A hollow sphere of mass ' M ' and radius ' R ' is rotating with a… - Vidyadip

MCQ

A hollow sphere of mass ' $M$ ' and radius ' $R$ ' is rotating with angular frequency ' $\omega$ '. It suddenly stops rotating and $75 \%$ of kinetic energy is converted to heat. If ' $S$ ' is the specific heat of the material in $J / kg K$ then rise in temperature of the sphere is (M.I. of hollow sphere $=\frac{2}{3} M R^2$ )

A
$\frac{R \omega}{4 S}$
✓
$\frac{R^2 \omega^2}{4 S}$
C
$\frac{R \omega}{2 S}$
D
$\frac{R^2 \omega^2}{2 S}$

✓

Answer

Correct option: B.

$\frac{R^2 \omega^2}{4 S}$

(b) : Kinetic energy of hollow sphere,
$
\begin{aligned}
& K=\frac{1}{2} I \omega^2=\frac{1}{2}\left(\frac{2}{3} M R^2\right) \omega^2 \quad\left[\because \quad I=\frac{2}{3} M R^2\right] \\
& K=\frac{1}{3} M R^2 \omega^2
...(i)\end{aligned}
$
According to question,
Heat in the hollow sphere $=75 \%$ of $K$
$
\begin{aligned}
& M S \Delta T=\frac{75}{100} K \\
& M S \Delta T=\frac{3}{4} \times \frac{1}{3} M R^2 \omega^2 \\
& \therefore \quad \Delta T=\frac{R^2 \omega^2}{4 S}
\end{aligned}
$

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