LAB Rotation and the Rolling Race

PHY 105 Name____________________

Rotation and the Rolling Race 1

LAB: Rotation and the Rolling Race

Objective In this experiment, you will roll round objects down an incline and record the time it takes them to reach the bottom. Using Newton's second law and its rotational analog along with 1-D kinematics, you will predict the time it will take for each object to roll down the incline.

Materials

✔ Tape measure or meter stick

✔ Stopwatch or app

✔ A long flat board as a ramp (or you can put one end of a table up on blocks)

✔ Blocks or something else to raise one end of the ramp

✔ Round objects that you can roll. Note that these must be smooth and round:  BAD: baseballs, golf balls, or balls of string  GOOD: solid high-bounce ball, hollow handball, a full can of beans, an empty

can with the lid and bottom cut off, a pill bottle filled with water, an empty pill bottle, a toy wheel, roll of tape without dispenser

Theory In this experiment, you will roll round objects down a straight ramp. Newton’s second law and its rotational analog can be used to predict their accelerations. A free-body diagram for the round object consists of a weight force down, a normal force perpendicular to the ramp, and a static friction force up the ramp. Newton’s second law in a rotated coordinate system can be applied as follows:

ΣFx = max

W*sin(θ)+ -fs = max

[1] mg*sin(θ) + -fs = max The above equation cannot be solved for acceleration (and don’t even think about using fs =

μsN as we do not know if that is true here). Use the rotational analog to Newton’s second law where the only force that exerts a non-zero torque is the friction (normal force is

opposite the r vector so sinϕ = 0 in the torque definition and weight force is exerted at the axis of rotation so r = 0).

Στz = Iαz

rFsinϕ = Iαz

[2] Rfs = Iαz R = fixed radius of round object

Round objects have the following formula for moments of inertia about their centers:

[3] I = NmR2

Rotation and the Rolling Race 2

N = dimensionless constant describing the distribution of matter N = 2/5 for a solid sphere N = 2/3 for a hollow sphere N = ½ for a solid disc or cylinder N = ½[1 + (Rin/Rout)

2] for a ring Note that there is not a standard symbol for the constant in the moment of inertia formula; here we are using the symbol “N”, but you may see other symbols elsewhere, or no symbol at all. . The relationship between linear acceleration and angular acceleration for a rolling object is the following:

ax = Rαz

[4] αz = ax/R

Equations [1] through [4] can be solved for the acceleration in terms of sin(θ), g, and N. This is left as an exercise for the student. Once you have a formula for the acceleration, you can use 1-d kinematics to predict the time it takes to roll down a ramp. The 1-d kinematics solution is left as an exercise. Exercise is fun! Physics is fun!

Procedure Part I: Properties of Round Objects

1) If you happen to have a postal scale, weigh and record the masses of the objects. If not, then weigh them in your hands and roughly rank them from most to least massive.

2) Measure the diameters and calculate the radii of the above objects. Determine both the inner and outer radii of any rings you have chosen to test.

3) Record the shapes of the objects (solid sphere, ring, etc.). 4) For each object, calculate the constants in their moments of inertia, N. Record the

constants for the other objects. Do not calculate moments of inertia. Part II: Ordinal Tests (Rolling Race) Note: DO NOT skip this part of the experiment in your report. DO NOT use a stopwatch in this part of the experiment.

5) Set up your inclined plane at an at an angle of about 5° (or prop one leg of your table up to achieve the same angle). Provide evidence in your report that you used such an angle in the form of an inverse trigonometry function.

6) Using the round objects, repeatedly test (and record the results) which object of a pair wins a simultaneous race down a ramp You can get creative here and have some sort of tournament or a regular season and playoffs. Include results of all races in the raw data.

7) Provide a ranking of your objects. Which variable or variables appear to determine the ranking and how? Is it mass, radius, shape, coaching, effort, etc.?

Rotation and the Rolling Race 3

Part III: Derivations of Symbolic Solution 8) Use equations [1] to [4] to derive a symbolic solution (a.k.a., equation or formula)

for the theoretical acceleration in terms of sin(θ), g, and N. You should only have one division symbol. Hint: it should look similar to the formula for the acceleration on a frictionless inclined plane. Do not calculate acceleration.

9) Use trigonometry to replace sin(θ) with measurable distances: height, y, and length of the ramp, d, in your symbolic solution for acceleration. You should only have one division symbol. Do not calculate acceleration.

10) Use 1D constant-acceleration kinematics to derive a symbolic solution for the time in terms of y, N, g, and d. You must continue to use a rotated coordinate system as in the derivation of acceleration. You should only have one division symbol and one appearance of each variable.

11) Check your symbolic solution with your instructor by sending a message. Show your work if you want anything more than a yes/no response.

Part IV: Cardinal Tests (Individual Time Trials)

12) Measure the length of the ramp, d. 13) Measure or calculate the height difference between the beginning and end of the

ramp, y. It cannot be emphasized enough that this is by far the most important measurement you will make. Do not let the ramp move during your experiment.

14) Use your theoretical equation to predict the time it will take for five different objects to roll down the ramp.

15) Carefully measure the time it takes for the objects to roll down the ramp individually. 16) Compare theoretical and experimental values for time. Report the % error and

discuss. Notes:

● Do not skip the ordinal tests or fail to provide all of the raw data from these tests. ● The above lab instructions do not at any point state that you should make a graph.

So don’t make a graph. ● The above lab instructions do not at any point state that you should calculate

acceleration. So don’t calculate acceleration. ● The above lab instructions do not at any point state that you should calculate

moments of inertia. So don’t calculate moments of inertia. ● As usual, you must identify a specific physical principle you are testing (purpose) and

then confirm or deny this (conclusion) based on evidence.