Materials Included In Kit

Additional Materials Required

Safety Precautions

The materials in this lab are considered safe. 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.
• For Part 1, shallow angles produce better results because the timing measurements will include three significant figures. Also, at steep angles, the ball may initially slide down the ramp before it begins to roll. This will affect the “correction factor.” Do not raise the ramp above 35° for Part 1.
• Students can take data at more angles to obtain a “better” best-fit line for Part 1. However, do not exceed 35°.
• The PSworks™ Photogate Timer can also be clamped to the bottom edge of the ramp to determine the speed of the marble just before it launches and enters free fall (Part 2). The timer will need to be set in Gate Mode. Students can compare this value to the value obtained using Equations 7 and 15.
• Use calipers to obtain a more accurate value for the diameter of the marble. A carpenter’s level can be used to more accurately adjust the end of the ramp to horizontal.

Teacher Tips

• This laboratory activity should be performed after students have studied topics such as kinematics, projectile motion and potential and kinetic energy.
• As a result of the ball rolling down a track with rails, instead of sliding on a near-frictionless surface, a correction factor is required to account for the rotation as well as the off-center contact points with the track in order to determine the experimental value for the acceleration due to gravity (Part 1, Post-Lab Question 6). This may be beyond the scope of the class, but the derivation is as follows:
  {12520_Tips_Equation_16}

  m = mass of the ball
  a_y = linear acceleration down the inclined plane
  I = moment of inertia of the ball
  α = rotational acceleration of the ball (= a_y/R_c)
  R = radius of the ball
  R_c = radius of the contact point with track (see Figure 6).
  g = acceleration due to gravity (to be determined)

  {12520_Tips_Figure_6}
  The moment of inertia of a solid ball is equal to (2/5)mR². Substituting into Equation 16:
  {12520_Tips_Equation_17}
  R_c is approximately 0.9R, so Equation 17 reduces to approximately
  {12520_Tips_Equation_18}
  (a_y)/(sin θ) is the calculated slope of the best-fit line.
• 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). 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 Part 2, a significant source of error occurs when the ball does not leave the end of the Marble Ramp horizontally. If the bottom edge points up, this will reduce the distance the ball is expected to travel because the ball will have both horizontal and vertical speed components.
• For Part 2, a different correction factor is required to account for the ball rolling down a track. Unfortunately, Equation 15 is a simple case in which the bottom of the ball is the only part that makes contact with the track. In this experiment, the ball rolls down the track similar to Figure 6. The correct “multiplication” factor is then:
  {12520_Tips_Equation_19}
  Which reduces to approximately
  {12520_Tips_Equation_20}
  Therefore, 4/3 should be used as the multiplication factor in Part 2, Post-Lab Question 3 instead of 10/7.
• For advanced classes, provide students with the laboratory objective and procedure without the Background information. Have students solve the equations of motion by referring to their textbooks in order to determine the speed of the launched ball. Then have students determine what the theoretical value should be. Many times students will forget that the ball is rolling and will not 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

Part 1. Forces and Gravity

Diameter of the marble: ___1.90 cm___

Part 2. Projectile Motion

Release Point Height
Release Point 1: ___49.1 cm___ 2: ___43.7 cm___ 3: ___33.2 cm___
Height of the bottom of ramp: ___5.2 cm___
Height of the edge of the Marble Ramp to the floor: ___95.8 cm___
Marble Ramp Angle: ___35°___

Answers to Questions

Part 1. Forces and Gravity

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

   Sample calculation
   (0.6621 s + 0.6628 s + 0.6625 s + 0.6625 s + 0.6618 s)/5 = 0.66234 s

2. Calculate the average speed of the marble as it passes through each photogate by dividing the diameter of the marble (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 Marble Ramp Angle.

   Sample calculation
   1.90 cm/0.0293 s = 64.846 cm/s

   {12520_Answers_Table_3}
3. Calculate the average acceleration of the marble as it travels down the Marble 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 angle of the Marble Ramp.

   Sample calculation
   (202 cm/s – 64.8 cm/s)/0.6623 s = 207.16 cm/s²

   {12520_Answers_Table_4}
4. How does the angle of the inclined plane affect the acceleration of the marble? Explain.

