Materials Included In Kit

Additional Materials Required

Prelab Preparation

Assembly

1. Attach the pulley to the ramp as shown in Figures 7 and 8.
   {13339_Preparation_Figure_7}
   {13339_Preparation_Figure_8}
2. Press the rubber tubing into the thin slot at the end of the ramp opposite to the pulley. Use the end of a pencil or pen to push it into place. This will act as the end-stop for the carriage (see Figure 9).
   {13339_Preparation_Figure_9}
3. Insert the ramp support foot into the plain, non–silk-screened side of the ramp.

Safety Precautions

The materials in this lab are considered safe. Remind students to use care when adding and removing masses to the slotted-mass hanger. Do not allow the masses to drop on the floor. Please follow all other 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. Each part of this laboratory activity can reasonably be completed in one 50-minute class period.
• Use thin, light-weight fishing line for Part 1 to reduce the error in the measurements. Heavy pulley cord may cause a significant mass shift as the carriage moves up the ramp, and more cord adds to the total mass pulling the carriage up the ramp.
• Students can collect data for more than three angles, if desired, to obtain a “better” best-fit line.
• For Part 1, a slotted-mass hanger may be too heavy when the ramp is positioned at a shallow angle. If this occurs, students can tie a paper clip to the end of the string and then hang the slotted masses on the “paper clip hook” (see Figure 10). This will also work if slotted-mass hangers are not available. Students should measure the mass of the paper clip before using and add this mass to the total “minimum mass.”
  {13339_Hints_Figure_10}
• For Part 1, spring scales can be used instead of hanging masses. In order to determine the Force Needed to Raise the Carriage, students must pull on the spring scale at a constant speed. When the carriage is being raised with a constant speed, the force registered on the spring scale will remain relatively constant (the pulling force and force due to gravity will be balanced). Students should record the force on the spring scale only when the carriage is moving at a constant speed.
• For Part 2, use calipers to obtain a more accurate value for the width of the carriage wing.
• For Part 2, students can quickly raise the carriage to the 7-cm mark, without affecting the photogates, by rotating the carriage 180° so the side wing points away from the measurement scale on the ramp. Students should then push the carriage up the ramp and rotate it 180° again, so it’s in the proper orientation to block the photogates on its trip down the ramp.

Teacher Tips

• This laboratory activity should be performed after students have studied topics such as kinematics, projectile motion and potential and kinetic energy.
• Use a lubricant such as WD-40^® to reduce the friction in the wheels. Wipe up excess fluid before using the carriage.
• Extra masses can be added to the carriage compartment as necessary.
• The correction factor of 1.2 in Post-Lab Question 6 in Part 2 was derived as follows: As a result of the carriage rolling down the inclined plane, instead of sliding on a near-frictionless surface, a correction factor is required to account for the rotation of the wheels in order to determine the experimental value for the acceleration due to gravity (Part 2). (This may be beyond the scope of class.)
  {13339_Tips_Equation_5}
  m_t = total mass of the carriage a_y = linear acceleration down the inclined plane I = moment of inertia of the carriage wheel (factor of 2 because there are two wheels) α = rotational acceleration of the wheel (= a_y/R_w) R_w = radius of the carriage wheel g = acceleration due to gravity (to be determined) The moment of inertia of a solid cylinder is equal to (½)m_wR_w², where m_w is the mass of the wheel. The mass of the wheel is a fraction of the mass of the entire carriage, so m_w = δm_t, where δ is the mass fraction of the wheel. Substituting into Equation 5:
  {13339_Tips_Equation_6}
  Which reduces to:
  {13339_Tips_Equation_7}
  The mass of one wheel is approximately 26.2 g. Dividing this value by the total mass of the carriage (152.16 g), and δ is 0.17, or approximately 0.2, which leads to Equation 8. [(a_y)/(sin θ) is the calculated slope of the best-fit line.]
  {13339_Tips_Equation_8}
  The δ value will vary slightly from carriage to carriage and can be calculated for each carriage by the students, if desired. However, to calculate this value more accurately, the wheels will need to be removed from the carriage, weighed separately, and then the carriage will need to be reassembled properly. To avoid this, use the δ value 0.2 which is a good approximation.
• For Part 2, students can use the Photogate Timer to calculate the instantaneous speed of the carriage along the track by dividing the width of the wing by the time it takes the wing to transit the photogate (in Gate Mode). Students can then compare this experimental speed at certain positions along the track to the theoretical speed using the principle of the conservation of energy. Students will need to measure the height the carriage falls from its starting (rest) position.
  {13339_Tips_Equation_9}
  {13339_Tips_Equation_10}
  {13339_Tips_Equation_11}
  δ is the mass fraction of the wheel compared to the total mass of the carriage.
• For advanced classes, provide students with the laboratory objective and procedure without the Background information.
• Refer to the Photogate Timer instructions for more detailed descriptions of the modes and how to select the various modes of the timer.

