Materials Included In Kit

Additional Materials Required

Safety Precautions

The materials in this lab are considered safe. Students should follow all normal laboratory safety guidelines.

Disposal

The materials should be saved and stored for future use.

Lab Hints

• Enough materials are provided in this kit for one student group. This laboratory activity can reasonably be completed in one 50-minute class period.
• For the best results, attach the Roller Coaster Track to the fifth or sixth hole from the bottom of the Support Stand.
• Use calipers to obtain a more accurate value for the diameter of the marble.

Teacher Tips

• This laboratory activity should be performed after students have studied topics such as potential and kinetic energy.
• As a result of the marble rolling down a track with rails (see Figure 4), instead of sliding on a near-frictionless surface, a correction factor is required to account for the rotational velocity in relation to the linear velocity due to the off-center contact points with the track.
  {13337_Tips_Figure_4}
  Equations 7 and 8 are for the simple case in which the bottom of the marble makes contact with the track. Due to the rails, the angular velocity (ω) is related to the linear velocity by a smaller fraction of the radius of the marble (R_c), the radius at which the marble makes contact with the track. This may be beyond the scope of the class, but the derivation is as follows:
  {13337_Tips_Equation_9}

  m = mass of the marble
  R = radius of the marble
  R_c = radius of the contact point with track (Figure 4)

  R_c is approximately 0.9 R, so Equation 9 becomes
  {13337_Tips_Equation_10}
  Which reduces to approximately
  {13337_Tips_Equation_11}
• Solving for v², Equation 11 reduces to approximately
  {13337_Tips_Equation_12}
  For a marble rolling down a flat surface, Equation 8 reduces to v² = (10/7)gh. So, there is a small correction necessary to account for the marble rolling on rails.
• Unfortunately, due to the curvature of the marble, and the diameter of the light beam emitted by the photogates, there is generally about 9–12% error associated with this experiment (systematic error). This error may be realized in Post-Lab Question 9 when students calculate the total energy of the marble. However, the experiment does develop excellent skills for graphing and manipulation of the kinematic equations. Due to the size of the marble, the timing precision alone will be at least 5% error.
• For advanced classes, provide students with the laboratory objective and procedure without the background information. Have students solve the potential and kinetic energy questions by referring to their textbooks. Many times students will forget that the marble is rolling and won’t include the rotational kinetic motion term into their energy equation. Their actual value and theoretical values will vary greatly if this term is forgotten. This is a great topic-reinforcement activity to perform before an exam on projectile motion, or kinetic and potential energy.

Sample Data

Observations
The marble rolls quickly just before the bottom of the first valley, and at the end of the track. It appears to have the fastest speed at the end of the track. The marble rolls slowly at the top of the first hill. The marble appears to have a similar speed as it rises up the hill and as it rolls down the other side.

Data Table
Diameter of the marble: ___1.90 cm___
Mass of the marble: ___28.2 g___
Height of Release Point (0 cm): ___45.0 cm___

Results Table

Answers to Questions

1. Calculate the average speed of the marble at each Track Position (in cm/s). Show all calculations and enter results in the Results Table. Hint: Divide the marble diameter by the average “Transit Time” at each position.
   Sample calculations
   1.90 cm/[(0.0099 s + 0.0099 s + 0.0100 s + 0.0099 s + 0.0100 s)/5] = 191.1 cm/s = 190 cm/s (two significant figures)
   1.90 cm/[(0.0190 s + 0.0193 s + 0.0193 s + 0.0191 s + 0.0191 s)/5] = 99.16 cm/s = 99.2 cm/s (three significant figures)
2. Graph the Average Speed versus the Track Position.
   {13337_Answers_Figure_5}
3. Where along the track does the marble roll with the greatest speed?
   The marble rolls with the greatest speed at the end of the track.
4. Graph the Average Speed versus the Height above the Tabletop. What conclusions can be made from the graph?
   {13337_Answers_Figure_6}
   The marble rolls with the greatest speed at the lowest height (10 cm), and with the slowest speed at the greatest height (34.8 cm). The marble’s speed at the 30-cm height is approximately the same for two positions, but the first 30-cm height position is much faster than the other two.
5. Theoretically, on a frictionless surface, the marble should have the same speed at positions that are at the same height. How does the speed of the marble compare at the three different 30-cm height positions? Explain any discrepancies.
   The speed of the marble at two of the 30-cm height positions is relatively close (137 cm/s and 145 cm/s). The first 30-cm height position has a significantly greater speed (186 cm/s). The most reasonable explanation for the discrepancy in speed at the different 30-cm height positions is due to increased frictional forces as the marble rolls through the curved upslope of the track. The speed is more closely related to the second and third 30-cm heights because the friction from the track slowed the marble down as it began to rise. There is very little frictional force at the beginning of the track as it rolls down the incline, so the marble rolls with a greater speed.
6. Imagine riding in a roller coaster or traveling on a bicycle that follows a similar path as the Roller Coaster Track. Would you feel heavier or lighter traveling down the inclined portion? Would you feel heavier or lighter in valley as you begin to travel up the curved path? Relate these experiences to the forces that affect the marble traveling along the Roller Coaster Track and explain why the marble does not travel at the same speed for each 30-cm height position.
   As a person travels down an incline, similar to the marble’s motion at the first 30-cm height position, the person will feel lighter (as a result of a lower Normal force). As one begins to travel up the curved path, the person will feel heavier and may actually feel like he/she is being “pushed” into the seat (if the person is in a car). This additional “weight” is caused by the centripetal force of traveling along a curved path. The “heavier” person will experience more frictional forces than the “lighter” person because the frictional forces are based on the “weight” of the person acting on the surface of the path (the Normal force). The increased frictional forces in the upward slope of the valley of the track cause the person (and the marble) to slow down. This causes the person (marble) to travel with a slower speed at the second 30-cm height position. As mentioned in Question 5, if no frictional forces existed, the person (marble) would travel at the same speed at each 30-cm height position because the “weight” of the person would not affect the speed of the marble.

