Question

Calculate the moment of inertia of a thin ring of mass m and radius R about an axis passing through its centre and perpendicular to the plane of the ring.

Answer 1

Hint: Inertia is the resistance to change. Whether it can be change of state or change of direction. Usually we call mass as the measurement of inertia because it tells how much resistance is given for linear acceleration when force is applied. Similarly moment of inertia tells us the resistance given for the angular acceleration when torque is applied.

Formula used:
$I = m{r^2}$

Complete answer:
Since the moment of inertia is the resistance for the angular acceleration there should be a rotating body. When a body is rotating there will be an axis of rotation. The axis about which the body is rotating is called the axis of rotation and we determine the moment of inertia about the axis of rotation. Moment of inertia of a mass about the axis of rotation is the product of mass and its perpendicular distance from the axis of rotation.
If we consider below ring we have
Let the mass of the ring be ‘m’ and the length of the ring is $2\pi R$. So the mass per unit length will be $\lambda $
We have $I = m{r^2}$
For a small element of mass ‘dm’ the length will be $Rd\theta $.
So dm in terms of length will be $dm = \lambda Rd\theta $. Hence moment of inertia will be
$\eqalign{
  & I = m{r^2} \cr
  & \Rightarrow dI = dm{R^2} \cr
  & \Rightarrow dI = \left( {\lambda Rd\theta } \right){R^2} \cr
  & \Rightarrow \int\limits_0^I {dI} = \int\limits_0^{2\pi } {\left( {\lambda Rd\theta } \right){R^2}} \cr
  & \Rightarrow I = \int\limits_0^{2\pi } {\left( {\lambda {R^3}d\theta } \right)} \cr
  & \Rightarrow I = 2\pi \lambda {R^3} \cr
  & \Rightarrow I = 2\pi \left( {\dfrac{m}{{2\pi R}}} \right){R^3} \cr
  & \therefore I = m{R^2} \cr} $
So the moment of inertia of the ring will be $I = m{R^2}$ where R is radius and ‘m’ is mass.

Note:
Geometrical center and the center of mass of a system are completely different. Geometrical center of the system is independent of the mass distribution while the center of mass is the point where the entire mass is assumed to be concentrated and this depends upon the mass distribution whether it is uniform or non uniform.