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0502
Experiment P26: Rotational Inertia 
(Smart Pulley)

PURPOSE
The purpose of this laboratory activity is to measure the rotational inertia of a ring experimentally and to compare this value to the theoretical value.

THEORY
Theoretically, the rotational inertia, I, of a ring is given by

I = 1/2 M ( R1 2 + R2 2 )    Equation 1

where M is the mass of the ring, R1 is the inner radius of the ring, and R2 is the outer radius of the ring.

To find the rotational inertia of the ring experimentally, a known torque is applied to the ring and the resulting angular acceleration is measured. 

Since t = Ia

I = t / a      Equation 2

where a is the angular acceleration and t is the torque.

Now,

t = r x F           Equation 3

where r is the distance from the center of the ring to the point where a force is applied, and F is the applied force. The value of r x F is r F sin theta where theta is the angle between r and the direction of F, the applied force. The torque is maximum when r and F are perpendicular. 

In this case, the applied force is the tension (T) in a string that is tied to a step pulley that is part of a rotational apparatus. The string is pulled by a hanging mass m. The value of r is the radius of the step pulley on the apparatus. The radius is perpendicular to the applied force (Tension). 

Therefore, the torque is:

t = r T      Equation 4

Applying Newton's Second Law for the hanging mass, m, results in: 

SUM F = m a = mg - T

Solving for the tension in the string gives 

T = m ( g -a )

The torque is:

t = r T = r m ( g - a )      Equation 5

The linear acceleration a of the hanging mass is the tangential acceleration, aT, of the rotating apparatus. 
The angular acceleration is related to the tangential acceleration as follows:

a = at / r     Equation 6

Substituting Equation 5 and Equation 6 into Equation 2 gives:

I = t / a = r m ( g - a ) / ( at / r ) = r m ( g - a ) r / at =
      mgr2 / at - mr2 = mr2 ( g / at - 1 )

The rotational inertia, I, can be calculated from the tangential acceleration, aT.

PROCEDURE
For this activity, the Smart Pulley measures the motion of a hanging mass that is connected by a string to a step pulley on the rotational apparatus. The Science Workshop program calculates and displays velocity versus time. The slope of the best fit line of velocity versus time is the value of acceleration. Please refer to the lab manual for a detailed procedure.

Science - Stretching the Boundaries
(Science Workshop 2.2)
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