From a uniform sphere of mass M and radius R, a cavity of diameter R is created as shown

Question

From a uniform sphere of mass M and radius R, a cavity of diameter R is created as shown. Find the ratio of moment of inertia of the sphere left about AA′ and BB′.

Smart Achievers · Accepted Answer

(4)

$I_{A'} = \frac{2}{5} M R^{2} - \frac{2}{5} \frac{M}{8} \times \frac{R^{2}}{4}$

$I_{A'} = \frac{2}{5} M R^{2} \times \frac{31}{32}$

$I_{B'} = \frac{2}{5} M R^{2} - \frac{7}{5} \frac{M}{8} \frac{R^{2}}{4}$

$I_{B'} = \frac{2}{5} M R^{2} [1 - \frac{7}{64}]$

$I_{B'} = \frac{2}{5} M R^{2} \times \frac{57}{64}$

$\frac{I_{A'}}{I_{B'}} = \frac{31}{32} \times \frac{64}{57} = \frac{62}{57}$

Solution

Q.
From a uniform sphere of mass M and radius R, a cavity of diameter R is created as shown. Find the ratio of moment of inertia of the sphere left about AA′ and BB′.

1 $\frac{15}{28}$

2 $\frac{28}{15}$

3 $\frac{31}{30}$

4 $\frac{62}{57}$

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Solution

Q. From a uniform sphere of mass M and radius R, a cavity of diameter R is created as shown. Find the ratio of moment of inertia of the sphere left about AA′ and BB′.

1 1528

2 2815

3 3130

4 6257