Materials Included In Kit

Additional Materials Required

Safety Precautions

Remind students to quickly retrieve the ball once it hits the floor. Wear safety glasses. Please follow all laboratory safety guidelines.

Disposal

The materials should be saved and stored for future use.

Lab Hints

• Enough materials are provided in this kit for eight student groups. All materials are reusable.
• It is important for the students to create a smooth curve with the metal sheet so that the ball does not bounce when it reaches the end of the inclined plane. Place the curved metal sheet approximately 1–2 cm from the edge of the tabletop so the ball does not roll along the tabletop for an extended period of time.
• If the inclined plane is tilted and the ball does not roll in a straight line down the inclined plane, use several folded sheets of paper to shim the bottom of the inclined plane until it is level and the ball rolls in a straight line.

Teacher Tips

• This laboratory activity should be performed after students have studied topics such as kinematics, projectile motion and potential and kinetic energy.
• The most significant source of error occurs if the ball bounces instead of rolling smoothly off the edge of the table. It is important that the metal sheet have a smooth curve between the end of the inclined plane and the tabletop. If the curve is not smooth or the metal sheet is curved too much the ball will bounce off the tabletop, reducing the distance the ball is expected to travel because the ball will have both horizontal and vertical speed components. The ball’s momentum may bend the metal when it rolls over it, especially if it is released from a great height. A small amount of crumpled paper can be placed under the curved metal junction in order to give it some more support.
• 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 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

Height of the tabletop from the floor (H): ___83.3 cm___

Answers to Questions

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

   Experiment 1

   (65.3 cm + 65.0 cm + 64.7 cm + 65.4 cm + 64.9 cm + 64.7 cm)/6 = 65.0 cm

   Experiment 2

   (52.1 cm + 52.3 cm + 52.6 cm + 52.4 cm + 52.3 cm + 52.3 cm)/6 = 52.3 cm

   Experiment 3

   (63.8 cm + 62.3 cm + 61.2 cm + 61.6 cm + 62.5 cm + 61.4 cm)/6 = 62.1 cm

   Experiment 4

   (54.3 cm + 54.0 cm + 54.4 cm + 54.5 cm + 54.5 cm + 54.8 cm)/6 = 54.4 cm

2. Substitute the average launch distance calculated in Question 1 into Equation 5 (see the Background section) to determine the launch speed of the ball for each experiment.
   {12650_Background_Equation_5}
   {12650_Background_Equation_14}
3. Use Equation 13 to calculate the theoretical launch speed the ball should have for each experiment.
   {12650_Background_Equation_13}
   {12650_Background_Equation_18}
4. In some trials, the ball may bounce slightly on the table before it leaves the tabletop. How would this affect the horizontal speed of the ball as it leaves the tabletop? Would the experimentally determined launch speed of the ball be higher or lower than the theoretical value as a result of this error? Explain.

   The most significant source of error in the experiment comes from the bounce of the ball when it reached the tabletop. The curved metal sheet and crumpled paper supporting the sheet helped to make a smooth transition between the inclined plane and the tabletop but the bounce was not completely eliminated. When the ball bounces, it will attain an initial vertical velocity component as well as the horizontal velocity component. The total speed of the ball will be a sum of the vertical and horizontal components. Therefore, the horizontal speed will be slightly less than total speed of the ball when the ball bounces. Equation 5 assumes that all the speed the ball achieves as it rolls down the inclined plane will only be in the horizontal direction—the vertical velocity will be zero. However, when the ball bounces, the horizontal speed will be less than the total speed, and therefore the ball will travel a shorter distance than expected. (Reducing the ball speed and lowering the angle appeared to reduce some of the ball bounce because the experimentally determined ball speed for Experiments 2 and 4 was closer to the expected result.)
   Note: Other minor sources of error could include: the ball rolled down the inclined plane at an angle and therefore did not roll off the tabletop perpendicular to the edge (this would decrease the launch distance of the ball compared to perpendicular edge of the tabletop—averaging measurements from several trials will minimize this error); not releasing the ball cleanly; small speed decrease when the ball travels between the end of the metal sheet and the edge of the tabletop; air resistance (will have a very minor effect on the ball’s motion); and systematic measuring error.

5. How did the angle of the inclined plane affect the launch speed of the ball? Explain.

   The angle of the inclined plane did not affect the speed of the ball at the bottom of the inclined plane. The only variable that affected the speed of the ball was the initial height of the ball on the inclined plane. Potential energy is only affected by the height of the ball, not by the angle at which it rolls.

