Density, Compressibility & Pascal's Law

In a hydraulic press, the piston in the small cylinder has a diameter of $d_1$ cm, while the larger piston has a diameter of $d_2$ cm. If a force $F_1$ is applied to the small piston, then the force $F_2$ on the large piston is given by

In a hydraulic press, the small cylinder has a diameter of $d_1$ cm, while the large cylinder has a diameter of $d_2$ cm. If a force $F_1$ is applied to a piston of small cylinder, then the force on the piston of large cylinder $F_2$ is

A piston of cross-sectional area $2.5 \times 10^{-2}$ m$^{2}$ is used in a hydraulic lift to exert a force of $250$ N on water. The cross-sectional area of the other piston which supports a car of mass $3000$ kg is ($g = 9.8$ m/s$^{2}$)

The increase in the pressure required to decrease the volume ( $\Delta V$) of water is $6.3 \times 10^{7}\text{ N/m}^{2}$. The percentage decrease in the volume is ______.

(Bulk modulus of water $= 2.1 \times 10^{9}\text{ N/m}^{2}$.)

A sample of a liquid is kept at 1 atm. It is compressed to 5 atm which leads to a change of volume of 0.8 cm3. If the bulk modulus of the liquid is 2 GPa, the initial volume of the liquid was _______ litre. (Take 1 atm = 105 Pa)

The increase in pressure required to decrease the volume of a water sample by $0.2\%$ is $P \times 10^{5}\text{ N m}^{-2}$. Bulk modulus of water is $2.15 \times 10^{9}\text{ N m}^{-2}$. The value of P is _________.

In a hydraulic lift, the surface area of the input piston is 6 cm2 and that of the output piston is 1500 cm2. If 100 N force is applied to the input piston to raise the output piston by 20 cm, then the work done is _______ kJ.

The fractional compression $\left(\frac{\Delta V}{V}\right)$ of water at the depth of 2.5 km below the sea level is __________ %. Given, the Bulk modulus of water = $2 \times 10^{9}$ N m$^{-2}$, density of water = $10^{3}$ kg m$^{-3}$, acceleration due to gravity $g = 10$ m s$^{-2}$.

A hydraulic press containing water has two arms with diameters as mentioned in the figure. A force of $10\text{ N}$ is applied on the surface of water in the thinner arm. The force required to be applied on the surface of water in the thicker arm to maintain equilibrium of water is _________ N.

The depth below the surface of sea to which a rubber ball be taken so as to decrease its volume by $0.02\%$ is _______ $m$.

(Take density of sea water $= 10^{3}\text{ kg m}^{-3}$, Bulk modulus of rubber $= 9 \times 10^{8}\text{ N m}^{-2}$, and $g = 10\text{ m s}^{-2}$)

If average depth of an ocean is $4000\text{ m}$ and the bulk modulus of water is $2 \times 10^{9}\text{ N m}^{-2}$, then fractional compression $\frac{\Delta V}{V}$ of water at the bottom of ocean is $\alpha \times 10^{-2}$. The value of $\alpha$ is _______ (Given, $g = 10\text{ m s}^{-2},\ \rho = 1000\text{ kg m}^{-3}$)

A certain pressure '$P$' is applied to 1 litre of water and 2 litre of a liquid separately. Water gets compressed to $0.01\%$ whereas the liquid gets compressed to $0.03\%$. The ratio of Bulk modulus of water to that of the liquid is $\frac{3}{x}$. The value of $x$ is ____________.

A hydraulic automobile lift is designed to lift vehicles of mass $5000\text{ kg}$. The area of cross section of the cylinder carrying the load is $250\text{ cm}^{2}$. The maximum pressure the smaller piston would have to bear is $[$ Assume $g = 10\text{ m/s}^{2}]$

Given below are two statements: one is labelled as Assertion $\mathbf{A}$ and the other is labelled as Reason $\mathbf{R}$

Assertion A: When you squeeze one end of a tube to get toothpaste out from the other end, Pascal's principle is observed.

Reason R: A change in the pressure applied to an enclosed incompressible fluid is transmitted undiminished to every portion of the fluid and to the walls of its container.

In the light of the above statements, choose the most appropriate answer from the options given below

The bulk modulus of a liquid is 3 $\times$ 1010 Nm${}^{-2}$. The pressure required to reduce the volume of liquid by 2% is:

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 object is located at 2 km beneath the surface of the water. If the fractional compression $\frac{\Delta V}{V}$ is 1.36%, the ratio of hydraulic stress to the corresponding hydraulic strain will be ____________. [Given: density of water is 1000 kgm${}^{-3}$ and g = 9.8 ms${}^{-2}$]

The normal density of a material is $\rho$ and its bulk modulus of elasticity is K. The magnitude of increase in density of material, when a pressure P is applied uniformly on all sides, will be:

When a rubber ball is taken to a depth of __________m in deep sea, its volume decreases by 0.5%. (The bulk modulus of rubber = 9.8 $\times$ 108 Nm${}^{-2}$, Density of sea water = 103 kgm${}^{-3}$, g = 9.8 m/s2)

A solid sphere of radius r made of a soft material of bulk modulus K is surrounded by a liquid in a cylindrical container. A massless piston of area a floats on the surface of the liquid, covering entire cross section of cylindrical container. When a mass m is placed on the surface of the piston to compress the liquid, the fractional decrement in the radius of the sphere, $\left(\frac{dr}{r}\right)$ is:

The approximate depth of an ocean is 2700 m. The compressiblity of water is 45.4 $\times$ 10$-$11 Pa${}^{-1}$ and density of water is 103 kg/m3. What fractional compression of water will be obtained at the bottom of the ocean?

The Bulk moduli of Ethanol, Mercury and water are given as 0.9, 25 and 2.2 respectively in units of 109 Nm${}^{-2}$. For a given value of pressure, the fractional compression in volume is $\frac{\Delta V}{V}$. Which of the following statements about $\frac{\Delta V}{V}$ for these three liquids is correct?

A ball is made of a material of density $\rho$ where $\rho_{oil} < \rho < \rho_{water}$ with $\rho_{oil}$ and $\rho_{water}$ representing the densities of oil and water, respectively. The oil and water are immiscible. If the above ball is in equilibrium in a mixture of this oil and water, which of the following pictures represents its equilibrium position?