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Question

# A solid cube is first floating in a liquid. The coefficient of linear expansion of the cube is α and the coefficient of volume expansion of the liquid is γ. On increasing the temperature of the (liquid + cube) system, the cube will sink if

A
γ=3α
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B
γ>3α
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C
γ<3α
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D
γ=2α
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Solution

## The correct option is B γ>3αFrom thermal expansion of a liquid, we can say that, γreal=γapp+γcube Let the linear expansion of the cube be α Then, volume expansion coefficient of cube (γcube)=3α .......(1) ∴γreal=γapp+3α ........(2) Given, Initially cube is floating in the liquid. ⇒ Density of cube is less than the density of liquid i.e ρcube<ρliquid Cube will sink if density of cube is greater than the density of liquid i.e ρ′cube>ρ′liquid .....(3) From the formula of variation of density with temperature ρ′=ρ(1−γΔT), we can write, For cube: ρ′cube=ρcube(1−γcube ΔT) ........(4) For liquid : ρ′liquid=ρliquid(1−γreal ΔT) ........(5) Applying condition (3) and substituting (4) and (5), we get ρcube(1−γcube ΔT)>ρliquid(1−γreal ΔT) Since ρcube<ρliquid, 1−γcube ΔT>1−γreal ΔT ⇒γreal>γcube i.e γreal>3α [using (1)] Hence, option (b) is the correct answer.

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