PhysicsLAB: Target Lab: Ball Bearing Rolling Down an Inclined Plane

Purpose: To predict the landing point of a projectile after it has rolled down a ramp.

Equipment needed: Each group needs: a ramp, target paper, carbon paper, meter stick, and plumb line.

Resource Lessons: 2D projectiles (horizontal release), Conservation of Energy, and Rotational Kinetic Energy

Background

Remember that when analyzing two-dimensional projectile motion, the horizontal and vertical motions are independent of each other. Horizontally, projectiles in freefall travel at a constant velocity; while vertically, they experience uniform acceleration resulting in a classic parabolic trajectory. Our secret to working projectile problems was to build a chart in which we delineated the Horizontal | Vertical properties in each situation.

Horizontally, the only equation available to us was R = v_Ht, where v_H represents the projectile's constant horizontal velocity. Vertically, in the above illustration, the projectile's initial velocity equaled zero, since it was launched straight forward. Usually, in this situation, we let v_o = 0, a = -9.8 m/sec², and s = -h and then used the kinematics equation s = v_ot + ½at² to solve for the time that the projectile spent in the air.

Your goal in this experiment is to predict where a steel ball will land on the floor after having rolled down an incline plane. The final test of your measurements and computations will be to position a bull's-eye on the floor so that the ball lands in its center circle on the first attempt. Make sure that ALL measurements and calculations are reported with three significant figures.

Part I: The Experiment

Step 1: Assemble your ramp. Make it as sturdy as possible so the steel ball bearing rolls smoothly and consistently. The ramp should not sway or bend. Since the ball bearing must leave the table horizontally, make sure that the horizontal part of the ramp is level with the surface of the table. The vertical height, h, of the ramp should be no less than 7 cm.

Step 2: Calculate the ball bearing's horizontal velocity at the base of the ramp using conservation of energy principles. At the top of the ramp, if the ball bearing is released from rest, it will only have potential energy, PE, which equals the product of its mass (in kilograms) times the acceleration due to gravity (9.8 m/sec²) and its height (in meters) above an arbitrary reference line. At the base of the ramp, the ball has both translational kinetic energy, KE = ½mv², and rotational kinetic energy, KE_rot = ½Iw². Recall that the moment of inertia for a solid sphere equals I = (2/5)mr² and that v = rw.

PE_top = Total KE_base
mgh = ½mv² + ½Iw²

This velocity at the base of the incline will remain the ball bearing's horizontal velocity when it leaves the table. Remember that you will need to release the ball at the very top of the ramp and not put any pressure against the ramp that might result in it "springing" forward when the ball is released.

How high (in cm) was the back of your ramp (ruler) above the top of the table?

What will be your ball bearing's horizontal velocity (in m/sec) at the base of its ramp? Show your calculations for the ball's horizontal velocity in the space provided below on your answer sheet.

Why did you not need to measure the ball bearing's mass for these calculations?

Step 3: Using a plumb line, string, and meter stick to measure and record in blank below the vertical height of the lab table above the floor in centimeters.

Step 4: Using the appropriate equation from the background information given above, calculate the time, t, that the ball bearing will take to fall from the base of the ramp on the table's surface to the floor.

t (in sec) =

Step 5: The range is the horizontal distance a projectile once it is leaves the table until it strikes the floor. Calculate the range of the ball bearing. Show your equation and any necessary calculations used in predicting the ball's range.

R (in m) =

Teacher certification that you have calculated your experimental range.

Step 6: Now tape your target paper on the floor so that its target line is at the prediected range. When you are ready to release the ball bearing, call your instructor over to witness your trial. Remember to make sure that the ball is released from the top of the ramp. (You will be allowed a maximum of three releases.) You may remove your target paper from the floor to measure how far the ball's impact point was located from your predicted range.

End of Part I: If the ball bearing overshot the target then report your answer as a + x number of centimeters. If it fell short of the target report your answer as - x number of centimenters.

Step 7. Our ball missed the center of the bullseye by ___ cm.

Part II: Analysis of Experimental Results

Obtain from your teacher a glass marble. Roll the glass marble down your ramp and observe the ball-ramp system. Catch the ball when it lease the ramp so that it doesn't strike the ground.

Step 8. What do you notice about the ramp as the glass marble rolls down the track?

Step 9. Are there ramifications to your previous observation that might explain your ball bearing's actual experimental rang reported in Step 7?

Remember that all of your calculations are to be done in meters, kilograms, and seconds,

projectile motion

flight time
s = v_ot + ½at²(sec)

vertical v_f
v_f = v_o + at
(m/sec)

actual v_H
R_exp = v_Ht
(m/sec)

v_R resultant
impact velocity
(m/sec)

Step 10. State the mass of your ball bearing in kilograms.

energy calculations

total PE
ramp + table
mg(h + H)
(J)

total KE_trans
(1/2)mv_R²
(J)

total KE_rot
(1/5)mv_H²
(J)

total KE
(J)

Step 11. Explain why you think that you were asked to use v_R when calculating the ball's final translational KE at impact but were only asked to use v_H when calculating the ball's total rotational KE at impact?

Step 12. How much total mechanical energy was lost during the experiment?

Step 13. What percentage of the ball's total PE was transformed into rotational kinetic energy?

Part III: Graphical Analysis of the ball bearing's translational velocity

During this lab, the marble changed both its horizontal and vertical motion as it moved along the path from the top of the ramp to the point just where it struck the ground. The path can be broken into three parts: rolling down the angled portion of the ramp, rolling along the flat section of the ramp, and leaving the table as a projectile in two-dimensions.

Given below are nine "general" curves for you to form the best three combinations for the velocity graph requested. You may use a "general" curve more than once if neessary.

Step 14. Complete the following graph of horizontal velocity vs time by choosing the best "general" curve for each section.

A to B

B to C

C to D

Step 15. Complete the following graph of vertical velocity vs time by choosing they best "general" curve for each section.

A to B

B to C

C to D

After submitting your results, each group is to turn in your "target paper" and all of your calculations.