Question 9: Units , Dimensions and Error Analysis

Question

A wooden block floats with $\frac{3}{5}$ of its volume submerged in a tank of water. If a denser liquid is poured into the tank, the wooden block floats with half its volume in the liquid and the remaining half in water. The relative density of the liquid is:

Accepted Answer

First, from the first equilibrium (only water) we have
Ã¢ÂÂ¢ submerged fraction = 3/5 Ã¢ÂÂ
$$ho_{m wood}=\frac{3}{5}\,ho_{m water}.$$
Next, when the block floats half in water and half in the other liquid, ArchimedesÃ¢ÂÂ principle gives
$$ho_{m wood}Vg  = ho_{m water}\,\frac{V}{2}\,g\;+\;ho_{m liquid}\,\frac{V}{2}\,g$$
Divide by $Vg$:
$$ho_{m wood}=\frac{1}{2}\,ho_{m water}+\frac{1}{2}\,ho_{m liquid}$$
Solve for $ho_{m liquid}$:
$$ho_{m liquid}=2ho_{m wood}-ho_{m water} =2\cdot\frac{3}{5}ho_{m water}-ho_{m water} =\frac{1}{5}\,ho_{m water}.$$
Thus the relative density is $\displaystyle\frac{ho_{m liquid}}{ho_{m water}}=	frac15$.
Answer: Option D.

Answer

$\frac{1}{2}$

Answer

$\frac{1}{3}$

Answer

$\frac{1}{6}$

Answer

$\frac{1}{5}$

Units , Dimensions and Error Analysis

A wooden block floats with \frac{3}{5} of its volume submerged in a tank of water. If a denser liquid is poured into the tank, the wooden block floats with half its volume in the liquid and the remaining half in water. The relative density of the liquid is:

Choose the correct answer: