## Chapter 20

## Q. 20.8

Use the parallel axes theorem to find the moments of inertia of

i) a thin uniform rod of mass *M* and length *h* about a perpendicular axis through its end

ii) a thin uniform solid disc of mass* M* and radius *r* about

a) an axis perpendicular to its plane through a point on its circumference

b) a tangent.

iii) Which of the above is equally applicable to a solid cylinder?

## Step-by-Step

## Verified Solution

i) For the rod I_{G} = \frac{1}{3}M\left(\frac{h}{2}\right)^{2} and the axes are a distance \frac{h}{2} apart. Hence

I_{A} = I_{G} + M\left(\frac{h}{2}\right)^{2}

= \frac{1}{3}M\left(\frac{h}{2}\right)^{2} + M\left(\frac{h}{2}\right)^{2}

= \frac{1}{3}Mh^{2}.

ii) a) In this case I_{G} = \frac{1}{2}Mr^{2} and the axes

are a distance *r* apart.

Hence I_{A} = I_{G} + Mr^{2}

= \frac{1}{2}Mr^{2} + Mr

= \frac{3}{2}Mr^{2}.

b) Now I_{G} = \frac{1}{4}Mr^{2}

⇒ I_{A} = \frac{1}{2}Mr^{2} + Mr

= \frac{5}{4}Mr^{2}.

iii) The result is applicable to a cylinder so long as this is formed by elongating the body in a direction parallel to the axis of rotation. It therefore applies in part ii) a) but not in the other cases.