**NEET Physics Notes Mechanics-Properties of Matter-Stoke’s Law**

**Stoke’s Law**

**Stokes’ Law**

Stokes proved that for a small spherical body of radius r moving with a constant speed v called terminal velocity through a fluid having coefficient of viscosity η the viscous force F is given by **F = 6πηrv** It is known as the Stokes’ law.

**Terminal Velocity**

If a small spherical body is dropped in a fluid, then initially it is accelerated under the action of gravity. However, with an increase in speed, the viscous force increases and soon it balances the weight of the body.

Now, the body moves with a constant velocity, called the terminal velocity. Terminal velocity vt is given by

**Streamline and Turbulent Flow**

Flow of a fluid is said to be streamlined, if each element of the fluid passing through a particular point travels along the same path, with exactly the same velocity as that of the preceding element. A special case of streamline flow is laminar flow.

A turbulent flow is the, one in which the motion of the fluid particles is disordered or irregular.

For a fluid, the critical velocity is that limiting velocity of the fluid flow upto which the flow is streamlined and beyond which the flow becomes turbulent. Value of critical velocity for the flow of liquid of density ρ and coefficient of viscosity η, flowing through a horizontal tube of radius r is given by \(V_{c} \propto \frac{\eta}{\rho r}\)

**Reynolds’ Number (NR)**

It is a unitless and dimensionless number given by

A smaller value of Reynolds’ number (generally NR < 1000) indicates a streamline flow but a higher value (NR > 1500) indicates that the flow is turbulent and between 1000 to 1500, the flow is unstable.

**Equation of Continuity**

Let us consider the streamline flow of an ideal, non-viscous fluid through a tube of variable cross-section. Let at the two sections, the cross-sectional areas be At and A2, respectively and the fluid flow velocities are and v2, then according to the equation of continuity \(A_{1} v_{1} \rho_{1}=A_{2} v_{2} \rho_{2}\)

**Energy of a Flowing Liquid**

There are three types of energies in a flowing liquid.

**Pressure Energy**

If p is the pressure on the area A of a fluid, and the liquid moves through a distance / due to this pressure,

then

**Kinetic Energy**

If a liquid of mass m and volume V is flowing with velocity v, then the kinetic energy is = \(\frac{1}{2} m v^{2}\)

**Potential Energy**

If a liquid of mass m is at a height h from the reference line (h = 0), then its potential energy is mgh.

Potential energy per unit volume of the liquid

**Bernoulli’s Theorem**

According to the Bernoulli’s theorem for steady flow of an incompressible, non-viscous fluid through a tube/pipe, the total energy (i.e. the sum of kinetic energy, potential energy and pressure energy) per unit volume (or per unit mass too) remains constant at all points of flow provided that there is no source or sink of the fluid along the flow.

If a liquid is filled in a vessel up to a height H and a small orifice O is made at a height h, then from Bernoulli’s theorem it can be shown that velocity of efflux v of the liquid from the vessel is

The flowing fluid describes a parabolic path and hits the base level at a horizontal distance (called the range)

**Limitations of Bernoulli’s Theorem**

- When a fluid is at rest, i.e. its velocity is zero everywhere, Bernoulli’s equation becomes

- Bernoulli’s equation ideally applies to fluids with zero viscosity or non-viscous fluids.
- Bernoulli’s equation applies to fluids which must be incompressible, as the elastic energy of the fluid is also not taken into consideration.
- Bernoulli’s equation does not hold for steady or turbulent flows, because in that situation velocity and pressure are constantly fluctuating in time.
- Bernoulli’s equation for flowing liquid is

**Surface Tension**

Surface tension is the property of a liquid diie to which its free surface behaves like a stretched elastic membrane and tends to have the least possible surface area.