PURPOSE

You will use the conservation of energy principle applied to both translational and rotational energy of motion to experimentally determine the moment of inertia for a solid and to compare the result with that calculated from the masses and dimensions.

EQUIPMENT  rotation platform, disc or ring, pulley, smart pulley, mass, 5-gram mass holder, meter stick, and vernier caliper.

RELEVANT EQUATIONS

Radian measure
Angular velocity
Angular acceleration
Torque and Angular Acceleration
Moment of Inertia

DISCUSSION

♦♦♦DO NOT MOVE THE ROTATION PLATFORM TO ANOTHER POSITION ON THE TABLE SINCE IT HAS BEEN LEVELED IN ITS PRESENT POSITION!♦♦♦

The radian is a unit of angular measure. It is a "natural" way of measuring angles because it has no physical units, and therefore does not depend on some arbitrary definition of what a "degree" is. The angle in radians is defined as the arc length on a circle that subtends the angle, divided by the radius of the circle:

(1)

This definition is illustrated in Figure 13-1.

Consider a point on the edge of a circular wheel, and suppose that it covers a distance Δs in a time Δt. The average linear speed of the point is the rate at which distance along the circular arc is covered:

(2)

The quantity ("omega bar") is the average angular velocity.

If the wheel is revving up or revving down, then the angular velocity ω, changes with time. The rate of change of the angular velocity is the angular acceleration α. The relation between the linear acceleration of a point at the periphery of the wheel and the angular acceleration of the wheel as a whole is:

(3)

where the angular acceleration is defined as the rate of change of the angular velocity:

(4)

Consider the system illustrated in Figure 13-2. The suspended mass m applies a force to the outer edge of the hub via the pulley and string system. The force applied through the string applies a torque to the hub, which is attached to the rotating platform.

According to Newton's Second law for rotational motion, the net torque is:

(5)

where I is the moment of inertia of the rotating system (platform + hub + anything placed on the platform), and α is the angular acceleration of the rotating system expressed in rad/s².

From equation (3), the angular acceleration of the rotating system is related to the linear acceleration of the suspended mass: α = a/r, where r is the hub radius. The linear acceleration can be measured directly with the "smart" pulley, which is interfaced to a computer. The net torque applied is equal to the force exerted by the cord m g, times the lever arm r minus any frictional torque that is produced by the bearings.

Combining these relations, we can solve for the moment of inertia:

(6)

In this experiment, you will measure the acceleration for different values of the suspended mass after correcting for the frictional torque. Using equation (6), values for the moment of inertia will be obtained for the platform alone and for the platform + disk system. From these data, a value for the moment of inertia for the disk can be obtained and compared with the theoretical value.