Advanced Post-Lab

7. Calculate the potential energy of the marble at each Track Position. Hint: Convert grams to kilograms and centimeters to meters to obtain the unit of joules (J).
   Sample calculation
   PE = mgh = (0.0282 kg) x (9.81 m/s²) x (0.450 m) = 0.124 J
   {13337_Answers_Table_3}
8. Because the marble rolls down a tract, or rail, Equation 7 must be modified slightly to account for this condition. The total kinetic energy (KE_T) of the marble will be equal to approximately ¾ mv². Use this equation to calculate the kinetic energy of the marble at each Track Position. Hint: Convert grams to kilograms and centimeters per second to meters per second to obtain the unit of joules (J).
   Sample calculation
   KE = ¾mv² = (3/4)(0.0282 kg)(1.90 m/s)² = 0.07635 J
   {13337_Answers_Table_4}
9. Compare the total energy at each Track Position with the initial potential energy of the marble. What relationship exists between the kinetic and potential energy of the marble?
   Sample calculation
   PE_initial = 0.124 J PE + KE = 0.0570 J + 0.0764 J = 0.1334 J {13337_Answers_Table_5} The total energy fluctuates but the average is about 0.13 J which is close to the original 0.124 J. The first 30-cm height position does show an unusual increase in total energy, but this is most likely due to timing errors. The total energy is the smallest at the top of the second hill of the track, which makes sense because the marble experienced a large amount of friction as it rolled up the hill. The kinetic and potential energy showed an inverse relationship. As the potential energy decreases, the kinetic energy of the marble increases and vice versa.

Introduction

Riding a bicycle down a hill can sure be fun. However, if you want to get back to where you started you must go up the hill eventually, and that requires work. In this experiment, use the Roller Coaster Track to model the “ups and downs” of riding on a hilly road or riding on a roller coaster.

Concepts

• Conservation of energy
• Potential energy
• Kinetic energy
• Speed

Background

Work is the act of using a force to move an object through a distance. In order to reach the highest point on a roller coaster, energy (work) must be used. The energy expended to raise a roller-coaster car to a higher position is “stored” in the car—the car now has potential energy (PE). The potential energy of the roller-coaster car is related to its height and weight, and is equal to the mass (m) of the car multiplied by the acceleration due to gravity (g) multiplied by the relative height (h) of the car above the ground (Equation 1).

In this experiment, a rolling marble will simulate a roller-coaster car. When the marble rolls downward along the Roller Coaster Track, its potential energy is converted into kinetic energy, or energy of motion. Its kinetic energy is converted back into potential energy as the marble rolls up the track. This is due to the conservation of energy principle. The conservation of energy principle states that energy cannot be created or destroyed—energy can only be converted from one form to another. Therefore, the initial potential energy the marble has at the release point will be completely converted into kinetic energy at the bottom of the roller coaster (neglecting frictional forces).

When the marble rolls it has two different types of kinetic energy—linear and rotational. Linear kinetic energy (KE_l) is related to the mass (m) and linear speed (v) of the object (Equation 2). Rotational kinetic energy (KE_r) is related to the moment of inertia (I) of the object and the rotational speed (ω; the Greek letter “omega”). See Equation 3.

The moment of inertia of an object is its “resistance” to being rotated. This “resistance” is based on the mass of the object and the spatial distribution of the mass about the point of rotation. The point of rotation for the marble is its center. The total kinetic energy (KE_T) of a rolling marble is therefore equal to the linear kinetic energy plus the rotational kinetic energy (Equation 4). The conservation of energy principle is represented by Equations 5 and 6. Equation 6 is used to determine the theoretical speed of the rolling marble at any point along the roller coaster. Remember, h is the difference between the starting height and the height of the “point of interest.” In other words, it is only the change in height that matters, not the path that the marble takes. The moment of inertia of a solid sphere rotating about its center is equal to 2/5 mR², where R is equal to the radius of the sphere. Rotational speed, ω, is related to linear speed of the marble, v, and the radius, R, of the object: ω = v/R. Substituting these values into Equation 6 Equation 7 reduces to

Materials

Safety Precautions

The materials in this lab are considered safe. Follow all laboratory safety guidelines.