Sample Data

Part 1. Mechanical Advantage
Distance between “start line” and “finish line”: ___70.0 cm___
Mass of carriage: ___153.16 g___
Additional mass added to carriage (if necessary): ___NA___

Results Table Part 2. Forces and Gravity
Length of the carriage wing: ___5.14 cm___

Answers to Questions

Part 1. Mechanical Advantage

1. For each inclined plane angle, calculate the Force Needed to Raise the Carriage. Multiply the Minimum Mass Needed to Raise Carriage (in kilograms) by the acceleration due to gravity constant, 9.81 m/s². The resulting force will have units known as newtons (N). Enter the results, as well as the units, in the results table.

   Sample calculation
   54 g x (1 kg/1000 g) x 9.81 m/s² = 0.5297 N

2. Multiply the mass of the carriage plus any additional mass (in kilograms) by the acceleration due to gravity constant to determine the weight of the carriage.

   153.16 g x (1 kg/1000 g) x 9.81 m/s² = 1.50 N

3. Use Equation 1 to calculate the actual mechanical advantage for each experimental angle. Record these results in the results table. Sample calculation 1.50 N/0.53 N = 2.83
4. Use Equation 2 to calculate the ideal mechanical advantage for each experimental angle of the inclined plane. Record these results in the results table.

   Sample calculation
   70.0 cm/(35.7 cm – 14.6 cm) = 3.3

5. Calculate the amount of energy that was needed to raise the carriage for each Ramp Angle. Multiply the force needed to raise the carriage at a specific angle by how far the carriage traveled along the inclined plane (in meters). The resulting energy will have units known as joules (J). Record these results, including units, in the results table.

   Sample calculation
   0.53 N x 0.700 m = 0.37 J

6. Calculate the “ideal” energy required to raise the carriage from the start line height to the finish line height by multiplying the weight of the carriage by total height the carriage was raised. Record this in the results table.

   Sample calculation
   1.50 N x (35.7 cm – 14.6 cm) x (1 m/100 cm) = 0.317 J

7. What angle of the ramp required the least effort (force) to raise the carriage?

   The smallest angle (7°) required the least amount of force to raise the carriage.

8. How does the mechanical advantage compare to the ease of raising the carriage?

   The larger the mechanical advantage, the easier it is to raise the carriage. It requires less force. More force is required to lift the carriage when the mechanical advantage is small.

9. What is an advantage of an inclined plane? What is a disadvantage of an inclined plane? Briefly describe the principles of a simple machine.

   An advantage of an inclined plane (simple machine) is that it helps reduce the force (effort) needed to raise a heavy object. A disadvantage of an inclined plane (simple machine) is that, although a smaller force is required, the force must be applied over a longer distance compared to how far (or high) the object moves.

10. Explain why the energy required in each case is similar to the “ideal” energy even though the needed force was less than the weight of the carriage. If any “extra” energy was needed to raise the carriage up the ramp, compared to the ideal case, why was it needed?

    The energy required to raise the carriage with a ramp was similar to the ideal case of simply lifting the carriage straight up because the force used to raise the carriage to a certain, smaller, height was used for a longer distance along the ramp. Energy is a force multiplied by the distance.
    For each case, the actual energy required was slightly larger than the ideal case. It required more energy to raise the carriage using the inclined plane because “extra” energy was needed to overcome the forces of friction in the wheels of the carriage and in the spinning pulley. (Simple machines can never be 100% efficient.)

