1. Tardigrade
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  3. Physics
  4. A sinker of weight W0 has an apparent weight W1 when weighed in a liquid at a temperature t1 and W2 when weight in the same liquid at temperature t2. The coefficient of cubical expansion of the material of sinker is β. What is the coefficient of volume expansion of the liquid?

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Q. A sinker of weight $W_0$ has an apparent weight $W_1$ when weighed in a liquid at a temperature $t_1$ and $W_2$ when weight in the same liquid at temperature $t_2$. The coefficient of cubical expansion of the material of sinker is $\beta$. What is the coefficient of volume expansion of the liquid?

Thermal Properties of Matter

Solution:

$W_0-W_1=V \times d_{\ell} \times g$......(i)
$W_0-W_2=V^{\prime} \times d^{\prime \prime} \times g$......(ii)
Also, $V^{\prime}=V(1+\beta \Delta T)$.......(iii)
and $d_{\ell}=d^{\prime}{ }_{\ell}\left(1+\gamma_{\ell} \Delta T\right)$.......(iv)
From (ii), (iii) and (iv)
$W_0-W_2=\frac{V(1+\beta \Delta T) \times d_{\ell}}{1+\gamma_{\ell} \Delta T} \times g$.......(v)
Dividing (i) and (v), we get
$\frac{W_0-W_1}{W_0-W_2}=\frac{V d_{\ell} g\left(1+\gamma_{\ell} \Delta T\right)}{V(1+\beta \Delta T) d_{\ell} g}$
$\Rightarrow \frac{W_0-W_1}{W_0-W_2}=\frac{1+\gamma_{\ell} \Delta T}{1+\beta \Delta T} $
$\Rightarrow \frac{W_0-W_1}{W_0-W_2}=\frac{1+\gamma_{\ell}\left(t_2-t_1\right)}{1+\beta\left(t_2-t_1\right)}$
$\Rightarrow\left(W_0-W_1\right)\left[1+\beta\left(t_2-t_1\right)=\left(W_0-W_2\right)\left[1+\gamma_{\ell}\left(t_2-t_1\right)\right]\right.$
$\Rightarrow \gamma_{\ell}=\frac{W_2-W_1}{\left(W_0-W_2\right)\left(t_2-t_1\right)}+\frac{\beta\left(W_0-W_1\right)}{\left(W_0-W_2\right)}$