Introduction

Use knowledge of projectile motion and the law of conservation of energy to determine the launch speed of a steel ball after rolling down an inclined plane.

Concepts

• Projectile motion
• Rolling objects
• Potential and kinetic energy

Background

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). 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 (on average) 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 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. This means that a vertical force will have no effect on the magnitude of the horizontal force component, and vice versa.

In this experiment, a ball will roll down an inclined plane and launch off the end of the table. When a ball launches horizontally from the end of the tabletop and falls toward the ground, the only force acting on the ball is the downward pull due to gravity. After the ball leaves the table, there is no force in the horizontal direction (neglecting frictional forces). Since there is no horizontal force, the horizontal speed of the ball as it leaves the tabletop will remain constant throughout its flight.

The vertical velocity will experience a change however. The instant the ball leaves the tabletop traveling horizontally, its vertical speed is zero—the ball is not moving up or down for a very brief moment after leaving the table. After this very short instant, the ball begins to drop due to the downward pull of gravity. The force due to gravity is constant—the ball falls with a constant downward acceleration. This downward acceleration causes the ball's vertical speed to change at a constantly increasing rate while its horizontal speed remains the same. These principles of projectile motion will be used to determine the initial horizontal speed of the ball when it leaves the tabletop.

In this experiment, the distance the ball travels horizontally and the height of the fall will be used to determine the initial speed of the ball. Since acceleration due to gravity is constant for all objects, the time it takes for any object, initially at rest, to fall a specific distance will be the same. The distance any falling object travels in a given amount of time can be determined using Equation 1.

H = height (sometimes referred to as Δy)
g = acceleration due to gravity
t = time

Rearranging Equation 1 to solve for t produces Equation 2, which can be used to calculate the time it will take for any object to fall a certain distance. The time calculated in Equation 2 is the total time it takes for the ball to fall from the edge of the tabletop to the floor. Since the horizontal speed is constant, the distance the ball travels horizontally can be determined by multiplying the horizontal speed by the total flight time of the ball (Equations 3 and 4).

D = horizontal distance (sometimes referred to as Δx)
v_h = horizontal speed
t = time

Substituting Equation 2 into Equation 3 yields The initial horizontal speed can then be evaluated by rearranging Equation 4 to solve for v_h. Equation 5 will be used to calculate the actual speed of the ball as it leaves the tabletop.

How will the actual speed compare to the theoretical speed? To determine the theoretical speed at which the ball should travel as it leaves the edge of the tabletop after rolling down the inclined plane, the potential and kinetic energy of the ball on the inclined plane need to be evaluated.

In order to raise the ball to the release point on the inclined plane, one must exert energy. Work, another term for energy, is being performed on the ball in order to raise it. Work is defined as a force used through a distance. It requires work to move any object. If a force is used on an object that doesn't move, such as pushing on a brick wall, then no work is being done. The energy used to raise the ball to a higher position is "stored" in the ball—the ball is said to have potential energy (PE). This potential energy may be used at a later time to do work. The potential energy of the ball is related to its height and weight. The potential energy 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 bottom of the inclined plane (Equation 6). As the ball begins to roll down the inclined plane, its potential energy is converted into kinetic energy, or energy of motion. The ball experiences two different types of motion—it is traveling in a straight path down the inclined plane and it is rotating. Therefore it has both linear kinetic energy and rotational kinetic energy. Linear kinetic energy (KE_l) is related to the mass (m) and linear speed (v) of the object (Equation 7). Rotational kinetic energy (KE_r) is related to the moment of inertia (I) of the ball and the rotational speed (ω; the Greek letter “omega”). See Equation 8. (Notice the similarity between Equation 7 and Equation 8.) The moment of inertia is the “resistance” an object has to being rotated. It 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 the center of the ball. The total kinetic energy (KE_T) of a rolling ball is therefore equal to the linear kinetic energy plus the rotational kinetic energy (Equation 9). The law of conservation of energy states that energy cannot 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 inclined plane (neglecting frictional forces). This is represented by Equation 10. Substituting Equation 6 and Equation 9 into Equation 10 yields Equation 11 can be used to determine the theoretical speed of the rolling ball when it reaches the bottom of the inclined plane.

