Rotational Motion Question 206
Question: The moment of inertia of a hollow thick spherical shell of mass M and its inner radius $R _{1}$ and outer radius $R _{2}$ about its diameter is
Options:
A) $\frac{2M}{5}\frac{(R _{2}^{5}-R _{1}^{5})}{(R _{2}^{5}-R _{1}^{3})}$
B) $\frac{2M}{5}\frac{(R _{2}^{5}-R _{1}^{5})}{(R _{2}^{3}-R _{1}^{3})}$
C) $\frac{4M}{5}\frac{(R _{2}^{5}-R _{1}^{5})}{(R _{2}^{3}-R _{1}^{3})}$
D) $\frac{4M}{3}\frac{(R _{2}^{5}-R _{1}^{5})}{(R _{2}^{3}-R _{1}^{3})}$
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Answer:
Correct Answer: A
Solution:
[a]
$\rho =\frac{M}{\frac{4}{3}\pi (R _{2}^{3}-R _{1}^{3})}$ $I _{shell}=\frac{2}{5}M _{2}R _{2}^{2}-\frac{2}{5}M _{1}R _{1}^{2}$ $ –(1)
$M _{2}=\rho \times \frac{4}{3}\pi R _{2}^{3}$ $=\frac{MR _{2}^{3}}{(R _{2}^{3}-R _{1}^{3})};M _{1}=\frac{MR _{1}^{3}}{R _{2}^{3}-R _{1}^{3}}$
Putting values of 1 $M_{1}$ and $M_{2}$ in eq.(1),
$I _{shell}=\frac{2M}{5}\frac{(R _{2}^{5}-R _{1}^{5})}{(R _{2}^{3}-R _{1}^{3})}$
BETA
