Revision: Properties of Bulk Matter >> Mechanical Properties of Fluids Physics Science (English Medium) Class 11 CBSE

The force which produces compression is called thrust. Its S.I unit is the newton.

SI unit of pressure is the pascal (Pa) or Nm⁻²
One Pascal: When a force of one newton acts normally on an area of one square metre (1 m²) then the pressure acting on the surface acting on the surface is called one Pascal.

The maximum constant velocity acquired by a body while falling freely through a viscous medium is called the terminal velocity V_T.

The coefficient of viscosity of a liquid is defined as the viscous force acting tangentially per unit area of a liquid layer having a unit velocity gradient in a direction perpendicular to the direction of flow of the liquid.

The rate of change of velocity (dv) with distance (dx) measured from a stationary layer is called velocity gradient.

∴ Velocity gradient = `(dv)/dx`

Surface tension is defined as the force per unit length acting at right angles to an imaginary line drawn on the free surface of the liquid. 

The angle between tangents drawn at the point of contact to the liquid surface and the solid surface inside the liquid is called the angle of contact for a pair of solid and liquid. It is denoted by θ.

Surface tension is defined as the force acting on a unit length of an imaginary line drawn on the free surface of the liquid, the direction of the force being perpendicular to the line so drawn and acting parallel to the surface.

When a liquid is in contact with a solid, the angle between the tangent drawn to the free surface of the liquid and the surface of solid at the point of contact measured inside the liquid is called the angle of contact.

The constant maximum velocity acquired by a body while falling through a viscous fluid is called terminal velocity.

The resultant of all the forces exerted by a fluid on a body partly or wholly dipped in it, due to hydrostatic pressure, is called upthrust or buoyancy.

The principle which states that for a non-viscous liquid in streamline flow passing through a tube of varying cross-section, the product of the area of cross-section and the velocity of flow remains constant at every point is called the Equation of Continuity.

If \[\lim_{x\to a^{+}}f\left(x\right)\neq\lim_{x\to a^{-}}f\left(x\right),\] then f(x) is said to be non-removable discontinuous.

If \[\lim_{x\to a^{-}}f\left(x\right)=\lim_{x\to a^{+}}f\left(x\right)\neq f\left(a\right),\] then f(x) is said to be removable discontinuous.

A function f(x) is said to be continuous at a point x = a, if the following three conditions are satisfied

1. f is defined at every point on an open interval containing a.
2. \[\lim_{x\to a}f\left(x\right)\] exists.
3. \[\lim_{x\to a}f\left(x\right)=f\left(a\right)\].

v = \[\frac{2}{9}\cdot\frac{r^2(\rho-\sigma)g}{\eta}\]

where:

• v = terminal velocity
• r = radius of the body
• ρ = density of the body
• σ = density of the fluid
• g = acceleration due to gravity
• η = coefficient of viscosity of the fluid

For a non-viscous liquid in streamline flow passing through a tube of varying cross-section:

av = constant

or equivalently:

a ∝ \[\frac {1}{v}\]

where: 

• a = area of cross-section of the tube
• v = velocity of flow of the liquid

(b) There is no gravitational force acting downwards. However, when the starting velocity is 20 m/s, the viscous force, which is directly proportional to velocity, becomes maximum and tends to accelerate the ball upwards.

\[\text{ When the ball falls under gravity, }\]

\[\text{ neglecting the density of air: } \]

\[\text{ Mass of the sphere = m }\]

\[\text{ Radius = r }\]

\[\text{ Viscous drag coeff . }= \eta\]

\[\text{Terminal velocity is given by}: \]

\[\text{ mg  }= 6\pi\eta r v_T \]

\[ \Rightarrow \frac{6\pi\eta r v_T}{m} = g . . . (1)\]

\[\text{ Now, at terminal velocity, the acceleration of the ball due to the viscous force is given by: } \]

\[a = \frac{6\pi\eta r v_T}{m}\]

\[\text{ Comparing equations (1) and (2), we find that : } \]

\[ \text{ a = g }\]

Thus, we see that the initial acceleration of the ball will be 9.8 ms - 2  .

(c) The velocity of the ball will decrease with time because of the upward viscous drag. As the force of viscosity is directly proportional to the velocity of the ball, the acceleration due to the viscous force will also decrease.

(d) When all the kinetic energy of the ball is radiated as heat due to the viscous force, the ball comes to rest. 

"When a body is partly or wholly dipped in a fluid, the fluid exerts a force on the body due to hydrostatic pressure. At any small portion of the surface of the body, the force exerted by the fluid is perpendicular to the surface and is equal to the pressure at that point multiplied by the area. The resultant of all these constant forces is called upthrust or buoyancy."

• Hydraulic Press — Two cylinders (C & D) filled with liquid; small force applied on piston P₁ (smaller area A₁) is converted into a very large upward force on piston P₂ (larger area A₂), since A₂ > A₁.
• Hydraulic Lift — Works on Pascal's Law to lift or support heavy objects such as cars and trucks using liquid pressure.
• Hydraulic Brakes — Small force on the brake pedal is instantly transmitted equally through brake fluid to all cylinders, producing a large thrust on the wheels to stop the vehicle.

• A highly soluble impurity increases surface tension, while a partially soluble impurity (e.g., detergent) decreases it; a waterproofing agent increases it.
• Surface tension decreases with increase in temperature, given by T = T₀(1 − αθ), where T₀​ is surface tension at 0°C and α is the temperature coefficient.
• When a soap bubble is charged (positively or negatively), force acts outward on the surface, increasing its radius — thus electrification always decreases surface tension.