Question

A test tube of thin walls has some lead shots in it at its bottom and the system floats vertically in water, sinking by a length $l_0=10cm$. A liquid of density less than that of water, is poured into the tube till the levels inside and outside the tube are even. If the tube now sinks to a length $l=40cm$, the specific gravity of the liquid is $\frac{(70+x)}{100}$. Find $x$.

Answer 1

When the test tube sinks by a length $l_0=10cm$ and is in equilibrium, upward thrust is equal to the weight of the test tube plus the lead shots.
Let $A$ be the cross sectional area of the test tube.
Weight of water displaced: $W=A{l}_0{\rho }_{w}g=W_{lead}$ .....(1)
When a liquid is poured into the test tube, the tube sinks 40 cm.
Hence, Weight of water displaced: $W_1=Al{\rho }_{w}g$
Also, $W_1$ is equal to the weight of the lead shots plus the weight of the liquid poured.
Hence, $W_1=W_{lead}+W_{liquid}=W_{lead}+Al{\rho }_{l}g$
$\therefore W_{lead}=W_1-Al{\rho }_{l}g=Al{\rho }_{w}g-Al{\rho }_{l}g$ ....(2)
From eqns(1) and (2)
$\therefore Al{\rho }_{w}g-Al{\rho }_{l}g=A{l}_0{\rho }_{w}g$
$\therefore l{\rho }_w-l{\rho }_l=l_0{\rho }_w$
$\therefore \frac{{\rho }_l}{{\rho }_w}=1-\frac{l_0}{l}=1-\frac{10}{40}=\frac{3}{4}=0.75$