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Area and Volume Expansion

A glass sinke...

Question

A glass sinker has a mass $M$ in the air. When weighed in a liquid at temperature $t_{1}$ , the apparent mass is $M_{1}$ and when weighed in the same liquid at temperature $t_{2}$ , the apparent mass is $M_{2}$ . If the coefficient of cubical expansion of the glass is $γ_{g}$ , then coefficient of expansion of the liquid is

\frac{γ_{g} (M - M_{1})}{M - M_{2}} + (\frac{M_{2} - M_{1}}{M - M_{2}}) . \frac{1}{(t_{2} - t_{1})}

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\frac{γ_{g} (M - M_{1})}{M - M_{2}} - (\frac{M_{2} - M_{1}}{M - M_{2}}) . \frac{1}{(t_{2} - t_{1})}

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\frac{γ_{g} (M - M_{1})}{M - M_{2}} - (\frac{M - M_{2}}{M_{2} - M_{1}}) . \frac{1}{(t_{2} - t_{1})}

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\frac{γ_{g} (M - M_{1})}{M - M_{2}} + (\frac{M_{2} - M_{1}}{M_{2} + M_{1}}) \frac{1}{(t_{2} - t_{1})}

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Solution

The correct option is A $\frac{γ_{g} (M - M_{1})}{M - M_{2}} + (\frac{M_{2} - M_{1}}{M - M_{2}}) . \frac{1}{(t_{2} - t_{1})}$
Given:
Mass of the glass sinker in the air $= M$
Its apparent mass in the liquid at temperature $t_{1} = M_{1}$
and at temperature $t_{2} = M_{2}$
Coefficient of cubical expansion of the glass $= γ_{g}$
To find:
Coefficient of cubical expansion of the liquid $γ_{l} = ?$

We know that, reduction in weight $=$ upthrust
$\Rightarrow M g - M_{1} g = V_{1} ρ_{l 1} g$ ...........(1)
where $V_{1}$ is the volume of glass sinker and $ρ_{l 1}$ is the density of liquid at temperature $t_{1}$ .
Also, $M g - M_{2} g = V_{2} ρ_{l 2} g$
where $V_{2}$ is the volume of glass sinker and $ρ_{l 2}$ is the density of liquid at temperature $t_{2}$ .
$\Rightarrow M g - M_{2} g = V_{1} [1 + γ_{g} (t_{2} - t_{1})] \frac{ρ_{l 1}}{[1 + γ_{l} (t_{2} - t_{1})]} g$
[from formula of change in volume and density with change in temperature]
$\Rightarrow M g - M_{2} g = \frac{(M g - M_{1} g) (1 + γ_{g} (t_{2} - t_{1}))}{1 + γ_{l} (t_{2} - t_{1})}$
[ from (1), $V_{1} ρ_{l 1} g = M g - M_{1} g$ ]
$\Rightarrow \frac{M - M_{2}}{M - M_{1}} = \frac{γ_{g} + x}{γ_{l} + x}$
$[where x = \frac{1}{t_{2} - t_{1}}]$
On applying componendo and dividendo, if $\frac{a}{b} = \frac{c}{d}$ then $\frac{a}{b - a} = \frac{c}{d - c}$
$\Rightarrow \frac{M - M_{2}}{M_{2} - M_{1}} = \frac{γ_{g} + x}{γ_{l} - γ_{g}}$
$\Rightarrow γ_{l} - γ_{g} = (γ_{g} + x) \frac{M_{2} - M_{1}}{M - M_{2}}$
$\Rightarrow γ_{l} = γ_{g} + \frac{(M_{2} - M_{1}) γ_{g}}{M - M_{2}} + \frac{(M_{2} - M_{1}) x}{M - M_{2}}$
$\Rightarrow γ_{l} = \frac{(M - M_{1}) γ_{g}}{M - M_{2}} + \frac{(M_{2} - M_{1}) x}{M - M_{2}}$
$\Rightarrow γ_{l} = \frac{(M - M_{1}) γ_{g}}{M - M_{2}} + \frac{(M_{2} - M_{1})}{(M - M_{2}) (t_{2} - t_{1})}$

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