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Question

# If the radius of the earth contracts to half of its present value without a change in its mass, then new duration of the day will be;

A
12 h
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B
6 h
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C
4 h
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D
3 h
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Solution

## The correct option is B 6 hIf R1 is the initial radius and M is the mass of the earth revolving with angular speed ω1 about its own axis, then the present angular momentum of the earth with respect to its center is, L1=Iω1=25MR21ω1 Let, the angular speed of the earth be ω2 when its radius becomes half of its initial radius. New angular momentum of the earth, L2=Iω2=25MR22ω2 ⇒L2=25M(R12)2 ⇒L2=110MR21ω2 As the external torque is zero we can say that, L1=L2 25MR21ω1=110MR21ω2 ∴ω2=4ω1 As, T1=2πω1 and T2=2πω2 T2T1=ω1ω2=14 ⇒T2=T14 As, the actual duration of the day, T1=24 h so, T2=244=6 h Hence, option (b) is the correct answer.

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