The radius of gyration of a body about a given axis is the perpendicular distance of a point P from the axis, where if whole mass of the body were concentrated, as the body shall have the same moment of inertia as it has with the actual distribution of mass. 

Radius of gyration is represented by K. 

Consider a liquid drop of radius R and surface tension T. 

Due to surface tension the molecules on the surface film experience the net force in inward direction normal to the surface.

Therefore there is more pressure inside than outside.

Let p_i be the pressure inside the liquid drop and p_o be the pressure outside the drop. 

Therefore excess of pressure inside the liquid drop is,

                        p = p₁– p₀

Due to excess of pressure inside the liquid drop the free surface of the drop will experience the net force in outward direction due to which the drop will expand.

Let the free surface displace by dR under isothermal conditions. 

Therefore excess of pressure does the work in displacing the surface and that work will be stored in the form of potential energy.

The work done by excess of pressure in displacing the surface is, 

dW  = Force x displacement

       = (Excess of pressure x surface area) x displacement of surface
       
       

Increase in the  potential energy is,

dU = surface tension x increase in area of the free surface

    

The above expression gives us the pressure inside a liquid drop. 

Derive an expression for work done in isothermal process

Statement: For the streamline flow of non-viscous and incompressible liquid, the sum of potential energy, kinetic energy and pressure energy is constant. 

Proof: 
 

Let us consider the ideal liquid of density ρ flowing through the pipe LM of varying cross-section.

Let P₁ and P₂ be the pressures at ends L and M and A₁ and A₂ be the areas of cross-sections at ends L and M respectively.

Let the liquid enter with velocity V₁ and leave with velocity v₂. 

Let A₁ > A₂. 

Now, by equation of continuity,

                      

Since, A₁ > A₂

Therefore, 

v₂ > v₁ and P₁ > P₂ 

Let, m be the mass of liquid enetring at end L in time t. 

The liquid will cover a distance = v₁t 

Therefore, the work done by pressure on the liquid at end L in time t is, 

W₁ = Force  

      = P₁A₁v₁t           ... (1) 

Since same mass m leaves the pipe at end M in same time t, in which liquid will cover the distance given by v₂t.

Therefore, work done by liquid against the force due to pressure P₁ is given by, 
 
        W₂ = P₂A₂v₂t        ... (2) 

Net ork done by pressure on the liquid in time t is, 

W = W₁ - W₂ 

    =P₁A₁v₁t  - P₂A₂v₂t        ... (3) 

This work done on liquid by pressure increases it's kinetic energy and potential energy. 

Increase in K.E of liquid is, 


That is, 

 

i) It is an orderly type of motion in which the liquid flows in parallel layers while, turbulent motion is disorderly type of motion.
ii) Every particle of the liquid follows the path of it's preceeding particle and travel with the same velocity in magnitude and direction whereas, the motion of particles of the liquids becomes different at different points in turbulent flow.
iii) The velocity of streamline flow is less than critical velocity but, in turbulent motion the liquid moves with a velocity greater than the critical velocity of the liquid.
iv) Streamline flow is laminar whereas, turbulent flow is non-laminar.

STP : Standard temperature and pressure, abbreviated STP, refers to nominal conditions in the atmosphere at sea level. This value is important to physicists, chemists, engineers, and pilots and navigators.
Standard temperature is defined as zero degrees Celsius (0 0C), which translates to 32 degrees Fahrenheit (32 0F) or 273.15 degrees kelvin (273.15 0K). This is essentially the freezing point of pure water at sea level, in air at standard pressure.
NTP : Normal Temperature and Pressure - is defined as air at 20oC (293.15 K, 68oF) and 1 atm ( 101.325 kN/m2, 101.325 kPa, 14.7 psia, 0 psig, 29.92 in Hg, 760 torr). Density 1.204 kg/m3 (0.075 pounds per cubic foot