Question

A disc of mass m and radius ${}^{\prime }{R}^{\prime }$ is attached to a rectangular plate of the same mass, breadth $R$ and length $2R$ as shown in the figure. Find the moment of inertia of this system about the axis $X{X}^{\prime }$ passing through the centre of the disc and along the plane.

Answer 1

Given,

$Mass=m,radius=R,bredth=R,length=2R$

So, MI of the disc about the axis XX' is

$I=\left(\frac{m{R}^2}{A}\right)$

MI of the plate about AB

$I_{AB}=\frac{m{R}^2}{12}$

From parallel axis theorem,

$IX{X}^{\prime }=I_{AB}+m\times {(R+\frac{R}{2})}^2=\frac{m{R}^2}{2}+m\times {\left(\frac{BR}{2}\right)}^2=\frac{m{R}^2}{2}+\frac{9}{4}m{R}^2=\frac{11}{4}m{R}^2$

Thus, $I_{total}=\frac{m{R}^2}{2}+\frac{11mR62}{4}=3m{R}^2$