First slide

Moment of intertia

Question

Four spheres each having mass m and radius r are placed with their centres on the four corners of a square of side a. Then the moment of inertia of the system about an axis along one of the sides of the square is :

Moderate

A

$\frac{8}{5} {mr}^{2}$
B

$\frac{8}{5} {mr}^{2} + {ma}^{2}$
C

$\frac{8}{5} {mr}^{2} + 2 {ma}^{2}$
D

$\frac{4}{5} {mr}^{2} + 4 {ma}^{2}$

Solution

$I_{AB} = I_{1 AB} + I_{2 AB} + I_{SAB} + I_{4 AB}$
$= \frac{2}{5} {mr}^{2} + \frac{2}{5} {mr}^{2} + (\frac{2}{5} {mr}^{2} + {ma}^{2})$
$+ (\frac{2}{5} {mr}^{2} + {ma}^{2})$
$= \frac{8}{5} {mr}^{2} + 2 {ma}^{2}$

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