Question

A solid sphere of density $\alpha \left(> 1\right)$ times lighter than 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. $\left(\frac{\alpha - 1}{\alpha }\right)mg$
• B. $\alpha mg$
• C. $\frac{m g}{\alpha - 1}$
• D. $\left(\alpha - 1\right)mg$

Answer 1

Answer: (D)

Density of sphere $\sigma =\frac{\rho _{\omega }}{\alpha }$

$\frac{\rho _{\omega }}{\sigma }=\alpha $
$F_{B}=mg+T$
$T=F_{B}-mg=\rho _{\omega }Vg-mg$
$=\rho _{\omega }\frac{m}{\sigma }g-mg$
$=\left(\frac{\left(\rho \right)_{\omega }}{\sigma } - 1\right)mg=\left(\alpha - 1\right)mg$