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

5. On graph paper, plot the average acceleration calculated for each angle on the y-axis versus 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.”
   {12520_Answers_Figure_7}
6. Since the marble rolls down a track, or rail, a correction factor is needed to account for the rotation. Multiply the calculated value of the slope of the best-fit line by 1.5. This value new 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 slope of the best-fit line is 734 cm/s² (per spreadsheet trendline function).
   734 cm/s² x 1.5 = 1101.6 cm/s² (1100 cm/s² to 3 significant figures)
   The value is higher than the literature value, with a percent error of 12%. This error is due mainly to timing errors based on the small size of the marble and the curvature of the marble. Often, the photogate will not “see” the entire length of the marble, depending on the position of the ball’s center with respect to the ray of light shining between the arms of the photogate. If the spacing is off by 1 mm, and the photogate “sees” the marble as 1.8 cm, instead of 1.9, then this would result in an error of 5%.

Part 2. Projectile Motion

1. Calculate the average flight distance for each experiment. Enter the results in the data table.

   Sample calculation
   (102.8 cm + 100.8 cm + 100.4 cm + 100.1 cm + 99.2 cm)/5 = 100.66 cm

2. Use Equation 7 and the average flight distance to calculate the initial launch speed of the marble for each trial.

   Sample calculation

   {12520_Answers_Equation_21}
3. Use a modified Equation 15 to calculate the theoretical launch speed of the marble for each trial. Since the marble rolls down a track, or rail, the multiplication factor is 4/3, instead of 10/7.

   Sample calculation

   {12520_Answers_Equation_22}
4. Compare the theoretical and calculated launch speed values.

   The experimental launch speeds are all lower than the theoretical launch speed (between 3–5% error). This can be reasonably explained by friction and timing errors.

5. If the bottom of the ramp were curved upward instead of horizontal, how would this affect the flight distance? Would the experimental launch speed of the marble be higher or lower than the theoretical value as a result of this setup error? Explain.

   The flight distance of the marble would be shorter because the marble would have risen up slightly and “hopped” so some of the marble’s speed would be in the vertical direction and the rest in the horizontal direction. The marble would not travel as fast horizontally and it would remain in the air only a fraction of a second longer so the marble will land closer to the Marble Ramp. The experimental launch speed would then be calculated to be lower than the theoretical speed because the experimental launch speed is based on the distance the marble travels.

Introduction

Experiment with inclined planes and projectiles to determine how the acceleration due to gravity affects falling objects.

Concepts

• Conservation of energy
• Kinetic energy
• Force
• Projectile motion
• Potential energy
• Vectors

Background

Gravity and Forces

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) due to 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 981 cm/s² (32 ft/s²).

Newton also demonstrated that forces can be separated into horizontal and vertical components that are independent of each other. 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 both 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 calculation of the horizontal and vertical forces 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). Equations 1 and 2 can be used for any vector quantity, including acceleration and velocity. A vector quantity is a value that has both a magnitude and a direction associated with it. For example, 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 of the ball traveling down the inclined plane. Orientating the axis in this manner allows one to use Equations 1 and 2. (Recall that F = ma.) Part 1 of this experiment will use an inclined plane (ramp) and a ball (marble) to determine the acceleration due to gravity. Projectile Motion

When a ball is in “free fall,” the only force acting on it is the downward pull due to gravity. Therefore, if the “free falling” ball has any initial horizontal speed, the horizontal speed will remain constant throughout the free fall. The vertical speed of the ball, however, will change the instant it begins to fall. The constant acceleration due to gravity will cause the ball’s vertical speed to change at a constantly increasing rate.

In Part 2 of this experiment, a ball will travel down an inclined plane and launch horizontally off the end of a table. The ball’s speed at the bottom of the inclined plane, just as it leaves the edge, will be calculated by measuring its flight distance. The first value that is needed is the time the ball is in free fall. Because the acceleration due to gravity is the same for all objects, the time it takes for any object, initially at rest, to fall a specific distance will be the same. Therefore, the distance any falling object travels in a given amount of time can be determined using Equation 3.

