Pressure and Pascal's law

When the chimney is closed using a cap at the top, a pressure difference $\Delta P$ develops between the top and the bottom surfaces of the cap. If the changes in the temperature and density of the hot air, due to the stoppage of air flow, are negligible then the value of $\Delta P$ is _____ $\mathrm{N\,m^{-2}}$.

The area of cross-section of a large tank is 0.5 m$^2$. It has a narrow opening near the bottom having area of cross-section 1 cm$^2$. A load of 25 kg is applied on the water at the top in the tank. Neglecting the speed of water in the tank, the velocity of the water, coming out of the opening at the time when the height of water level in the tank is 40 cm above the bottom, will be ______ cms$^{-1}$. [Take $g=10$ ms$^{-2}$]

Two cylindrical vessels of equal cross-sectional area 16 cm$^2$ contain water upto heights 100 cm and 150 cm respectively. The vessels are interconnected so that the water levels in them become equal. The work done by the force of gravity during the process, is [Take, density of water = $10^3$ kg/m$^3$ and g = 10 ms$^{-2}$] :

Consider a cylindrical tank of radius 1 m is filled with water. The top surface of water is at 15 m from the bottom of the cylinder. There is a hole on the wall of cylinder at a height of 5 m from the bottom. A force of $5\times10^5$ N is applied an the top surface of water using a piston. The speed of efflux from the hole will be :

(given atmospheric pressure $P_A=1.01\times10^5$ Pa, density of water $\rho_w$ = 1000 kg/m$^3$ and gravitational acceleration g = 10 m/s$^2$)

A cylindrical tube, with its base as shown in the figure, is filled with water. It is moving down with a constant acceleration $a$ along a fixed inclined plane with angle $\theta = 45^\circ$. $P_1$ and $P_2$ are pressures at points 1 and 2, respectively, located at the base of the tube. Let $\beta = (P_1 - P_2)/(\rho g d)$, where $\rho$ is density of water, $d$ is the inner diameter of the tube and $g$ is the acceleration due to gravity. Which of the following statement(s) is(are) correct?

A hydraulic press can lift 100 kg when a mass 'm' is placed on the smaller piston. It can lift _______ kg when the diameter of the larger piston is increased by 4 times and that of the smaller piston is decreased by 4 times keeping the same mass 'm' on the smaller piston.

An open-ended U-tube of uniform cross-sectional area contains water (density $10^3\ \mathrm{kg\,m^{-3}}$). Initially the water level stands at 0.29 m from the bottom in each arm. Kerosene oil (a water-immiscible liquid) of density $800\ \mathrm{kg\,m^{-3}}$ is added to the left arm until its length is 0.1 m, as shown in the schematic figure below. The ratio $\left(\dfrac{h_1}{h_2}\right)$ of the heights of the liquid in the two arms is

When water is filled carefully in a glass, one can fill it to a height $h$ above the rim of the glass due to the surface tension of water. To calculate $h$ just before water starts flowing, model the shape of the water above the rim as a disc of thickness $h$ having semicircular edges, as shown schematically in the figure. When the pressure of water at the bottom of this disc exceeds what can be withstood due to the surface tension, the water surface breaks near the rim and water starts flowing from there. If the density of water, its surface tension and the acceleration due to gravity are $10^3\ \mathrm{kg\,m^{-3}}$, $0.07\ \mathrm{N\,m^{-1}}$ and $10\ \mathrm{m\,s^{-2}}$, respectively, the value of $h$ (in mm) is ______.

A cubical solid aluminium (bulk modulus $= -V\dfrac{dP}{dV} = 70$ GPa) block has an edge length of 1 m on the surface of the earth. It is kept on the floor of a 5 km deep ocean. Taking the average density of water and the acceleration due to gravity to be $10^3\ \mathrm{kg\,m^{-3}}$ and $10\ \mathrm{m\,s^{-2}}$, respectively, the change in the edge length of the block in mm is ______.

A spherical body of radius $R$ consists of a fluid of constant density and is in equilibrium under its own gravity. If $P(r)$ is the pressure at $r\,(r<R)$, then the correct option(s) is(are)

There is a circular tube in a vertical plane. Two liquids which do not mix and of densities $d_1$ and $d_2$ are filled in the tube. Each liquid subtends 90° angle at centre. Radius joining their interface makes an angle $\alpha$ with vertical. Ratio $\dfrac{d_1}{d_2}$ is :

An open glass tube is immersed in mercury in such a way that a length of 8 cm extends above the mercury level. The open end of the tube is then closed and sealed and the tube is raised vertically up by additional 46 cm. What will be length of the air column above mercury in the tube now ?

(Atmospheric pressure $=76$ cm of Hg)

A tank with a small hole at the bottom has been filled with water and kerosene (specific gravity 0.8). The height of water is 3 m and that of kerosene 2 m. When the hole is opened the velocity of fluid coming out from it is nearly : (take $g=10$ ms$^{-2}$ and density of water $=10^3$ kg m$^{-3}$)

An air bubble of radius 0.1 cm is in a liquid having surface tension 0.06 N/m and density $10^3$ kg/m$^3$. The pressure inside the bubble is 1100 Nm$^{-2}$ greater than the atmospheric pressure. At what depth is the bubble below the surface of the liquid ? ($g=9.8$ ms$^{-2}$)

A cylindrical vessel of height 500 mm has an orifice (small hole) at its bottom. The orifice is initially closed and water is filled in it up to height H. Now the top is completely sealed with a cap and the orifice at the bottom is opened. Some water comes out from the orifice and the water level in the vessel becomes steady with height of water column being 200 mm. Find the fall in height (in mm) of water level due to opening of the orifice. [Take atmospheric pressure $=1.0\times10^5\ \mathrm{N/m^2}$, density of water $=1000\ \mathrm{kg/m^3}$ and $g=10\ \mathrm{m/s^2}$. Neglect any effect of surface tension.]

When the gas bubble is at a height $y$ from the bottom, its temperature is

The piston is now pulled out slowly and held at a distance $2L$ from the top. The pressure in the cylinder between its top and the piston will then be

While the piston is at a distance $2L$ from the top, the hole at the top is sealed. The piston is then released, to a position where it can stay in equilibrium. In this condition, the distance of the piston from the top is

The piston is taken completely out of the cylinder. The hole at the top is sealed. A water tank is brought below the cylinder and put in a position so that the water surface in the tank is at the same level as the top of the cylinder as shown in the figure. The density of the water is $\rho$. In equilibrium, the height $H$ of the water column in the cylinder satisfies

Water is filled up to a height $h$ in a beaker of radius $R$ as shown in the figure. The density of water is $\rho$, the surface tension of water is $T$ and the atmospheric pressure is $P_0$. Consider a vertical section ABCD of the water column through a diameter of the beaker. The force on water on one side of this section by water on the other side of this section has magnitude