Properties of Matter 2 Question 10
10. A metal ball immersed in alcohol weighs $w _1$ at $0^{\circ} C$ and $w _2$ at $50^{\circ} C$. The coefficient of cubical expansion of the metal is less than that of the alcohol. Assuming that the density of the metal is large compared to that of alcohol, it can be shown that
$(1980,2 M)$
(a) $w _1>w _2$
(b) $w _1=w _2$
(c) $w _1<w _2$
(d) All of these
Passage Based Questions
Passage
A wooden cylinder of diameter $4 r$, height $h$ and density $\rho / 3$ is kept on a hole of diameter $2 r$ of tank, filled with liquid of density $\rho$ as shown in the figure.
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Answer:
Correct Answer: 10. Most appropriate option is (d)
Solution:
$$ w _{\text {app }}=w _{\text {actual }}-\text { Upthrust } $$
Upthrust $F=V _S \rho _L g$
Here, $\quad V _S=$ volume of solid, $\rho _L=$ density of liquid
At higher temperature $F^{\prime}=V _S^{\prime} \rho _L^{\prime} g$
$$ \therefore \quad \frac{F^{\prime}}{F}=\frac{V _S^{\prime}}{V _S} \cdot \frac{\rho _L^{\prime}}{\rho _L}=\frac{\left(1+\gamma _S \Delta \theta\right)}{\left(1+\gamma _L \Delta \theta\right)} $$
Since,
$$ \gamma _S<\gamma _L $$
$$ \therefore \quad F^{\prime}<F \text { or } w _{\text {app }}^{\prime}>w _{\text {app }} $$
