# Inertia - Moment Of Inertia And Its Application

Newton’s First law of motion states that a body at rest or uniform motion will remain to be at rest or uniform motion until an external force acts on it. This is also known as Law of inertia. It introduced us to a new term. Now the question arises what is inertia? Is moment of inertia and inertia same? Suppose if we try to push two different bodies one having larger mass (like an iron box) as compared to other (Wooden block). We find that larger force is required to change the state of the body with larger mass. In other words when force is applied then the body tries to resists change in its state of uniform motion or rest. So the definition of inertia can be given as property of the body to resist change. The next question is how do we measure inertia? Which body is said to have higher inertia? As we can observe one with larger mass requires larger force hence we can say it has greater inertia. So in other words mass can be taken as measure of inertia. Greater the mass greater will be the inertia.

Using the above definition we can explain why does a person tend to fall backwards when a bus starts to move? Initially both person and the bus are at rest. When the bus suddenly starts moving, due to inertia the person resists change in its state of rest and hence tends to fall backwards. We can explain similarly why a person tends to fall forward when a bus stops suddenly. We can have several other examples from our daily life where we can see the effect of inertia.

This was just an introduction to inertia. To know more about Newton’s first law of motion and its use in practical application join us here at Byjus.com and learn in a simplified way.

What is inertia ?

The first thing that we will answer here is what is inertia? In a simple term it can be said that inertia in rotational motion is analogous to mass in linear motion. To explain it in a better let’s consider the example of a rotating disc. The kinetic energy of the disc can be represented as

$K.E$ = $\frac{1}{2}\sum m_i {v_i}^2$

But as we know in this case each mass does not have same linear velocity hence it is easier to realize using angular velocity which is equal for all the masses. We also know,

$v$ = $ωr$

Substituting the value we get,

$K.E$ = $\frac{1}{2} ∑{m_i} {r_i}^2 ω^2$

$K.E$ = $\frac{1}{2}(∑{m_i} {r_i}^2)ω^2$

From the above equation we get, $∑{m_i}{r_i}^2$ is the moment of inertia of the system.

Unit of moment of inertia is $kg-m^2$ .

Moment of Inertia

There are two theorems which are used to calculate moment of inertia about any arbitrary axis. They are parallel axis and perpendicular axis theorem. According to parallel axis if the moment of inertia of a body about a particular axis is I. Then the moment of inertia of the body about an axis having perpendicular distance of d from the given axis is:

$I_d$ = $I + Md^2$

Now according to perpendicular axes theorem if the moment of inertia is I about z-axis. If we select two perpendicular axes such that these two are mutually perpendicular to the original axis i.e. x-axis and y-axis then we can state that,
$I_z$ = $I_x + I_y$

$I_z$ = $I_x + I_y$
Generally we choose axes such that $I_x$ = $I_y$ so that calculations become easy.

Also we can represent all moment of inertia as follows,

$I$ = $MK^2$

Here, K is known as radius of gyration about the considered axis. Values of moment of inertia for some of the common bodies are:

Solid Cylinder about central axis = $\frac{1}{2} MR^2$

Solid Sphere about a diameter = $\frac{2}{5} MR^2$

Moment of inertia of ring about perpendicular axis = $MR^2$

Moment of inertia of disc about perpendicular axis = $\frac{1}{2}MR^2$

So now that we have learnt about inertia. To learn more on rotational motion stay connected to Byjus.com

#### Practise This Question

Two particles having masses m1 and m2 are situated in a plane perpendicular to line AB at a distance of r1and r2 respectively as shown.

Find the moment of intertia about axis passing through centre of mass and perpendicular to line joining m1 and m2