
11. Without changing the voltage, turn off the power supply.
12. Press the “TARE” or “ZERO” button.
13. Turn on the power supply.
14. Press the Start button. Observe the velocity data. If it does not maintain a constant value return to step 9.
15. Allow data collection to occur for approximately 5 seconds. Press the Stop button.
16. Decrease the voltage to 0 V. Turn off the power supply.
17. Enter the value of the Mean Force into the Force v. Radius data table.
18. From the Experiment Menu, select “Delete All Data Runs.”
19. Return to step 4 and increase the radius by approximately 0.010 m (1.0cm). Important: The spoke arc length must be recalculated each time the radius is adjusted.
20. Repeat data collection until at least 6 data pairs are recorded.

ANALYSIS – FORCE VS. RADIUS
1. Observe your Force v Radius Graph and Data Table.
2. Enter your Force values from the Force v Radius Data Table into the “Force v 1/Radius” Data Table.
3. Inverse the Radius values and enter them into the “Force v 1/Radius ” Data Table.
4. Observe the Force v 1/Radius Graph. Select the “fit” button and choose the appropriate fit.

FINAL ANALYSIS

1. Using words and a mathematical expression, describe the relationship between force and mass in uniform circular motion.
2. Using words and a mathematical expression, describe the relationship between force and velocity in uniform circular motion.
3. Using words and a mathematical expression, describe the relationship between force and radius in uniform circular motion.
4. Combine the three relationships above to create one relationship for force, mass, velocity, and radius.
5. How would you convert this expression into an equation?
6. What is the constant of proportionality for this equation? Explain.
7. How could such an equation be used?
8. The figure above is an overhead view of the rotating mass. For each of the 4 points, draw the direction and relative magnitude of the force.