A homogeneous solid cylinder of length $L$$(L < H/2)$. Cross-sectional area $A/5$ is immersed such that it floats with its axis vertical at the liquid-liquid interface with length $L/4$ in the denser liquid as shown in the fig. The lower density liquid is open to atmosphere having pressure ${P_0}$. Then density $D$  of solid is given by

A slender homogeneous rod of length $2L$ floats partly immersed in water, being supported by a string fastened to one of its ends, as shown. The specific gravity of the rod is $0.75$. The length of rod that extends out of water is : 

A small spherical monoatomic ideal gas bubble $\left(\gamma=\frac{5}{3}\right)$ is trapped inside a liquid of density $\rho_{\ell}$ (see figure). Assume that the bubble does not exchange any heat with the liquid. The bubble contains n moles of gas. The temperature of the gas when the bubble is at the bottom is $\mathrm{T}_0$, the height of the liquid is $\mathrm{H}$ and the atmospheric pressure is $\mathrm{P}_0$ (Neglect surface tension).

Figure: $Image$

$1.$ As the bubble moves upwards, besides the buoyancy force the following forces are acting on it

$(A)$ Only the force of gravity

$(B)$ The force due to gravity and the force due to the pressure of the liquid

$(C)$ The force due to gravity, the force due to the pressure of the liquid and the force due to viscosity of the liquid

$(D)$ The force due to gravity and the force due to viscosity of the liquid

$2.$ When the gas bubble is at a height $\mathrm{y}$ from the bottom, its temperature is

$(A)$ $\mathrm{T}_0\left(\frac{\mathrm{P}_0+\rho_0 \mathrm{gH}}{\mathrm{P}_0+\rho_t \mathrm{gy}}\right)^{2 / 5}$

$(B)$ $T_0\left(\frac{P_0+\rho_t g(H-y)}{P_0+\rho_t g H}\right)^{2 / 5}$

$(C)$ $\mathrm{T}_0\left(\frac{\mathrm{P}_0+\rho_t \mathrm{gH}}{\mathrm{P}_0+\rho_t \mathrm{gy}}\right)^{3 / 5}$

$(D)$ $T_0\left(\frac{P_0+\rho_t g(H-y)}{P_0+\rho_t g H}\right)^{3 / 5}$

$3.$ The buoyancy force acting on the gas bubble is (Assume $R$ is the universal gas constant)

$(A)$ $\rho_t \mathrm{nRgT}_0 \frac{\left(\mathrm{P}_0+\rho_t \mathrm{gH}\right)^{2 / 5}}{\left(\mathrm{P}_0+\rho_t \mathrm{gy}\right)^{7 / 5}}$

$(B)$ $\frac{\rho_{\ell} \mathrm{nRgT}_0}{\left(\mathrm{P}_0+\rho_{\ell} \mathrm{gH}\right)^{2 / 5}\left[\mathrm{P}_0+\rho_{\ell} \mathrm{g}(\mathrm{H}-\mathrm{y})\right]^{3 / 5}}$

$(C)$ $\rho_t \mathrm{nRgT} \frac{\left(\mathrm{P}_0+\rho_t g \mathrm{H}\right)^{3 / 5}}{\left(\mathrm{P}_0+\rho_t g \mathrm{~g}\right)^{8 / 5}}$

$(D)$ $\frac{\rho_{\ell} \mathrm{nRgT}_0}{\left(\mathrm{P}_0+\rho_{\ell} \mathrm{gH}\right)^{3 / 5}\left[\mathrm{P}_0+\rho_t \mathrm{~g}(\mathrm{H}-\mathrm{y})\right]^{2 / 5}}$

Give the answer question $1,2,$ and $3.$

A cube of external dimension $10\  cm$ has an inner cubical portion of side $5\  cm$ whose density is twice that of the outer portion. If this cube is just floating in a liquid of density $2\  g/cm^3$, find the density of the inner portion

An ornament weighs $10\, g$ in air and $6\, g$ in water. Density of material of ornament is $20\,\frac{g}{{cc}}$. The  volume of cavity in ornament is ........ $cc$

An object suspended by a wire stretches it by $10 \,mm$. When object is immersed in a liquid the elongation in wire reduces by $\frac{10}{3} \,mm$. The ratio of relative densities of the object and liquid is ............