Every object that has mass in the universe has inertia loads. Anything that has mass has inertia. Inertia is the resistance to a change in velocity and relates to Newton's first law of motion.

## Understanding Inertia With Newton's Law of Motion

**Newton's first law of motion** states that an object at rest stays at rest unless acted upon by an unbalanced external force. An object undergoing constant velocity motion will remain in motion unless acted upon by an unbalanced external force (such as friction).

Newton's first law is also referred to as the **law of inertia**. Inertia is the resistance to a change in velocity, which means the more inertia an object has, the more difficult it is to cause a significant change in its motion.

## Inertia Formula

Different objects have different moments of inertia. Inertia is dependent on mass and the radius or length of the object and the axis of rotation. The following indicates some of the equations for different objects when calculating load inertia, for simplicity, the axis of rotation will be about the center of the object or central axis.

Hoop about the central axis:

*I = MR ^{2 }*

Where *I* is the moment of inertia, *M* is mass, and *R* is the radius of the object.

Annular cylinder (or ring) about the central axis:

*I = 1/2M(R _{1}^{2}+R_{2}^{2})*

Where *I* is the moment of inertia, *M* is mass, *R _{1}* is the radius to the left of the ring, and _R

_{2 }_is the radius to the right of the ring.

Solid cylinder (or disk) about the central axis:

*I = 1/2MR ^{2}*

Where *I* is the moment of inertia, *M* is mass, and *R* is the radius of the object.

## Energy and Inertia

Energy is measured in joules (J), and moment of inertia is measured in kg x m^{2} or kilograms multiplied by meters squared. A good way of understanding the relationship between the moment of inertia and energy is through physics problems as follows:

Calculate the moment of inertia of a disk that has a kinetic energy of 24,400 J when rotating 602 rev/min.

The first step in solving this problem is to convert 602 rev/min to SI units. To do this, 602 rev/min has to be converted to rad/s. In one complete rotation of a circle is equal to 2π rad, which is one revolution and 60 seconds in a minute. Remember the units must cancel out to get rad/s.

*602 rev/min x 2_π */60s = 63 rad/s_

The moment of inertia for a disk as seen in the previous section is *I = 1/2MR ^{2}*

Since this object is rotating and moving, the wheel has kinetic energy or the energy of motion. The kinetic energy equation is as follows:

*KE = 1/2Iw ^{2 }*

Where *KE* is kinetic energy, *I* is the moment of inertia, and *w* is the angular velocity which is measured in *rad/s.*

Plug 24,400 J for kinetic energy and 63 rad/s for angular velocity into the kinetic energy equation.

*24,400 = 1/2I(63 rad/s ^{2 })^{2}*

Multiply both sides by 2.

*48,800 J =I (63 rad/s ^{2 })^{2}*

Square the angular velocity on the right side of the equation and divide by both sides.

*48,800 J /3,969 rad ^{2}/s^{4} = I*

*Therefore the moment of inertia is as follows:*

*I = 12.3 kgm ^{2}*

## Inertial Load

The inertial load or *I* can be calculated depending on the type object and the axis of rotation. A majority of objects that have mass and some length or a radius have a moment of inertia. Think of inertia as the resistance to change, but this time, the change is velocity. Pulleys that have a high mass and very large radius will have a very high moment of inertia. It may take a lot of energy to get the pulley going, but after it starts moving, it will be tough to stop the inertial load.