The moment of inertia of a solid sphere rotating about its center is equal to 2/5mR², where R is equal to the radius of the sphere. Rotational speed, ω, is related to linear speed of the ball, v, and the radius, R, of the object: ω = v/R. Substituting these values into Equation 11 yields Rearranging Equation 12 to solve for v² Notice that Equation 13 shows that the speed of the ball at the bottom of the inclined plane is independent of the size or the mass of the solid ball. This means that any size solid ball will have the same speed at the bottom of the inclined plane so long as it is released from the same height. Equation 13 will be used to calculate the theoretical speed of the ball at the bottom of the inclined plane.

Experiment Overview

Roll a ball down an inclined plane off the end of a tabletop and measure the distance the ball falls. Use these measurements to calculate the initial speed of the ball, and compare this value to the theoretical speed calculated using the conservation of energy principle.

Materials

Procedure

Preparation

1. Obtain the 4" x 4" metal sheet.
2. Set up the inclined plane as shown in Figure 1. The angle of the inclined plane should be between 15° and 45°. Position the bottom of the inclined plane approximately 10 cm from the edge of the table. Make sure there is at least 2–3 meters of open space on the floor around the edge of the tabletop so the steel ball will have enough space to launch off the end of the table.
   {12650_Preparation_Figure_1}
3. Use transparent tape to tape the edge of the 4" x 4" metal sheet to the bottom end of the inclined plane.
4. Bend the metal sheet slightly to form a smooth transition curve between the end of the inclined plane and the tabletop surface. The free end of this sheet should be within a centimeter or two from the edge of the tabletop. When the metal sheet forms a smooth curve, tape the other end of the sheet to the tabletop using transparent tape. Make sure the tape is flat and smooth at both ends of the metal sheet (see Figure 2).
   {12650_Preparation_Figure_2}
5. Practice rolling the ball down the inclined plane and off the edge of the tabletop. Make sure the ball does not hit the tabletop hard, causing the ball to bounce upward. Adjust the curve in the metal sheet until the ball makes a smooth transition between the inclined plane and the tabletop. If the metal appears to bend down when the ball rolls over it, small amounts of crumpled paper can be used under the sheet to help support it.

Experiment

6. 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.
7. Hang the plumb-bob from the end of the tabletop where the inclined plane will launch the steel ball. The fishing line only needs to be held in place with a hand. It does not need to be tied. Make sure the plumb-bob hangs close to the floor without touching.
8. Place a strip of transparent tape, or use erasable chalk or pencil, to mark the spot on the floor directly below the hanging plumb-bob. If tape is used, make sure one edge of the tape signifies the location directly below the plumb-bob.
9. Use a meter stick to measure the height of the top edge of the tabletop from the floor. Make sure to measure from the mark made below the hanging plumb-bob. Record this height in the data table.
10. Use a pencil to lightly mark on the inclined plane approximately 10 cm from the high end of the inclined plane. This will be the release position of the ball.
11. Use a ruler or meter stick to measure the height of this release point above the tabletop. Record this height in the data table.
12. Use a protractor to measure the angle the inclined plane makes with respect to the tabletop. Record this angle in the data table.
13. When performing the experiments, one lab partner will place the steel ball on the release line marked on the inclined plane and then release it. The other lab partner needs to be in a position to catch the ball after it launches off the end of the tabletop and hits the floor.
14. Place the center of the steel ball at the release line. (One person should be ready to catch the ball after the first bounce.) Gently release the ball, making sure not to give it any additional push. The ball should roll straight down the inclined plane. Take note of the general area on the floor where the ball lands.
15. Securely tape 3 or 4 sheets of white printer paper along the horizontal path that the launched ball followed during the test run in step 14. Place a sheet or two beyond the point where the ball hit during the practice run.
16. Repeat step 14 six or more times. Release the ball from the same release height for each trial. The ball should leave a dark mark 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.
17. Use a meter stick to measure 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 first ball mark only—the mark closest to the inclined plane. Do not measure additional marks left by the bouncing ball.
18. Repeat steps 10–17 for a release height approximately halfway up the inclined plane.
19. Change the angle of the inclined plane. Draw a ball release line that is at the same height as the ball release height in Experiments 1 or 2. Repeat steps 10–17. It may be necessary to replace several ball-marked sheets of paper.
20. Consult your instructor for appropriate storage procedures.

Student Worksheet PDF

12650_Student1.pdf