Calculations and Post-Lab

1. Calculate the average values for the Transit Time between Photogates 1 and 2 for each Ramp Angle. Record these results in the data table.

   Sample calculations
   (0.4418 s + 0.4425 s + 0.4419 s + 0.4426 s + 0.4423 s)/5 = 0.44222 s
   (0.0595 s + 0.0598 s + 0.0595 s + 0.0595 s + 0.0598 s)/5 = 0.05962 s

2. Calculate the average speed of the carriage as it passes through each photogate by dividing the length of the carriage wing (in cm) by the average Transit Time (in seconds) through the individual photogate. Calculate the average speeds (in cm/s) through Photogates 1 and 2 for each Ramp Angle.

   Sample calculations
   (5.14 cm)/(0.0596 s) = 86.24 cm/s
   (5.14 cm)/(0.0208 s) = 247.12 cm/s

   {13339_Answers_Table_4}
3. Calculate the average acceleration of the carriage as it travels down the ramp. Subtract the average (calculated) speed at Photogate 2 by the average (calculated) speed at Photogate 1, and then divide this value by the average Transit Time between Photogates 1 and 2. Calculate the average acceleration (in cm/s²) for each Ramp Angle.

   Sample calculation
   (247 cm/s – 86.2 cm/s)/(0.4422 s) = 363.64 cm/s²

   {13339_Answers_Table_5}
4. How does the Ramp Angle affect the acceleration of the carriage? Explain.

   The acceleration of the carriage is lower when the angle of the ramp is shallow (low). A smaller fraction of the acceleration due to gravity is directed down the ramp.

5. On graph paper, plot the acceleration calculated for each angle on the y-axis, and the sin θ on the x-axis. Draw a straight best-fit line through the data points, including the origin (0,0). Then, calculate the slope of the best-fit line by dividing the “rise” by the “run.”
   {13339_Answers_Figure_11}
   Slope of best fit line = 835 cm/s²
   (using the trend-line equation function of the spreadsheet software)
6. Since the carriage rolls down the ramp, instead of sliding, a small correction factor is needed to account for the rotation of the wheels. Multiply the calculated slope of the best-fit line by 1.2. This new value is the experimentally determined value for the acceleration due to gravity. How does the value compare to the “true” value of 981 cm/s².

   The experimental value is 835 cm/s² x 1.2 = 1002 cm/s². This is approximately 2% above the “true” value of the acceleration due to gravity. This is reasonable experimental error when considering the “typical” error associated with the combination of timing, angle and length measurements.

Introduction

Simple machines provide us with a mechanical advantage—they make work easier to do. Experiment with the Carriage and Ramp (an inclined plane) to determine how to maximize its mechanical advantage.

Galileo and Newton used inclined planes to study the effects of gravity. In a second experiment, use the Carriage and Ramp and Photogate Timer to repeat one of their experiments and determine the acceleration due gravity.

Concepts

• Mechanical advantage
• Speed calculations
• Acceleration due to gravity
• Acceleration calculations

Background

Mechanical Advantage
A simple machine is a piece of equipment that changes the size or direction of an applied force. Examples of simple machines include pulleys, screws, gears, levers and wedges. These devices may appear “simple,” but by grouping various simple machines together, very “complex” machines can be created, such as engines or cranes. An inclined plane, also known as a ramp, is another example of a simple machine.

Simple machines are useful because they reduce the amount of force needed to move or lift an object. They provide the means for a normal person to lift a two-thousand pound car in order to change a tire (using a car jack) or for a 200-lb man to move a two-ton boulder (using a lever). The comparison between how much force (weight) is moved to how much force is applied is referred to as the mechanical advantage of the simple machine. For example, a simple machine with a mechanical advantage of five will provide five times more lifting force compared to the applied force—100 lb of applied force can lift a 500-lb object. The mechanical advantage of a simple machine is determined by calculating the ratio of the force required to move the object without the assistance of a simple machine to the actual force applied to the simple machine (Equation 1).

