Test: Pascal's Principle - 1 - EmSAT Achieve MCQ

Pascal’s principle: Pascal’s Law is the principle of transmission of fluid-pressure.

• It says that "a pressure exerted anywhere in a point of the confined fluid is transmitted equally in all directions throughout the fluid”.

Hydrostatic Law 

The law states that the rate of increase of pressure in a vertically downward direction must be equal to the specific weight of the fluid at that point.

Given:

Speed of stream of water (v) = 20 m/s

Area of cross-section of tube (A)= 10^-1m² (or 0.1 m²)

Concept:

Force exerted on the wall = Rate of loss in momentum of the water 

Formula:

The volume rate of flow of water: dV/dt = Av

The mass rate of flow of water: dm/dt = ρAv

Rate of loss in momentum of water = (dm/dt)v

Calculation:

Mass rate of flow of water : dm/dt = ρAv

⇒ dm/dt = 1000 × 0.1 × 20 

= 2000 kg/s

Rate of loss in momentum of water = (dm/dt)v

⇒ (dm/dt)v = 2000 × 20

= 40000 N

∴ Force exerted = 40000 N

CONCEPT:

• Pressure: Pressure is defined as the physical force exerted on an object. The unit of pressure is Pascal.

The basic formula for pressure is given by:

P = F/A, Where P is pressure, F = force, A = area.

The pressure exerted by a liquid on the wall of container can be given by

P = ρ h g, Where ρ = density of liquid, h = height of the liquid, g = gravity.

EXPLANATION:

• The pressure exerted by a liquid on the wall of container can be given by

P = ρ h g, Where ρ = density of liquid, h = height of the liquid, g = gravity.

• Thus it depends on the density of the liquid, height of the liquid in the container, and the acceleration due to gravity. So option 4 is correct.

Concept:

• Gauge pressure inside the water at the depth of ℎ m will be:

P_gauge = ρgh N/m²

• Derived using Bernoulli's equation

Calculation:

Given: P_atm = 10 N/cm² = 10× 10⁴ N/ m² , ρ = 1000 kg/m³, g = 9.8m/s²

P_atm= 10⁵Pa

• Now P_gauge  should be equal to three times P_atm

Pabs = P_atm + P_gauge

⇒ 3 × P_atm = P_atm + ρgh

⇒ ρgh = 2× P_atm

⇒ 1000 × 9.8 × h = 2 × 10⁵

⇒ h = 20.4m 

• Hence at the depth of 20.4m the pressure inside the water will be three times the atmospheric pressure.

Additional Information

• Gauge pressure or relative pressure (Pg) is the difference between the measured pressure and the local atmospheric pressure.
• Absolute pressure is the actual pressure at a point measured with respect to vacuum pressure i.e. zero pressure.

CONCEPT:

• When a body is submerged in a fluid, it experiences pressure due to the surrounding fluid.
• The resultant of the pressure due to the fluid acts at a point in the body.
• Centre of pressure: The point where the total sum of (or resultant) pressure due to the fluid acts on a body submerged inside it.

EXPLANATION:

• The resultant pressure on a body submerged in a fluid acts on a point on a body called the centre of pressure.

Additional Information

• Centre of mass: The point where all the mass of a body is assumed to be concentrated.
• Centre of buoyancy: The centre of gravity of the volume of fluid displaced by the body submerged in the fluid is called the centre of buoyancy.
• Metacentre: The point at which an imaginary vertical line passing through the centre of buoyancy and centre of gravity intersects the imaginary vertical line through a new centre of buoyancy created when the body is displaced, or tipped, in the fluid.

