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A glass sinker has a mass M in the air. When weighed in a liquid at temperature t1, the apparent mass is M1 and when weighed in the same liquid at temperature t2, the apparent mass is M2. If the coefficient of cubical expansion of the glass is γg, then coefficient of expansion of the liquid is

A
γg(MM1)MM2+(M2M1MM2).1(t2t1)
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B
γg(MM1)MM2(M2M1MM2).1(t2t1)
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C
γg(MM1)MM2(MM2M2M1).1(t2t1)
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D
γg(MM1)MM2+(M2M1M2+M1)1(t2t1)
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Solution

The correct option is A γg(MM1)MM2+(M2M1MM2).1(t2t1)
Given:
Mass of the glass sinker in the air =M
Its apparent mass in the liquid at temperature t1 =M1
and at temperature t2 =M2
Coefficient of cubical expansion of the glass =γg
To find:
Coefficient of cubical expansion of the liquid γl=?

We know that, reduction in weight = upthrust
MgM1g=V1ρl1g ...........(1)
where V1 is the volume of glass sinker and ρl1 is the density of liquid at temperature t1.
Also, MgM2g=V2ρl2g
where V2 is the volume of glass sinker and ρl2 is the density of liquid at temperature t2.
MgM2g=V1[1+γg(t2t1)]ρl1[1+γl(t2t1)]g
[from formula of change in volume and density with change in temperature]
MgM2g=(MgM1g)(1+γg(t2t1))1+γl(t2t1)
[ from (1), V1ρl1g=MgM1g]
MM2MM1=γg+xγl+x
[where x=1t2t1 ]
On applying componendo and dividendo, if ab=cd then aba=cdc
MM2M2M1=γg+xγlγg
γlγg=(γg+x)M2M1MM2
γl=γg+(M2M1)γgMM2+(M2M1)xMM2
γl=(MM1)γgMM2+(M2M1)xMM2
γl=(MM1)γgMM2+(M2M1)(MM2)(t2t1)

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