**UNIT 16 ****READING****
A**

**Moment of inertia and rotational kinetic energy**

Consider an object of mass m moving in a circle about an axis as in the picture below.

_{}

This can be written in terms of the angular speed of the
object, _{}, and the distance of the object from
the axis of rotation, r

_{}

The quantity mr^{2} is defined to be the moment of
inertia of the mass m. The kinetic energy can then be rewritten in terms on the
moment of inertia

_{}

For a large object, we can consider it made up of many smaller pieces of mass.

Each piece of mass, m_{i}, is at a different
distance, r_{i}, from the axis of rotation. The kinetic energy of the
object is found by summing up the kinetic energies of each the small pieces of
mass:

_{}

This can be written as

_{}

where _{} is the moment of
inertia of the object. This is the rotational kinetic energy of an object.

A rolling object has both rotational and translational kinetic energy, the total kinetic energy for a rolling object is

_{}

where I is the moment of inertia of the object, _{} is the angular speed
of the object, M is the mass of the object and _{} is the velocity of the
center of mass of the object.

_{ }