NEET Mechanical Properties of Fluids Notes by PW

The Mechanical Properties of Fluids chapter connects fluid pressure, flow and surface effects with important principles of Physics. Concepts such as hydrostatic pressure and buoyancy describe fluids at rest, while continuity and Bernoulli’s theorem help explain the behaviour of moving fluids.

To strengthen preparation for this chapter, Physics Wallah covers key Fluid Mechanics concepts by focusing on the physical meaning of each concept, the conditions in which a formula applies and numerical problems based on pressure, fluid flow, viscosity, surface tension and capillarity. Regular practice can help you identify given conditions, select the appropriate relation and apply it accurately to different Fluid Mechanics problems. 

Pressure in Fluids

Pressure is the normal force acting per unit area of a surface.

Pressure = Force/Area

For a small surface element:

dF = P dA

If pressure is uniform over the complete surface:

F = PA

Pressure force always acts perpendicular to the surface.

Hydrostatic Pressure

Pressure increases with depth in a liquid because of the weight of the liquid above the point.

The pressure difference between two points separated by a vertical depth h is:

ΔP = ρgh

Therefore, pressure at depth h below the liquid surface is:

P = P₀ + ρgh

where:

• P₀ = pressure at the liquid surface

• ρ = density of the liquid

• g = gravitational acceleration

• h = depth

When atmospheric pressure is not included, the pressure is called gauge pressure. Absolute pressure includes atmospheric pressure.

In the same continuous liquid at rest, points at the same horizontal level have the same pressure.

Applying the Dipping Method

The pressure at a point in a connected liquid can be found by starting from a point where pressure is known and moving through the liquid.

• Move downward: add ρgh.

• Move upward: subtract ρgh.

• Move horizontally at the same level in the same liquid: pressure remains unchanged.

• Use the density of the liquid through which the vertical movement takes place.

This approach is useful when pressure has to be compared at different points in connected liquids.

Pressure in Accelerating and Rotating Fluids

The pressure distribution changes when a liquid is accelerating.

Horizontally Accelerating Liquid

When a container of liquid accelerates horizontally, the free surface becomes inclined.

The angle of inclination satisfies:

tan θ = a/g

where a is the horizontal acceleration.

In the accelerating frame, a pseudo-force acts opposite to the direction of acceleration. Pressure increases in the direction of the pseudo-force.

For horizontal distance L, the pressure change is:

ΔP = ρaL

Therefore, pressure changes due to vertical and horizontal movements can be considered together.

Pressure in a Rotating Liquid

For a liquid rotating with angular velocity ω, pressure increases as the distance from the axis increases.

The pressure difference between radial positions R₁ and R₂ is:

ΔP = 1/2 ρω²(R₂² − R₁²)

The free surface of a rotating liquid takes a parabolic shape.

Quick Revision: In a rotating liquid, pressure increases away from the axis. In a horizontally accelerating liquid, pressure increases towards the direction of the pseudo-force.

Pascal’s Law and Hydraulic Lift

Pascal’s law states that pressure applied to a confined fluid is transmitted equally and without reduction to every part of the fluid and the walls of the container.

For a hydraulic lift:

F₁/A₁ = F₂/A₂

Therefore, a small force applied over a smaller area can produce a larger force over a larger area.

For circular pistons:

A ∝ d²

Hence, when piston diameters are given, first convert them into the corresponding area ratio.

If piston weights are also considered, the pressure on each side must include the appropriate external force and piston weight.

When pistons are at different heights, the hydrostatic pressure difference between them must also be considered.

Buoyant Force and Floating Bodies

A fluid exerts pressure on an immersed body from all directions. Since pressure increases with depth, the upward pressure force is greater than the downward pressure force. The resulting upward force is called buoyant force.

The buoyant force is:

FB = ρVg

where V is the volume of fluid displaced by the immersed portion of the body.

Only the submerged volume is used in this expression.

For a floating body:

Buoyant force = Weight of body

or:

ρliquid Vsubmerged g = Mg

The value of g cancels when finding the fraction of a floating body's volume that is submerged. However, the actual buoyant force changes if the effective gravitational acceleration changes.

Equation of Continuity

The equation of continuity follows from the conservation of mass.

For steady flow of an incompressible fluid:

A₁v₁ = A₂v₂

The volume flow rate is:

Q = Av

The mass flow rate is:

Mass flow rate = ρAv

Therefore, when the cross-sectional area of a pipe decreases, the velocity of an incompressible fluid increases.

Quick Revision: Smaller area → greater velocity; larger area → lower velocity.

Bernoulli’s Theorem

For steady flow of an ideal fluid:

P + ρgh + 1/2 ρv² = Constant

The three terms represent pressure energy per unit volume, gravitational potential energy per unit volume and kinetic energy per unit volume.

For a horizontal pipe, the height is the same at both points. Therefore:

P₁ − P₂ = 1/2 ρ(v₂² − v₁²)

This shows that, in a horizontal pipe, higher fluid velocity is associated with lower pressure.

How to Apply Bernoulli’s Theorem

A useful approach is:

1. Use the continuity equation to determine the unknown velocity.

2. Apply Bernoulli’s theorem between the required points.

3. Use Q = Av when the volume flow rate is required.

Venturimeter

A Venturimeter measures fluid flow using the pressure difference between a wider and narrower section of a pipe.

