Question

A thin rod of length 4l, mass 4m is bent at the points as shown in the figure. What is the moment of inertia of the rod about the axis passing through O and perpendicular to the plane of the paper?



(a) ml23 (b) 10 ml23 (c) ml212 (d) ml224

Answer 1

Since, the mass of the rod is 4 m and the length is 4 l

so, mass of AB = BO = OC = CD = m

and, length of AB = BO = OC = CD = l

We know, Moment of inertia of a rod about its end = (ml^2)/3

Moment of inertia of AB bout B (I 1) = (ml^2)/3

Moment of inertia of BO about O (I 2) = (ml^2)/3

Moment of inertia of OC about O (I 3)= (ml^2)/3

Moment of inertia of CD about C (I 4)= (ml^2)/3

Now, from parallel axis theorem,

Moment of inertia of AB about O (I 5)

= I 1 + ml^2 ————(parallel axis theorem)

= (ml^2)/3 + ml^2

= (4 ml^2)/3

Similarly, Moment of inertia of CD about O (I 6)

= (4 ml^2)/3

So, moment of inertia of the rod about O

= I 2 + I 3 + I 5 + I 6

= ( ml^2)/3 + ( ml^2)/3 + (4 ml^2)/3 + (4 ml^2)/3

= (10 ml^2)/3