A simple machine does not provide “extra force” without something in return. A simple machine with a mechanical advantage of five will provide five times more lifting force compared to the force that is applied. However, the lifting (applied) force must be used over a distance that is five times greater than the distance the heavier object moves. The ideal mechanical advantage of a simple machine is determined by comparing how far the applied force moves to how far the object moves. It is considered “ideal” because it is based only on distances. Actual mechanical advantage must account for the force needed to overcome friction, as well as other factors. Therefore, actual mechanical advantage will always be less than the ideal mechanical advantage. For the inclined plane, the ideal mechanical advantage can be calculated using Equation 2. Forces and Gravity
What Galileo (1564–1642) proposed and Newton (1643–1727) essentially proved is that all objects fall toward the Earth at the same increasing rate (if in a vacuum) as a result of the force known as gravity. That is, all objects will accelerate toward the Earth equally, regardless of their mass. At the surface of the Earth, the acceleration of all objects toward the center of the Earth is measured to be 9.81 m/s² (32 ft/s²).

Newton also demonstrated that forces can be separated into horizontal and vertical components that are independent of each other. For example, for a force that pushes a ball up at an angle with respect to the ground, the force is said to have force components in the vertical and horizontal directions. Both of the components are dependent on the total force and the angle of the force with respect to the ground, but they are independent of each other. The expressions for calculating the horizontal and vertical force components are: F_x and F_y are the horizontal and vertical components of the force, F, respectively. The Greek letter theta (θ) represents the angle between the force and the x-coordinate (see Figure 1). Recall that a force is equal to the mass of the object times the acceleration of the object (F = ma). Therefore, a ball traveling down an inclined plane will accelerate due to the force of gravity, and the forces are represented in Figure 2. Note: As a convention, the y-axis is positioned parallel to the direction the ball travels down the inclined plane. Positioning the axis in this manner allows one to use Equations 1 and 2.

Experiment Overview

In Part 1 of this experiment, use the Carriage and Ramp to investigate the mechanical advantage of an inclined plane positioned at different angles. In Part 2, use the Carriage and Ramp and a Photogate Timer to determine the acceleration due to gravity.

Materials

Safety Precautions

Use care when adding and removing masses to the slotted-mass hanger. Do not allow the masses to drop on the floor. The materials in this lab are considered safe. Please follow all other laboratory safety guidelines.

Procedure

Part 1. Mechanical Advantage

Set up

1. Measure the mass of the carriage to the nearest 0.1 g using a balance. Record the mass under Part 1 on the Carriage and Ramp Worksheet.
2. Set up the PSworks™ Carriage and Ramp as shown in Figures 3 and 4. To extend the pulley over the edge of the table, use the metal support rod.
   {13339_Procedure_Figure_3}
   {13339_Procedure_Figure_4}
3. Cut fishing line or string approximately 1.5 m long. Tie a “looping knot” at one end of the fishing line as shown in Figure 5.
   {13339_Procedure_Figure_5_Loop knot}
4. Securely tie the non-looped end of the fishing line to the eye screw on the carriage.
5. Place the carriage at the base of the ramp so the eye loop and fishing line point toward the pulley and the side arm wing points toward the measurement scale on the ramp. Pull the fishing line over the pulley so that it hangs down. Place the loop around the hook of the slotted-mass hanger. Note: If the mass of the slotted-mass hanger is heavy enough to move the carriage up the ramp, add 100 g to the carriage to make it heavier than the slotted-mass hanger. Record the amount of additional mass under Part 1 on the Carriage and Ramp Worksheet, if necessary.

