A metallic body of material with density of $8000\  kg/m^3$ has a cavity inside. A spring balance shows its mass to be $10.0\  kg$ in air and $7.5\  kg$ when immersed in water. The ratio of the volume of the cavity to the volume of the material of the body must be

A block of wood floats in water with $\frac{4}{5}$ th of its volume submerged, but it just floats in another liquid. The density of liquid is (in $kg / m ^3$ )

The vessel shown in the figure has two sections. The lower part is a rectangular vessel with area of cross-section $A$ and height $h$. The upper part is a conical vessel of height $h$ with base area $‘A’$ and top area $‘a’$ and the walls of the vessel are inclined at an angle $30^o$  with the vertical.A liquid of density $\rho$ fills both the sections upto a height $2h$. Neglecting atmospheric pressure. 

A body floats in a liquid contained in a beaker. The whole system as shown falls freely under gravity. The upthrust on the body due to the liquid is

The spring balance $A$ reads $2\, kg$ with a block m suspended from it. $A $ balance $B$ reads $5 \,kg $ when a beaker filled with liquid is put on the pan of the balance. The two balances are now so arranged that the hanging mass is inside the liquid as shown in figure. In this situation

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 $3.$