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Question

# A piece of metal floats on mercury. The coefficients of volume expansion of the metal and mercury are γ1 and γ2 respectively. If their temperature are increased by ΔT each, the fraction of the volume of metal submerged in mercury changes by a factor of

A
(γ2γ1)ΔT
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B
γ1ΔT
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C
(1+(γ2γ1)ΔT)
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D
γ2γ1
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Solution

## The correct option is A (γ2−γ1)ΔTWhen the temperature is increased by a ΔT New volume of metal will be (V1)=V0(1+γ1 ΔT) .....(1) New volume of mercury will be (V2)=V0(1+γ2 ΔT) .....(2) From (1) and (2), we get V2V1=(1+γ2 ΔT1+γ1 ΔT) ⇒V2V1=(1+γ2 ΔT)(1+γ1 ΔT)−1 Applying binomial expansion, V2V1=(1+γ2 ΔT)(1−γ1 ΔT) ⇒V2V1=(1+γ2 ΔT−γ1ΔT−γ1 γ2( ΔT)2) ⇒V2V1−1=γ2 ΔT−γ1ΔT [neglecting second order terms] ⇒ΔVV1=(γ2 ΔT−γ1 ΔT) [V2−V1=ΔV] ∴ The fraction of the volume of metal submerged in mercury changes by a factor of (γ2−γ1) ΔT Hence, option (a) is the correct answer.

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