A wooden block floating in a bucket of water has $\frac{4}{5}$ of its volume submerged. When certain amount of an oil is poured into the bucket, it is found that the block is just under the oil surface with half of its volume under water and half in oil. The density of oil relative to that of water is

A hydraulic automobile lift is designed to lift cars with a maximum mass of $3000\, kg$. The area of cross section a of piston carrying the load is $425\, cm ^{2}$. What is the maximum pressure () would smaller piston have to bear ?

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 vessel filled with water is kept on a weighing pan and the scale adjusted to zero. A block of mass $\mathrm{M}$ and density $\rho $ is suspended by a massless spring of spring constant $\mathrm{k}$. This block is submerged inside into the water in the vessel. What is the reading of the scale ?

A metallic block of density $5\,gm \,cm^{-3}$ and having dimensions $5 cm × 5 cm × 5 cm$ is weighed in water. Its apparent weight will be

A ball whose density is $0.4 \times 10^3\,kg/m^3$ falls into water from a height of $9\,cm$ . To what depth does the ball sink ? ....... $cm$