H = falling height
g = acceleration due to gravity
t = time

Rearranging Equation 3 to solve for t produces The time calculated in Equation 4 is the “free fall” time of the ball to fall from the bottom edge of the inclined plane to the floor (H). Since the ball’s horizontal speed will be constant, the distance the ball travels horizontally can be determined by multiplying the horizontal speed by the flight time of the ball (Equations 5 and 6).

D = horizontal distance
v_h = horizontal speed
t = time

Substituting Equation 4 into Equation 5 yields And finally, the horizontal speed of the ball in flight can be evaluated by rearranging Equation 6 to solve for v_h. Equation 7 will be used to calculate the actual speed of the ball as it leaves the edge of the inclined plane. Conservation of Energy

How does the experimental speed of the ball at the bottom of the inclined plane compare to the theoretical speed? The theoretical speed at which the ball should travel as it leaves the edge of the inclined plane can be calculated by evaluating the potential and kinetic energy changes of the ball as it travels down the inclined plane. Work is the act of using a force to move an object through a distance. In order to raise the ball to the “release point” on the inclined plane, one must exert energy (work) to lift it. The energy expended to raise the ball to a higher position is “stored” in the ball—the ball now has potential energy (PE). The potential energy of the ball is related to its height and weight, and is equal to the mass (m) of the ball multiplied by the acceleration due to gravity (g) multiplied by the relative height (h) of the ball above the ground, or in the case of this experiment the bottom edge of the inclined plane (Equation 8). As the ball rolls down the inclined plane, its potential energy is converted into kinetic energy, or energy of motion. This is due to the conservation of energy principle. When the ball 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 9). 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 10.

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 ball is its center. The total kinetic energy (KE_T) of a rolling ball is therefore equal to the linear kinetic energy plus the rotational kinetic energy (Equation 11). The conservation of energy principle states that energy can not be created or destroyed—energy can only be converted from one form to another. Therefore, the initial potential energy the ball has at the release point will be completely converted into kinetic energy at the bottom of the ramp (neglecting frictional forces). This is represented by Equations 12 and 13. Equation 13 will be used to determine the theoretical speed of the rolling ball when it reaches the bottom edge of the inclined plane just as it launches into free fall. The moment of inertia of a solid ball 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 the linear speed of the ball, v, and the radius, R, of the object: ω = v/R. Substituting these values into Equation 13 gives Equation 14. Reducing and rearranging Equation 14 to solve for v² generates Notice that Equation 15 shows that the speed of the ball at the bottom of the inclined plane is independent of the size or the mass of the ball. Any size solid ball will have the same speed at the bottom of the inclined plane as long as it released from the same height. Equation 15 will be used to calculate the theoretical speed of the ball at the bottom edge of the Marble Ramp just before it begins to free fall to the floor.

Experiment Overview

In Part 1 of this experiment, use the Marble Ramp and a Photogate Timer to determine the acceleration due to gravity. In Part 2, use the Marble Ramp to launch a marble (ball) horizontally off the end and calculate the speed of the marble using the distance it travels and height it falls. Then, compare this value to the theoretical value using the conservation of energy principle.

Materials

Safety Precautions

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

Procedure

Part 1. Forces and Gravity

1. Set up the PSworks™ Marble Ramp as shown in Figure 3. Connect the proper cords into the PSworks Photogate Timer. Clamp one photogate at the 10-cm mark and the second photogate at the 90-cm mark on the Marble Ramp. Make sure the centers (the light sensor slot) of the photogates line up with the appropriate marks on the ramp.
   {12520_Procedure_Figure_3}
2. Use a ruler to measure the diameter of the marble to the nearest 0.1 mm. Record this value under Part 1 on the Marble Ramp Worksheet.
3. Use a protractor to measure the angle (θ) of the Marble Ramp 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 the data table under Part 1 on the 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 bottom center of the marble at the 2-cm mark on the Marble Ramp.
6. Release the marble without giving it any “extra” push.
7. The transit time of the marble 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 the values in the data table. Note: Release the ball from the same position each time.
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 bottom center of the marble at the 2-cm mark on the Marble Ramp.
11. Release the marble without giving it any “extra” push.
12. The transit time of the marble 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 ball from the same position each time.
14. Raise the angle of the Marble Ramp (two or three 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 Marble Ramp so that it is smaller than the original angle (two or three holes lower on the Support Stand than the original position), and repeat steps 3–13. Record all the values in the appropriate columns of the data table.

