$A$ 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 make an angle of $45^{\circ}$ with the horizontal as shown in the figure. The tension $T_1$ in each wire is:

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

An iceberg floats in sea water with a part of it submerged. The percentage fraction of the iceberg which is unsubmerged is (Density of ice $= 0.9 \text{ g/cm}^3$, density of sea water $= 1.1 \text{ g/cm}^3$). (in $\%$)

$A$ body when fully immersed in a liquid of specific gravity $1.2$ weighs $44 \text{ gwt}$. The same body when fully immersed in water weighs $50 \text{ gwt}$. The mass of the body is (in $\text{ g}$)

$A$ hot air balloon is carrying some passengers and a few sandbags of mass $1 kg$ each,so that its total mass is $480 kg$. Its effective volume providing buoyancy is $V$. The balloon is floating at an equilibrium height of $100 m$. When $N$ number of sandbags are thrown out,the balloon rises to a new equilibrium height of $150 m$ with its volume $V$ remaining unchanged. If the variation of the density of air with height $h$ from the ground is $\rho(h) = \rho_0 e^{-\frac{h}{h_0}}$,where $\rho_0 = 1.25 kg m^{-3}$ and $h_0 = 6000 m$,the value of $N$ is:

The tension in a massless cable connected to an iron ball of $100 \,kg$ when it is submerged in sea water is (Given: $\rho_{\text{iron}} = 8 \times 10^3 \,kg/m^3$, $\rho_{\text{sea water}} = 1000 \,kg/m^3$, $g = 10 \,m/s^2$). (in $\,N$)