# Moment Of Inertia Of A Solid Cylinder

Moment of inertia of a solid cylinder about its centre can be found using the following equation or formula;

 I = $\frac{1}{2}$MR2

Here, M = total mass and R = radius of the cylinder.

## Derivation Of Moment Of Inertia Of Solid Cylinder

We will take a solid cylinder with mass M, radius R and length L. We will calculate its moment of inertia about the central axis.

Here we have to consider a few things:

• The solid cylinder has to be cut or split into infinitesimally thin rings.
• Each ring consists of the thickness of dr with length L.
• We have to sum up the moments of infinitesimally these thin cylindrical shells.

We will follow the given steps.

1. We will use the general equation of moment of inertia:

dI = r2 dm

Now we move on to finding the dm. It is normally given as;

dm = ρ dV

In this case, the mass element can be expressed in terms of an infinitesimal radial thickness dr by;

dm = 2r L dr

In order to obtain dm we have to calculate dv first. It is given as;

dV = dA L

Meanwhile, dA is the area of the big ring (radius: r + dr) minus the smaller ring (radius: r). Hence;

dA = $\pi$(r + dr)2 – $\pi$r2

dA = $\pi$(r2 + 2rdr + (dr)2) – $\pi$r2

Notably, here the (dr)2 = 0.

dA = 2$\pi$r dr

2. Substitution of dA into dV we get;

dV = 2$\pi$r L dr

Now, we substitute dV into dm and we will have;

dm = 2$\pi$r L dr

The dm expression is further substituted into the dI equation and we get;

dl = 2$\pi$r3 L dr

3. Alternatively, we have to find the expression for density as well. We use the equation;

p = $\frac{M}{V}$

Now,

p = $\frac{M}{\pi R^{2}L}$

4. The final step involves using integration to find the moment of inertia of the solid cylinder. The integration basically takes the form of a polynomial integral form.

I = 2p$\pi$r L $\int_{R1}^{R2}$r3 dr

I = 2p$\pi$L $\frac{R^{4}}{4}$

I = 2$\pi$[ $\frac{M}{\pi R^{2}L}$ ]L$\frac{R^{4}}{4}$

Therefore, I = $\frac{1}{2}$MR2