Part 2. Projectile Motion

Setup

1. Set up the PSworks Marble Ramp so that the curved end hangs off the end of the table and the bottom edge is horizontal to the floor (see Figure 4).
   {12520_Procedure_Figure_4}
   Attach the ramp to the Support Stand at the eleventh hole from the bottom. Extend a ruler off the end of the ramp to check the levelness (see Figure 5). Roll the ball off the edge of the ramp a couple times to make sure it travels horizontally off the end of the ramp and does not “hop.” Adjust the angle of the ramp, adjust the leveling feet on the PSworks Support Stand or place thin books under the bottom of the ramp, as necessary, to obtain the appropriate horizontal launching platform.
   {12520_Procedure_Figure_5}
2. Cut 1.5–2 m of fishing line (depending on the height of the table). Tie one end of the fishing line to a washer to create a plumb-bob.
3. Hang the plumb-bob from the bottom end of the ramp (Figure 4). Hold the fishing line in place with your hand and make sure the plumb-bob hangs as close to the floor as possible without touching.
4. Place a strip of masking tape, or use erasable chalk or pencil, to mark the spot on the floor directly below the hanging plumb-bob. If tape is used, mark the edge of the tape to signify the location directly below the plumb-bob.
5. Use a meter stick to measure, to the nearest 0.1 cm, the height between the mark made on the floor and the top edge of the ramp where the ball will roll off (see Figure 4). Record this value under Part 2 on the Marble Ramp Worksheet.
6. Use a protractor to measure the angle (θ) of the Marble Ramp with respect to the tabletop to the nearest degree, and record this value under Part 2 on the Marble Ramp Worksheet.

Experiment

When performing this experiment, one lab partner will place the marble at the specific release point on the Marble Ramp as noted in step 7, and then release it. A second lab partner needs to be in a position to catch the marble after it launches off the end of the ramp and hits the floor.

7. Determine a marble “release point” on the ramp and record the value on the scale corresponding to this release location in the data table under Part 2 on the Marble Ramp Worksheet.
8. Use a meter stick to measure, to the nearest 0.1 cm, the height of this release point above the tabletop. Record this height: Release Point 1: ________________ 2: ________________ 3: ________________
9. Use a meter stick to measure, to the nearest 0.1 cm, the height of the bottom end of the ramp (where the ball leaves the ramp) and the tabletop. Record this height: ________________
10. Subtract the height measured in step 9 from the value measured in step 8. Record this height in the data table under Marble Release Height.
11. (Partner 1) Place the center of the marble at the release point on the Marble Ramp. (Partner 2 should be ready to catch the ball after the first bounce.) Quickly release the marble without giving it any “extra” push. Take note of the general area on the floor where the marble lands.
12. Securely tape 3 or 4 sheets of white printer paper along the horizontal path that the launched marble followed during the test run in step 9. Place a sheet or two at and beyond the point where the ball hit during the practice run.
13. Repeat step 11 five more times. Release the marble from the same release point for each trial. The marble should leave a dark mark or indentation on the white paper where it lands. Between each trial, use a pencil to circle the mark on the sheet of paper and label it with the trial number.
14. Use a meter stick to measure, to the nearest 0.1 cm, the horizontal distance between the plumb-bob mark and the initial ball marks on the paper for each trial. Record these distances in the data table. Note: For each trial, measure the mark closest to the Marble Ramp. Do not measure additional marks left by the bouncing ball.
15. Repeat steps 7–14 twice at two different release heights. Record all the values in Part 2 of the worksheet.
16. Consult your instructor for appropriate storage procedures.

Student Worksheet PDF

12520_Student1.pdf