Fluid Pressure and Pascal's Law

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LEVELJEE Advanced

A long cylindrical vessel is half-filled with a liquid. When the vessel is rotated about its own vertical axis, the liquid rises up near the wall. If the radius of vessel is $5 cm$ and its rotational speed is $2$ rotations per second, then the difference in the heights between the centre and the sides (in cm) will be

A submarine experience a pressure of $5.05 \times 1 0^{6}$ Pa at a depth of $d_{1}$ in a sea. When it goes further to a depth of $d_{2}$ , it experiences a pressure of $8.08 \times 1 0^{6}$ Pa, then $d_{2} - d_{1}$ is approximately (density of water = $1 0^{3} kg / m^{3}$ and acceleration due to gravity = $10 ms^{- 2}$ )

LEVELJEE Advanced

A cylindrical vessel containing a liquid is rotated about its axis, so that the liquid rises at its sides as shown in the figure. The radius of vessel is $5 cm$ and the angular speed of rotation is $ω rad s^{- 1}$ . The difference in the height $h$ (in cm) of liquid at the centre of vessel and at the side will be

\frac{25 ω ^{2}}{2 g}

\frac{5 ω ^{2}}{2 g}

\frac{2 ω ^{2}}{25 g}

\frac{2 ω ^{2}}{5 g}

Two liquids of densities $ρ_{1}$ and $ρ_{2}$ ( $ρ_{2} = 2 ρ_{1}$ ) are filled up behind a square wall of side $10 m$ as shown in figure. Each liquid has a height of $5 m$ . The ratio of the forces due to these liquids exerted on upper part MN to that at the lower part NO is (assume that the liquids are not mixing)

The pressure acting on a submarine is $3 \times 1 0^{5} Pa$ at a certain depth. If the depth is doubled, the percentage increase in the pressure acting on the submarine would be (Assume that atmospheric pressure is $1 \times 1 0^{5} Pa$ , density of water is $1 0^{3} kg m^{- 3}$ , $g = 10 ms^{- 2}$ )

\frac{200}{3} %

\frac{200}{5} %

\frac{5}{200} %

\frac{3}{200} %

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.

LEVELJEE Advanced

Two identical cylindrical vessels are kept on the ground and each contains the same liquid of density $d$ . The area of the base of both vessels is $S$ but the height of liquid in one vessel is $x_{1}$ and in the other, $x_{2}$ . When both cylinders are connected through a pipe of negligible volume very close to the bottom, the liquid flows from one vessel to the other until it comes to equilibrium at a new height. The change in energy of the system in the process is

g d S (x_{2}^{2} + x_{1}^{2})

\frac{3}{4} g d S (x_{2} - x_{1})^{2}

g d S (x_{2} + x_{1})^{2}

\frac{1}{4} g d S (x_{2} - x_{1})^{2}

LEVELJEE Advanced

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 $9 0^{\circ}$ angle at centre. Radius joining their interface makes an angle $α$ with vertical. Ratio $\frac{d _{1}}{d _{2}}$ is

\frac{1 + s i n α}{1 - s i n α}

\frac{1 + c o s α}{1 - c o s α}

\frac{1 + t a n α}{1 - t a n α}

\frac{1 + s i n α}{1 - c o s α}

JEE Advanced (1981)

Two identical cylindrical vessels with their bases at the same level each contain a liquid of density $ρ$ . The height of the liquid in one vessel is $h_{1}$ and in the other is $h_{2}$ . The area of either base is $A$ . What is the work done by gravity in equalising the levels when the two vessels are connected ?

LEVELJEE Advanced

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 $9 0^{\circ}$ angle at centre. Radius joining their interface makes an angle $α$ with vertical. Ratio $d_{1} / d_{2}$ is

\frac{1 + s i n α}{1 - s i n α}

\frac{1 + c o s α}{1 - c o s α}

\frac{1 + t a n α}{1 - t a n α}

\frac{1 + s i n α}{1 - c o s α}

JEE Advanced 1983

A U-tube of uniform cross-section is partially filled with a liquid I. Another liquid II which does not mix with liquid I is poured into one side. It is found that the liquid levels of the two sides of the tube are the same, while the level of liquid I has risen by $2 cm$ on the other side. If the specific gravity of liquid I is $1.1$ , the specific gravity of liquid II must be

JEE Advanced (1999)

A closed compartment containing gas is moving with some acceleration in horizontal direction. Neglect effect of gravity. Then, the pressure in the compartment is

same everywhere

lower in front side

lower in rear side

lower in upper side

JEE Advanced 1978

LEVELJEE Advanced

A column of mercury of length 10 cm is contained in the middle of a horizontal tube of length 1 m which is closed at both the ends. The two equal lengths contain air at standard atmospheric pressure of 0.76 m of mercury. The tube is now turned to vertical position. By what distance will the column of mercury be displaced ? Assume temperature to be constant.

JEE Advanced 1981

A vessel containing water is given a constant acceleration $a$ towards the right along a straight horizontal path. Which of the following diagrams represents the surface of the liquid?

Tilted upwards to the right

Tilted downwards to the right (straight line)

JEE Advanced 2005

LEVELJEE Advanced

A U-shaped tube contains a liquid of density $ρ$ and it is rotated about the line as shown in the figure. Find the difference in the levels of liquid column.

JEE Advanced (2009)

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 upto 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 = $1.0 \times 1 0^{5} N m^{- 2}$ , density of water = $1000 kg m^{- 3}$ and $g = 10 ms^{- 2}$ . Neglect any effect of surface tension.]

JEE Advanced 2020

An open-ended U-tube of uniform cross-sectional area contains water (density $1 0^{3} kg m^{- 3}$ ). Initially the water level stands at $0.29 m$ from the bottom in each arm. Kerosene oil (a water-immiscible liquid) of density $800 kg m^{- 3}$ is added to the left arm until its length is $0.1 m$ , as shown in the schematic figure below. The ratio $(\frac{h _{1}}{h _{2}})$ of the heights of the liquid in the two arms is-

\frac{15}{14}

\frac{35}{33}

\frac{7}{6}

\frac{5}{4}

JEE Advanced 2021

LEVELJEE Advanced

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 $θ = 4 5^{\circ}$ . $P_{1}$ and $P_{2}$ are pressures at points 1 and 2, respectively, located at the base of the tube. Let $β = (P_{1} - P_{2}) / (ρ g d)$ , where $ρ$ 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 ?

* Multiple Correct Options

β = 0

a = g / 2

β > 0

a = g / 2

β = \frac{2 - 1}{2}

a = g /2

β = \frac{1}{2}

a = g /2