In the wider section:

• Area is larger.

• Fluid velocity is lower.

• Pressure is higher.

In the narrower section:

• Area is smaller.

• Fluid velocity is higher.

• Pressure is lower.

The pressure difference can be determined from the difference in liquid-column heights and related to the velocity difference using Bernoulli’s theorem.

Torricelli’s Law and Water Jet

For a small opening at depth h below the free surface of a liquid, the speed of efflux is:

v = √(2gh)

This result assumes that the area of the tank is much larger than the area of the opening, so the velocity of the liquid surface can be neglected.

Shape of a Falling Water Jet

As a water jet falls, its speed increases due to gravity. Since the volume flow rate remains constant:

A₁v₁ = A₂v₂

Therefore, the cross-sectional area of the falling jet decreases as its velocity increases.

Horizontal Range of a Water Jet

If the total vertical height from the liquid surface to the ground is H, and the hole is at depth h below the surface, the horizontal range is:

R = 2√[h(H − h)]

The range is maximum when:

h = H/2

Thus, the hole should be halfway between the liquid surface and the ground level for maximum range.

Viscosity and Stokes’ Law

Viscosity is the property of a fluid that opposes the relative motion between its layers.

For fluid layers:

F = ηA(v/h)

where:

• η = coefficient of viscosity

• A = area of the layers

• v = relative velocity

• h = separation between the layers

The corresponding shear stress is:

Shear Stress = F/A

For a small spherical body moving through a viscous liquid, Stokes’ law gives:

Fv = 6πηrv

where:

• η = viscosity

• r = radius of sphere

• v = velocity of sphere

Terminal Velocity

When a sphere falls through a viscous liquid, its speed initially increases. As the viscous resistance increases, the net force decreases. Eventually, the sphere moves with constant speed called terminal velocity.

At terminal velocity:

Net Force = 0

Therefore:

Weight = Buoyant Force + Viscous Force

The terminal velocity of a small sphere is:

vt = 2r²g(ρs − ρl)/(9η)

where:

• ρs = density of the sphere

• ρl = density of the liquid

• η = viscosity

Therefore:

vt ∝ r²

and

vt ∝ 1/η

A larger sphere has greater terminal velocity, while greater viscosity reduces terminal velocity.

Quick Revision: At terminal velocity, first write net force = 0 and then balance the forces.

Surface Tension and Surface Energy

Surface tension is the force acting per unit length along the surface of a liquid.

Surface Tension = Force/Length

Surface energy is related to surface tension by:

Surface Energy = Surface Tension × Area

A soap film has two surfaces. Therefore, the force acting on a movable rod of length l is:

F = 2Sl

where S represents surface tension.

Excess Pressure in Drops and Bubbles

For a liquid drop, there is one liquid-air interface. Therefore:

Excess Pressure = 2S/R

For a soap bubble, there are two surfaces:

Excess Pressure = 4S/R

For an air bubble inside water, there is one liquid-air interface:

Excess Pressure = 2S/R

Merging of Liquid Drops

When identical liquid drops merge, volume is conserved.

If n identical drops of radius r merge into one drop of radius R:

R = n^(1/3)r

Since terminal velocity is proportional to the square of radius:

vt ∝ r²

Therefore, the new terminal velocity becomes:

vt,new = n^(2/3) vt,old

When drops merge or split, first apply volume conservation and then compare the required quantities.

Capillary Rise

The height to which a liquid rises or falls in a capillary tube is:

h = 2S cos θ/(rρg)

where:

• S = surface tension

• θ = angle of contact

• r = radius of capillary tube

• ρ = density of liquid

• g = gravitational acceleration

Capillary rise:

• Increases with surface tension.

• Increases with cos θ.

• Decreases with tube radius.

• Decreases with liquid density.

• Decreases with effective gravitational acceleration.

The dependence is on cos θ, not directly on the value of θ.

If the calculated height of rise is greater than the available length of the capillary tube, the liquid can rise only up to the end of the tube.

Key Takeaways for NEET 2026

• Pressure is force per unit area and acts normally to a surface.

• Hydrostatic pressure increases with depth according to P = P₀ + ρgh.

• Pascal’s law forms the basis of hydraulic lifts.

• Buoyant force depends on the density of the fluid and the volume displaced.

• For steady incompressible flow, A₁v₁ = A₂v₂.

• Bernoulli’s theorem connects pressure, velocity and height in ideal fluid flow.

• Torricelli’s law gives the efflux speed as √(2gh).

• Stokes’ law is used for the viscous force on a small sphere.

• At terminal velocity, the net force on the falling sphere is zero.

• Excess pressure is 2S/R for a liquid drop and 4S/R for a soap bubble.

• Capillary rise depends on surface tension, contact angle, tube radius, liquid density and effective gravity.

Mechanical Properties of Fluids is an important Physics topic that covers pressure, buoyancy, fluid flow, viscosity, surface tension and capillarity. Revise hydrostatic pressure, Pascal’s law, continuity, Bernoulli’s theorem, Torricelli’s law, Stokes’ law, terminal velocity, excess pressure and capillary rise through regular formula revision, numerical practice and concept-based learning. Physics Wallah covers these concepts with focused explanations and question practice to help strengthen understanding and improve accuracy in NEET 2026 Physics preparation. 

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