Properties of Matter 2 Question 2
2. A wooden block floating in a bucket of water has $\frac{4}{5}$ of its volume submerged. When certain amount of an oil is poured into the bucket, it is found that the block is just under the oil surface with half of its volume under water and half in oil. The density of oil relative to that of water is (2019 Main, 9 April II)
(a) 0.6
(b) 0.8
(c) 0.7
(d) 0.5
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Answer:
Correct Answer: 2. (a)
Solution:
- (a) For a floating body,
upthrust $=$ weight of the part of object, i.e. submerged in the fluid.
In first situation,
So, weight of block of volume $V=$ weight of water of volume $\frac{4}{5} V \Rightarrow V \rho _b g=\frac{4}{5} V \rho _w g$
where, $\rho _b=$ density of block
and $\rho _w=$ density of water.
$\Rightarrow \quad \frac{\rho _b}{\rho _w}=\frac{4}{5}$
In second situation,
So, weight of block of volume $V=$ weight of oil of volume $\frac{V}{2}+$ weight of water of volume $\frac{V}{2}$.
$\Rightarrow \quad V \rho _b g=\frac{V}{2} \rho _o g+\frac{V}{2} \rho _w g$
where, $\quad \rho _o=$ density of oil.
$$ \Rightarrow \quad 2 \rho _b=\rho _o+\rho _w $$
$$ \begin{array}{rlrl} \Rightarrow & \frac{2 \rho _b}{\rho _w} & =\frac{\rho _o}{\rho _w}+1 \\ \Rightarrow & 2 \times \frac{4}{5} & =\frac{\rho _o}{\rho _w}+1 \\ \Rightarrow & & \rho _0 / \rho _w & =8 / 5-1 \\ & & =3 / 5=0.6 \end{array} $$