Procedure

Observations

1. Set up the PSWorks™ Roller Coaster Track as shown in Figure 1. Attach the track at the fifth or sixth hole from the bottom of the PSWorks Support Stand.
   {13337_Procedure_Figure_1}
2. Place the marble at the top of the track near the 0-cm mark.
3. Release the marble, without giving it an “extra” push, and observe the speed of the marble as it rolls along the track. Where along the track does the marble travel fast? Where along the track does the marble travel slowly? Are there different positions along the track where the marble appears to have the same speed? Record your observations in the Roller Coaster Track Worksheet.

Speed and Energy Determination

1. Set up the PSWorks Roller Coaster Track as shown in Figure 1. Attach the track at the fifth or sixth hole from the bottom of the PSWorks Support Stand.
2. Use a meter stick to measure the diameter of the marble to the nearest 0.1 mm. Record this value in the Roller Coaster Worksheet.
3. Use a balance to measure the mass of the marble to the nearest 0.1 g. Record this value in the Roller Coaster Worksheet.
4. Use a meter stick to measure the height of the top of the Roller Coaster Track from the tabletop at the 0-cm mark. Record the height to the nearest 0.1 cm in the Roller Coaster Worksheet. This will be the “Release Point” for the marble.
5. Use a meter stick to measure the height of the top of the Roller Coaster Track from the tabletop at the lowest point of the first valley (approximately the 45-cm mark on the track). Record the height to the nearest 0.1 cm, and record the position along the track to the nearest 0.1 cm in the Roller Coaster Worksheet.
6. Use a meter stick to measure the height of the top of the Roller Coaster Track from the tabletop at the highest point of the second hill (approximately the 80-cm mark on the track). Record the height to the nearest 0.1 cm, and record the position along the track to the nearest 0.1 cm in the Roller Coaster Worksheet.
7. Use a meter stick to measure the height of the top of the Roller Coaster Track from the tabletop at the 130-cm mark on the track (the approximate end of the track). Record the height to the nearest 0.1 cm in the Roller Coaster Worksheet.
8. Locate three different positions along the top of the Roller Coaster Track that are at a height of 30 cm from the tabletop. Record the three positions (according to the scale on the track) on the data table in the Roller Coaster Worksheet. Note: Measure from the mark on the track that is perpendicular to the curve of the track (see Figure 2).
   {13337_Procedure_Figure_2}
9. Clamp one photogate at the lowest point of the first valley and the second photogate at the highest point of the second hill on the Roller Coaster Track as shown in Figure 3. Note: Line up the centers (the light sensor slot) of the photogates with the appropriate marks on the track.

1. Turn on the Photogate Timer and select the Gate Mode. (Refer to the Photogate Timer instructions, if necessary.)
2. Place the center of the marble at the 0-cm mark.
3. Release the marble without giving it an “extra” push.
4. The transit time of the marble at each photogate “Track Position” will be registered by the timer separately. Record the first value registered by the timer under Lowest Point of the First Valley in the data table, and the second value registered by the timer under Highest Point of the Second Hill in the data table on the Roller Coaster Worksheet.
5. Repeat steps 11–13 four more times, recording both photogate times in the appropriate columns of the data table. Note: Release the marble from the same position each time.
6. Unclamp each photogate from the Roller Coaster Track.
7. Clamp one photogate at the 30-cm Height Track Position 1 and the second photogate at the 30-cm Height Track Position 2.
8. Place the center of the marble at the 0-cm mark.
9. Release the marble without giving it an “extra” push.
10. The transit time of the marble at each photogate “Track Position” will be registered by the timer separately. Record the first value registered by the timer under 30-cm Height Track Position 1 in the data table, and the second value registered by the timer under 30-cm Height Track Position 2 in the data table on the Roller Coaster Worksheet.
11. Repeat steps 17–19 four more times, recording both photogate times in the appropriate columns of the data table. Note: Release the marble from the same position each time.
12. Unclamp each photogate from the Roller Coaster Track.
13. Clamp one photogate at the 30-cm Height Track Position 3 and the second photogate at the End of the Track (130-cm mark).
14. Place the center of the marble at the 0-cm mark.
15. Release the marble without giving it an “extra” push.
16. The transit time of the marble at each photogate “Track Position” will be registered by the timer separately. Record the first value registered by the timer under 30-cm Height Track Position 3 in the data table, and the second value registered by the timer under End of the Track in the data table on the Roller Coaster Worksheet.
17. Repeat steps 23–25 four more times, recording both photogate times in the appropriate columns of the data table. Note: Release the marble from the same position each time.

Student Worksheet PDF

13337_Student1.pdf