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Question

Two metallic spheres S1 and S2 are made of the same material and have got identical surface finish. The mass of S1 is thrice that of S2. Both the spheres are heated to same high temperature and placed in the same room having lower temperature but are thermally insulated from each other. The ratio of the initial rate of cooling of S1 to that S2 is

A
13
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B
13
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C
31
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D
(13)1/3
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Solution

The correct option is D (13)1/3
The rate at which energy leaves the object is
ΔQΔt=eσAT4
Since, ΔQ=mCΔT, we get
ΔTΔt=eσAT4mC
Also, since m=43πr3ρ for a sphere we get
A=4πr2=4π(3m4πρ)2/3
Hence, ΔTΔt=eσT4mC[4π(3m4πρ)2/3] =K(1m)1/3
For the given two bodies
(ΔT/Δt)1(ΔT/Δt)2=(m2m1)1/3=(13)1/3

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