stokes law for terminal velocity formula derivation physics class 11 cbse example

know stokes law for terminal velocity formula derivation physics class 11 cbse example ?

Fluid Pressure

Pressure is defined as the force exerted normally on a unit area of the surface of a fluid and is given by

P = F/A

In the SI system, the unit of pressure is newton per square metre (Nm^–2) which is also called Pascal (Pa). Thus

1 Pa = 1 Nm^–2

Pascal’s Law

Blaise Pascal (1623–1662), a French scientist, discovered a principle which tells us how force (or pressure) can be transmitted in a fluid. Pascal’s law states that pressure in a fluid in equilibrium is the same eveywhere (if the effect of gravity can be neglected). The pressure difference between two points in a fluid is either zero (if they are at the same horizontal level) or is a definite quantity depending on their height difference. So if the pressure at some point in a fluid is changed, there will be an equal change in pressure at any other point. Thus fluids, especially liquids (because they are incompressible) are ideal for transmitting pressure. This fact is used in hydraulic machines, such as hydraulic press, hydraulic brakes, hydraulic jacks, etc.

Density and Relative Density

The density of a substance is defined as the mass per unit volume of the substance. The SI unit of density is kilogram per cubic metre (kg m^–3).

1 g cm^–3 = 1000 kg m^–3

The relative density of a substance is the ratio of its density to that of water, i.e.

Being a ratio of two similar quantities, relative density is just a number; it has no units.

Atmospheric Pressure

Like all gases, air also has weight and hence exerts pressure. Just as water pressure is caused by the weight of water, the weight of all the air above the earth causes an atmospheric pressure. The atmosphere exerts this pressure not only on the earth’s surface, but also on the surface of all objects on the earth including living beings. The atmospheric pressure at sea level is

P₀ = 1.01 x 10⁵ Pa

Hydrostatic Pressure Pressure exists everywhere within a fluid. The hydrostatic pressure at a depth h below the surface of a fluid is given by

where p is the density of the fluid and g, the acceleration due to gravity. The pressure is the same at all points at the same horizontal level. The pressure at any point in a fluid contained in a vessel is independent of the shape or size of the vessel

Gauge Pressure

The pressure at any point in a fluid is equal to the sum of the atmospheric pressure P₀  acting on its surface and the hydrostatic pressure hpg due to the weight of the fluid above that point which is at a depth h below the surface of the fluid. This is called the gauge pressure which is given by

P = P₀ + hpg

Since liquids are incompressible, the density of a liquid is constant throughout the liquid.

Viscosity

When a fluid flows , there exists a relative motion between the layers of the fluid. Internal force acts which destroys this relative motion. This force is called viscous force. The viscous force between two layers of a fluid is given by

F = – η A dv/dx

Where A is the area of the layer, dv/d x is the velocity gradient and η is called the coefficient of viscosity of the fluid. The SI unit of h is Nsm^–2 which is called poiseuilli (Pl) or pascal second (Pa-s). The dimensional formula of η is

[ML^–1 T^–1].

Stokes’ Law

The viscous force experienced by a small spherical body of radius r moving with a small velocity v through a homogeneous fluid of coefficient of viscosity η is given by

F = 6 π ηrv This relation is called Stokes’ law.

Terminal Velocity

If a body is released in a viscous fluid, it is accelerated due to gravity and its velocity begins to increase. Hence viscous force on it also increases. A stage is reached when the velocity is such that the viscous force F becomes equal to (W – U ), where W is the weight of the body and U is the up thrust (Fig. below). Then no net force acts on the body and it falls with a constant velocity called the terminal velocity (v_t ). For a spherical body of radius r and density p falling in a fluid of density σ and coefficient of viscosity η, the terminal velocity is given by

For a body moving in a given fluid, V_t ∝ r²

Poiseuilli’s Formula

The volume of a liquid flowing per second through a capillary tube of radius r when its ends are maintained at a pressure difference p is given by

Where l is the length of the tube and η is the coefficient of viscosity of the liquid.

Capillaries Connected in Series

If two capillaries of lengths l₁ and l₂ and radii r₁ and r₂ are connected in series across constant pressure difference p, then the fluid resistance R is given by

As the volume of liquid flowing per second is the same through both capillaries.

If p₁ and p₂ are the pressure differences across individual capillaries, then

p = p₁ + p₂

Capillaries Connected in Parallel

If two capillaries are connected in parallel across constant pressure difference p, then the fluid resistance is given by

The volume of liquid flowing per second through the capillary of radius r₁ is

For the capillary of radius r₂, we have

Streamline or Laminar Flow

Streamline or Laminar flow is the flow in which every particle of the liquid follows the same path and has exactly the same velocity in magnitude and direction as that of the preceding particle at a given point in the flow. The actual path followed by the particles in a regular flow is called a streamline, which can be straight or curved. The tangent at a point on a streamline gives the direction of the liquid flow at that point.

