Introduction

In the case of linear motion, a net non-zero force will result in linear acceleration, given by Newton’s Law F=ma. For rotational motion, an unbalanced, non-zero torque, t, will result in angular acceleration, a, according to the equation t =Ia , where I is the rotational inertia of the object。The rotational inertia (or moment of inertia) depends not only on the mass but also on how the mass is distributed about the axis of rotation.

Theory

Variously shaped objects are placed on the rotational apparatus. A system of air-curshion pendulum (figure 1) is used so that a mass, m, can be attached to a string (figure 2) which is wound around the small radius of the rotational stand. Rotating the pendulum a small angle produces tension in the string which ultimately applies a torque to the rotational apparatus and whatever is placed on top of it. This restoring torque makes the pendulum oscillate periodically. We can measure rotational inertia by looking at the period of the pendulum.

The equation is

                                (1)

From the above equation we can determine the rotational inertia of the pendulum platform ( J₀) by measuring the oscillation period. The modulus of rigidity D, which is characteristic for a spring, will be given by the instructor in the lab.

If one object is placed on the pendulum platform and the oscillation period is T, the rotational inertia of the object ( J ) around the symmetry axis can be derived by

                              (2)

Eq. 2 is the basic formula in this lab.

Theoretical moment of inertia of a ring rotating around its symmetry axis is given by

                             (3)

Where m₁ is the mass of ring, d_in and d_out are the inner and outer diameter respectively, which can be measured by the vernier callipers. Compare the measured and calculated values and determine the relative deviation (percentage error).

Equipment

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Air-cushion pendulum, compressed gas source, motion detector, vernier callipers

The air-cushion pendulum system includes a pendulum platform, wound spring (torsion spring). The pendulum and wound spring provide a harmonic oscillation system. The vents on the platform wall will let air flow inside the platform and float the pendulum so that the friction is minimized. The air flow pressure can be controlled by a valve, which has already been adjusted. Please do not adjust the flow pressure during the experiment.

Procedure

Note: All of the electrical connections have been made by the instructor before you start.

Turn the “measurement selection” knob to “harmonic”, turn “timer” knob to 10ms, turn on the power switch, warm up and reset to zero.

Power on the gas compressor, the platform will be floated by the air flow from the vents.

1, measure the moment of inertia of regular objects (steel ring) through its mass center.

Slowly rotate the platform to a small angle (about 10^o) and release it. Record the total time t₀ the platform (without anything on it) oscillates 20 times, ie., 20T₀. Repeat the measurement 3 times. Calculate the period T₀ of platform oscillation. Using the provided D, determine J₀ with Eq. 1. Compare the measured J₀ with the known one provided in the lab. If they deviate too much, try to find the cause and redo the measurement.

Add a ring on the center of the platform so that they are co-axial. Rotate the platform to a small angle and release. Record the total time t₁ the platform and the ring oscillate 20 times, ie., 20T₁. Repeat the measurement 3 times. Determine the moment of inertia of the ring with Eq. 2. Calculate the percentage error between them ().

2, measure the moment of inertia for irregular objects ( plane ) through the symmetry axis of platform.

Place the plane model on the platform so that the mass center axis is on the symmetry axis of the platform. The mass center axis of the plane model can be estimated by the center of its bottom supporter. With the same methods of above procedures, determine the moment of inertia J₂ of the plane model.

Record all measured data in the tables like below and calculate it on your paper.