CONCEPT:

• The normal force exerted by liquid at rest on a given surface in contact with it is called the thrust of liquid on that surface.
• The normal force (or thrust) exerted by liquid at rest per unit area of the surface in contact with it, is called pressure of liquid or hydrostatic pressure.
• If Po is the atmospheric pressure then for a point at depth h below the surface of a liquid of density ρ,
• Hydrostatic pressure P is given by

P = P_o + ρgh

EXPLANATION:

• Hydrostatic pressure depends on the depth of the point below the surface (h), nature of the liquid (ρ), and acceleration due to gravity (g).
•  Hydrostatic pressure is independent of the amount of liquid, the shape of the container, or the cross-sectional area.
• If a given liquid is filled in vessels of different shapes to the same height, the pressure at the base in each vessel's will be the same, though the volume or weight of the liquid in different vessels will be different. Therefore option 4 is correct.

CONCEPT:

• Pascals Law: In a fluid at rest in a closed container, the external static pressure applied on a confined liquid is transmitted without loss to every portion of the fluid and to the walls of the container.
• In Pascal's law, the effect of gravity is neglected.

CALCULATION:

Given that A₁ = 0.25 m2 and A₂ = 1 m2; F₂ = 40N F₁ = ?  

From Pascal's Law, the pressure inside the liquid will be equal at every place. So the pressure at A and B will be equal.

P_A = P_B

So the correct answer is option 2.

Concept:

Find the expression for both volume and pressure at the bottom and at the surface of the lake. Then at constant temperature try to find the solution with the help of P-V relation.

Hydrostatic Pressure:

Hydrostatic pressure can be defined as the pressure that is exerted by a fluid at equilibrium to an object at a depth due to the gravitational force. It can be given as,

p = ρgh

Calculation:

Given:

P_atm = 10 m (height of water column)

At the bottom of the lake, the radius of the air bubble is r.

At the surface of the lake, the radius of the bubble will be 2r.

Volume of the bubble at the bottom of the lake 

Volume of the bubble at the surface of the lake 

Now the atmospheric pressure at the surface is = 2 m.

Atmospheric pressure at the bottom will be

P₁ = P_atm + Pressure at the bottom of the water at height h

P1 = (10 + h) m

The temperature has been given constant, therefore

P₁V₁ = P₂V₂

(10 + h)V = 10 × 8V

10 + h = 80

h = 70 m

• Pressure in liquids is the multiplication of three quantities i.e. 
• Where:
  • P is the pressure at the depth,
  • ρ is the density of the fluid,
  • g is the acceleration due to gravity, and
  • h is the depth or height of the fluid column.
• Option 1 - Dependent on the density of the liquid
  • Pressure is directly proportional to the density (ρ) of the liquid.
  • As the density increases, the pressure at a given depth also increases.
• Option 2 - Equal in all directions
  • Pascal's Law states that when there is an increase in pressure at any point in a confined fluid, there is an equal increase in pressure throughout the fluid in all directions.
  • In the context of a liquid in a container, this means that the pressure at the bottom of the container is transmitted equally in all directions to the sides and top of the container.
• Option 3 - Not dependent on the area of cross-section of container
  • As per the formula, the pressure (P) is not directly dependent on the area of cross-section (A) of the container. 
  • The pressure is primarily influenced by the height, density, and acceleration due to gravity.

CONCEPT:

• Pascal’s principle: Pascal’s Law is the principle of transmission of fluid-pressure.
  • It says that "a pressure exerted anywhere in a point of the confined fluid is transmitted equally in all directions throughout the fluid”.

EXPLANATION:

1. Centrifuge: Centrifugation is used to separate the contents of mixtures according to their size and density. The centrifuge works on the principle of sedimentation.
2. Hydraulic Lift: The lift that uses pascal's law and used to lift objects with the help of fluid is called hydraulic lift.  It works on Pascal's law. So option 2 is correct.
3. ​Motor: A motor is an electrical device that converts electrical energy into mechanical energy. It works on the principle that a current-carrying conductor experiences a mechanical force when it is placed in a magnetic field.
4. Lever: A lever is a simple machine that is made up of two load arms and a fulcrum. The distance from the fulcrum can be used to determine how the input and output forces are related to each other.