# Inquiry: **Moment of Inertia**

## Purpose: In this Inquiry you will prove the moment of inertia formulas for a thin ring (I = mr^{2}) and a disk (I = ½mr^{2})

## Procedure:

In this Inquiry we will be dropping a mass, which will turn a wheel. Objects such as thin rings and disks may be placed upon this wheel to find their moments of inertia. We will start by deriving a formula, which we will be able to use for this Inquiry. To do this we will start with Newton's second law.

F = ma

for rotary motion this is:

T = Iα

Fr = Iα

Fr = I(ωf - ωi)/Δt

where r is the radius of the shaft that the string is wound on (r = 0.01m) and F is the force applied on the string to set the body in rotation.

When a falling object is used to accelerate a rotating wheel not all of its weight is applied to the rotation of the wheel, some of the force is used to accelerate itself in the relation F=ma, therefore the force that is applied to accelerate a rotating wheel is given by F = 9.81m - ma where a is the linear acceleration and m is the mass of the falling object (0.5 kg)

Knowing this information and starting from the equation above try to derive the following equation into your notebook, which we will be using for this Inquiry.

- When you drop the mass and it falls watch the computer and record the highest angular velocity reached.

- Use the pulse mode on the photo gates to measure the time of fall of the mass.

- In order to find the moment of inertia of a ring you must first find the moment of inertia of the wheel and ring and then subtract the moment of inertia of the wheel.

- The thin ring must be used to hold the disk on the wheel.

- Use subtraction to calculate the rotational inertias for the disk and the ring.

- Use a vernier caliper and scale to get the measured values for the disk and the ring. Use the measured value as the theoretical value to calculate % error.

## Inquiry Questions:

- If a sphere has a mass of 5.0 kg and has a radius of 0.060 m, find the moment of inertia?

- A force of 5.0 N is applied tangentially to the rim of a disk, having a 0.20 m radius, if the mass of the disk is 25 kg, what is the angular acceleration?