Question

A solid sphere of density $\eta (>1)$ times lighter than that of water is suspended in a water tank by a string tied to its base as shown in figure. If the mass of the sphere is $m$, then the tension in the string is given by

• A. $\left(\frac{\eta -1}{\eta }\right)$
• B. $\eta mg$
• C. $\frac{mg}{\eta -1}$
• D. $(\eta -1)mg$

Answer 1

The correct option is D $(\eta -1)mg$


Let density of sphere be $\rho$.
Then according to question, $\rho =\frac{{\rho }_w}{\eta }$
From FBD,
$T+mg=F_b$
$\therefore T=F_b-mg$
$\Rightarrow T=V{\rho }_{w}g-mg$
$\Rightarrow T=\frac{m}{\left({\rho }_w/\eta \right)}{\rho }_{w}g-mg$
$\Rightarrow T=(\eta -1)mg$