PHYS 1111L - Introductory Physics Laboratory I

Laboratory Advanced Sheet
Rolling Motion

1. Objective. To experimentally verify the value of the moment of inertia of a uniform, solid sphere and a uniform, solid cylinder about their centers of mass using conservation of mechanical energy applied to rolling motion.

2. Theory.

a. Conservation of mechanical energy applied to an object of circular cross section released from rest on an inclined planed and rolling without slipping results in the following relationship:

where

v_x is the speed of the object along the inclined plane's surface,
M is the mass of the object,
g is the acceleration due to gravity,
h₀ is the height of the center of mass of the object above some reference height when released from rest,
h₁ is the height of the center of mass of the object above the same reference height after having traveled a distance, d, down the plane.
I is the moment of inertia of the object, and
R is the radius of the object.

b. The acceleration of the object can be found by applying the kinematics equation

where, in this experiment,

c. The moments of inertia of a sphere and a cylinder are known from calculus to be given by

I_sphere = (2/5) M R²

I_cylinder = (1/2) M R²

Instead of employing these expressions for the moments of inertia, the general expression

I = k M R²

will used. In this equation k is a constant which will be determined by measurements and then compared to the actual value of 2/5  or 1/2 (the actual comparison will be to k + 1 = 1.400 or 1.500) as a test of conservation of mechanical energy. Solving for the quantity k after making substitutions for v_x and I yields

d. The acceleration can be determined by measuring the distance the object travels along the incline (d) and the time it takes to do so. Applying the kinematic relationship

yields the final form for k + 1:

e. In this experiment, the time, t, required for the object to roll a distance, d, along the inclined plane while falling a height, h₀ - h₁, will be measured, as will d, h₀, and h₁. These measured quantities will be used to determine a value for k + 1, which will then be compared to the actual value for k+1, which is 1.400 (exactly) for the sphere and 1.500 (exactly) for the cylinder..

3. Apparatus and experimental procedures.

a. Equipment.

1) Board.

2) Concrete blocks.

3) Photogates with stands and rods.

4) Metal sphere and cylinder.

5) Meter stick and ruler.

6) Computer interface and computer.

b. Experimental setup. The experimental setup is shown in Figure 1 (to be provided by the student).

c. Capabilities. To be provided by the student.

4. Requirements.

a. In the laboratory.

1) Using the assembled apparatus, conduct several trial runs to insure the sphere and cylinder follow a path parallel to the center line of the plane from the point of release to the bottom of the plane. Some shimming of the inclined plane may be required.

2) Measure the heights of the bottom of the sphere and cylinder at the top and bottom of the inclined plane.

3) Measure the distance traveled by the center of mass along the inclined plane as it falls between the two heights.

4) Perform five (5) trials each to measure the time required for the sphere and cylinder to travel down the plane.

b. After the laboratory. The items listed below will be turned in at the beginning of the next laboratory period. A complete laboratory report is not required for this laboratory.

Para 2. Theory. Beginning with the law of conservation of mechanical energy, provide a complete derivation of the expression for k + 1 in terms of the measured quantities, using the symbols defined in paragraph 2 of this advanced sheet.

Para 3. Apparatus and experimental procedures.

1) Provide a figure of the experimental setup.

2) Provide a description of the capabilities of the equipment used in the experiment.

Para 4. Data.

1) Provide a copy of your data (Annex A provides tables for data).

2) Provide the following calculations in a spreadsheet.

a) The measured value of k + 1.

b) The percent discrepancy in the measured value of k + 1.

Para 5. Results and conclusions.

a. Results.

1) Provide a statement of the value of k+1.

2) Provide a statement of the percent discrepancy in the measured value of k+1.

b. Conclusions.

1) Comment on the validity of the application of the law of conservation of mechanical energy to rolling motion.

2) List sources of systematic error in the experiment.

3) List sources of random error in the experiment.

Annex A
Data

1. Sphere.

    a.    Direct measurements of heights and distance.

    b.    Measured Times.

2.    Cylinder.

    a.    Direct measurements of heights and distance.

    b.    Measured Times.

Last update: November 08, 2002