Question

A hollow sphere of mass $M$ and radius $r$ is immersed in a tank of water (density ${\rho }_w$). The sphere would float if it were set free. The sphere is tied to the bottom of the tank by two wires which makes angle ${45}^{\circ }$ with the horizontal as shown in the figure. The tension $T_1$ in the wires is

• A. $\frac{\frac{4}{3}\pi R^3{\rho }_{w}g-Mg}{\sqrt{2}}$
• B. $\frac{2}{3}\pi R^3{\rho }_{w}g-Mg$
• C. $\frac{\frac{4}{3}\pi R^3{\rho }_{w}g-Mg}{2}$
• D. $\frac{4}{3}\pi R^3{\rho }_{w}g+Mg$

Answer 1

The correct option is A $\frac{\frac{4}{3}\pi R^3{\rho }_{w}g-Mg}{\sqrt{2}}$

When the sphere is tied to the strings, it is in equilibrium. net force on sphere is zero. i.e. net upward force= net downward force
$\Rightarrow F_b=2T_1\cos {45}^{\circ }+Mg$; here $F_b$ is the buoyant force
$\Rightarrow \frac{4}{3}\pi R^3{\rho }_{w}g=2T_1\cos {45}^{\circ }+Mg$
$\Rightarrow T_1=\frac{\frac{4}{3}\pi R^3{\rho }_{w}g-Mg}{\sqrt{2}}$