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Question

A metallic sphere of radius r and specific heat S is rotated about an axis passing through its centre of mass at an angular speed of n rotations per second. It is suddenly stopped and 50% of its energy is used in increasing its temperature. Then, the rise in temperature of the sphere is

A
2π2n2r25S
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B
π2n210r2S
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C
78πr2n2S
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D
5(πrn)214S
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Solution

The correct option is A 2π2n2r25S
Moment of Inertia of the solid metallic sphere, I=25Mr2
Given , ω=n rotations per second or 2πn rad/sec
Rotational kinetic energy, (K)=12Iω2=12(25Mr2)(2πn)2
or (K)=45Mπ2r2n2
Half of this kinetic Energy is converted into heat.
dQ=K2=25Mπ2r2n2
But we know that,
dQ=MSdT
dT=dQMS=25Mπ2r2n2MS
dT=2π2r2n25S
Thus, option (a) is the correct answer.

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