Critical Velocity and Reynold’s Number

The liquid flow remains steady or streamline if its velocity does not exceed a limiting value called the critical value, which is given by

where η = coefficient of viscosity of the liquid, p = density of the liquid, r = radius of the pipe in which the liquid flows and k is a dimensionless constant called Reynold’s number. If the velocity of the liquid exceeds the critical velocity, the flow becomes irregular causing the liquid to flow in a disorderly fashion. Such a flow is called turbulent flow. The value of k is usually very high. If k is less than 2000, the flow is streamline. If the value of k exceeds 2000, the flow becomes turbulent.

Equation of Continuity of Flow

If a1 and a2 are the areas of cross-section at two sections of a tube of a variable cross-section and v 1 and v2 are the velocities of flow crossing these sections, then

a₁v₁ = a₂v₂ or av = constant

This means that smaller the area of cross-section, higher is of the liquid flow and vice versa. This is called equation of continuity of flow and it holds only if the flow is streamline.

Bernoulli’s Theorem

Bernoulli’s theorem states that the total energy of an incompressible and non-viscous liquid in a streamline flow remains constant throughout the flow; the total energy being the sum of pressure energy, potential energy and kinetic energy of the liquid

Velocity of Efflux

A liquid is filled up to a height H in a vessel which has a small hole at a depth h below the surface of the liquid (Fig. below).

When the hole is unplugged, the velocity v with which the liquid comes out of the hole is called the velocity of efflux. Let V be the velocity with which the free surface of the liquid falls in the vessel. Applying Bernoulli’s theorem to points A and B

Since P_A = P_B = P₀ ( atmospheric pressure) and AV = av (equation of continuity) where A = cross-sectional area of the vessel and a = cross-sectional area of hole, we have

The time taken by the liquid emerging from the hole to hit the ground is

Surface Tension and Surface Energy

Surface tension is the force acting per unit length of an imaginary line on a liquid surface; the direction of the force being perpendicular to the line and tangential to the liquid surface. The SI unit of surface tension is N m^–1 and its dimensional formula is [M L⁰ T^–2]. Consider a frame ABCD having a wire PQ of length L which can slide along sides AB and CD. The frame is dipped in a liquid (e.g. soap solution) and taken out. We get a film of liquid within PBCQ (Fig. below). Since the film has two surfaces each of length L, the force due to surface tension acting on wire PQ is

This force is directed inwards to the left and has to be applied to the right to hold the wire PQ in place. Hence, if the area of the film has to be increased, work has to be done against the force of surface tension. This work is stored as potential energy called surface energy. Work done to move the wire from a position PQ to a position P’Q’ is

where ∇A = increase in the surface area of the film. Thus work done = surface tension x increase in surface area of the film. Another SI unit of surface tension is Jm^–2.

Excess Pressure

When the free surface of a liquid is curved, there is a difference of pressure between the liquid side and the vapour side of the surface.

(i) Excess pressure inside a liquid drop of radius r is given by

where σ is the surface tension of the liquid.

(ii) Excess pressure inside a liquid bubble of radius r is given by

(iii) Excess pressure inside an air bubble of radius r in a liquid of surface tension σ is given by

If the pressure outside is P, the total pressure inside

Work Done in Blowing a Soap Bubble

Suppose the radius of a soap bubble is increased from r1 to r2 by blowing. Then, since there are two surfaces of

(iii) Work done in splitting a drop of radius R into n identical drops, each of radius r, is obtained as follows:

Angle of Contact

The shape of meniscus of water in a narrow glass tube is concave upwards [Fig. below (a)] while the shape of meniscus of mercury in a narrow glass tube is convex upwards [Fig. below (b)]

The angle of contact (θ) between a liquid and a solid surface is defined as the angle between the tangent to the liquid surface at the point of contact and the solid surface inside the liquid. The value of angle of contact depends upon (i) the nature of the liquid and solid in contact, (ii) the nature of the medium above the free surface of the liquid and (iii) the temperature of the liquid.

note : Generally the angle of contact for liquids which wet glass is acute and obtuse for liquids which do not wet glass.

Capillarity

The rise or fall of a liquid in a capillary tube is known as capillarity. The height to which a liquid of surface tension σ and density p rises in a capillary tube of radius r is given by

where θ is the angle of contact. For pure water and clean glass, θ ≈ 0° in which case cos θ ≈ cos 0° = 1 and we have

For mercury and glass, θ ≈ 140° so that cos θ is negative. Hence mercury falls in a capillary tube, i.e. the level of mercury in the capillary tube is lower than the level outside.

note : Surface tension of a liquid decreases with increase in temperature.