Question

A solid sphere of mass $m=2\text{kg}$ and specific gravity $s=0.5$ is held stationary relative to a tank filled with water. The tank is accelerating vertically upwards with $2{\text{m/s}}^2$. Choose the correct option(s):

• A. Tension is the thread is $24\text{N}$
• B. Tension in the thread is $28\text{N}$
• C. If the thread snaps, acceleration of the sphere is $14{\text{m/s}}^2$
• D. If the thread snaps, acceleration of the sphere is $16{\text{m/s}}^2$

Answer 1

Answer: (A), (C)

The correct options are
A Tension is the thread is $24\text{N}$
C If the thread snaps, acceleration of the sphere is $14{\text{m/s}}^2$

Let us suppose tension in the thread is $T$.
According to question,


We know for accelerated fluid in vertical direction,
$F_b={\rho }_w{g}_{eff}V\left[\text{For water}\right]$
Here, $g_{eff}=|\overrightarrow{g}-\overrightarrow{a}$
On applying $\sum F=ma$ along vertical :
$F_b-T-mg=ma----\left(1\right)$
$\Rightarrow {\rho }_w{g}_{eff}V-T-mg=2\times 2$
$\Rightarrow {\rho }_w\times 12\times \frac{2}{{\rho }_s}-T-2\times 10=4$
$\left[\begin{array}{c}{g}_{eff}=-10-2=-12{\text{m/s}}^2\\ V=\frac{m}{{\rho }_s};{\rho }_s\to \text{density of sphere}\end{array}\right]$
$\Rightarrow T=\frac{1}{0.5}\times 12\times 2-20-4$
$[s=\frac{{\rho }_s}{{\rho }_w}=0.5]$
$\Rightarrow T=24\text{N}$

When thread snaps, Tension $T$ acting on sphere is removed $(T=0)$
$\therefore F_b-mg=ma$
[From (1), $a\to$ acceleration of sphere]
$\Rightarrow {\rho }_w{g}_{eff}V-mg=ma$
$\Rightarrow {\rho }_w\times 12\times \frac{2}{{\rho }_s}-2\times 10=2a$
$\Rightarrow \frac{12\times 2}{0.5}-20=2a$
$\Rightarrow a=14{\text{m/s}}^2$
Therefore, option A and C are correct.