First slide

Moment of inertia

Question

Four spheres, each of mass M and radius $r$ are situated at the four corners of square of side R. The moment of inertia of the system about an axis perpendicular to the plane of square and passing through its center will be

Moderate

A

$\frac{5}{2} M (4 r^{2} + 5 R^{2})$
B

$\frac{2}{5} M (4 r^{2} + 5 R^{2})$
C

$\frac{2}{5} M (4 r^{2} + 5 r^{2})$
D

$\frac{5}{2} M (4 r^{2} + 5 r^{2})$

Solution

MOI of sphere A about its diameter $I_{O'} = \frac{2}{5} M r^{2}$
Now MOI of sphere A about an axis perpendicular to the plane of square and passing through its center will be
$I_{O} = I_{O'} + M {(\frac{R}{\sqrt{2}})}^{2}$
$= \frac{2}{5} M r^{2} + \frac{M R^{2}}{2}$ [by the theorem of parallel axis]
Moment of inertia of system (i.e. four sphere)
$I_{s y s} = 4 I_{0} = 4 [\frac{2}{5} M r^{2} + \frac{M R^{2}}{2}] = \frac{2}{5} M [4 r^{2} + 5 R^{2}]$

Get Instant Solutions

When in doubt download our app. Now available Google Play Store- Doubts App

Recieve an sms with download link

Doubts App

OR

SCAN ME

Doubts App

Doubts App