Experiment

1. Use a protractor to measure, to the nearest degree, the angle of the ramp (θ) with respect to the tabletop. Record this value under Part 1 on the Carriage and Ramp Worksheet.
2. At the 80-cm mark on the ramp, use a ruler to measure the height, to the nearest 0.1 cm, of the top edge of the ramp from the tabletop. Record this height in the data table under Part 1 on the Carriage and Ramp Worksheet. The 80-cm mark will be the “start line.”
3. At the 10-cm mark on the ramp, use a ruler to measure the height, to the nearest 0.1 cm, of the top edge of the ramp from the tabletop. Record this height in the data table. The 10-cm mark will be the “finish line.”
4. Start with the bottom of the front wheels of the carriage at the “start line.”
5. Carefully add slotted masses to the slotted-mass hanger in 10-g increments until the carriage just begins to travel up the ramp.
6. Once the carriage begins to move, remove the last 10-g mass that was added.
7. Move the carriage back to the “start line.”
8. Add slotted masses in 1- or 2-g increments until the carriage just begins to move up the ramp. When it appears that the carriage is about to move, give the carriage a small “nudge” with a finger to overcome the initial frictional forces (static friction). After the small nudge, if the carriage stops, add an additional mass to the mass hanger. If the carriage continues to move up the ramp, this is the minimum mass needed to raise the carriage.
9. Stop the carriage when the bottom of the front wheels of the carriage crosses the “finish line.”
10. Record the total mass, to the nearest gram, of the slotted masses, including the mass of the hanger, needed to just move the carriage up the ramp.
11. Carefully lower the carriage to the base of the ramp and remove the slotted masses.
12. Raise the angle of the ramp (four to six holes higher on the Support Stand) and repeat steps 1–11. Add the same extra mass to the carriage if this was required for the initial angle. Record all the measurements and data in the appropriate columns of the data table.
13. Lower the angle of the ramp (three to four holes lower on the Support Stand than the original position), and repeat steps 1–11. Add the same extra mass to the carriage if this was required for the initial angle. Record all the measurements and data in the appropriate columns of the data table. Part 2.

Forces and Gravity

1. Set up the PSworks™ Carriage and Ramp as shown in Figure 6. Connect the proper cords into the PSworks Photogate Timer. Clamp one photogate at the 10-cm mark and the second photogate at the 80-cm mark on the Carriage and Ramp. Make sure the centers (the light sensor slot) of the photogates line up with the appropriate marks on the ramp.
   {13339_Procedure_Figure_6}
2. Use a ruler to measure the length of the wing on the carriage to the nearest 0.1 mm. Record this value under Part 2 on the Carriage and Ramp Worksheet.
3. Use a protractor to measure the angle of the ramp (θ) with respect to the tabletop to the nearest degree, and record this value in the first five empty cells of the left-most column of data table under Part 2 on the Carriage and Marble Ramp Worksheet.
4. Turn on the Photogate Timer and select the Two Gate Mode. (Refer to the Photogate Timer instructions, if necessary.)
5. Hold the front edge of the front wheels of the carriage at the 7-cm mark on the ramp.
6. Release the carriage without giving it any “extra” push.
7. The transit time of the carriage wing between both photogates will be registered by the timer. Record this value in the data table.
8. Repeat steps 5–7 four more times, recording all measurements in the data table. Note: Release the carriage from the same position each time. Also, for steep angles, be prepared to stop the carriage at the bottom of the ramp after it rebounds off the rubber end-stop.
9. Clear the memory in the Photogate Timer (press and hold Memory and Reset simultaneously until the memory clears). Then select the Gate Mode.
10. Hold the front edge of the front wheels of the carriage at the 7-cm mark on the ramp.
11. Release the carriage without giving it any “extra” push.
12. The transit time of the carriage wing through each photogate will be registered by the timer separately. Record the first value registered by the timer under Photogate 1 in the data table, and the second (smaller) value registered by the timer under Photogate 2 in the data table.
13. Repeat steps 10–12 four more times, recording both photogate times in the appropriate columns of the data table. Note: Release the carriage from the same position each time. For steep angles, be prepared to stop the carriage at the bottom of the ramp after it rebounds off the rubber end-stop.
14. Raise the angle of the ramp (three or four holes higher on the Support Stand) and repeat steps 3–13. Record all the values in the appropriate columns of the data table.
15. Lower the angle of the ramp (two to four holes lower on the Support Stand than the original position), and repeat steps 3–13. Record all measurements in the appropriate columns of the data table.
16. Consult your instructor for appropriate storage procedures.

Student Worksheet PDF

13339_Student1.pdf