# mechanics

## Gravitational Self energy of a Uniform hollow and solid sphere – Definition and Derivation .

It is the energy possessed by a body due to the interaction forces inside the body . This can be defined as the work done by external agent in assembling all the particles of a body in a definite shape and size or it is the work done in creating a body.

– gravitational self energy of a hollow sphere of mass M and radius R is given as
Uself = – GM²/2R

– gravitational self energy of the solid sphere of mass M and radius R is given as
Uself = – 3GM²/5R ## Torricelli’s theorem ( Velocity of efflux ) : Principle , Derivation and sloved examples

According to Torricelli’s theorem -The velocity of efflux of a liquid issuing out of an orifice is the same as it would attain if allowed to fall freely through the vertical height between the liquid surface and orifice .
velocity of efflux v = ( 2gh )1/2

Note – The velocity of efflux is the velocity of escaping liquid relative to the container (but not necessarily relative to ground when the container moves ) .

## Force of reaction due to ejection of liquid – Derivation, Equation and Solved examples

If the velocity of a liquid (or gas) of density p coming out through an opening of area of cross section a is v, then there will be a thrust force F = ρav² due to liquid coming out the opening. The direction of the thrust force will be just opposite to the velocity direction . ## Newton’s law of Universal Gravitation | Definition , Equation , Facts and Solved Examples.

The magnitude of the gravitational force F that two particles of masses  m1 and m2 , separated by a distance r ,  there exists an attraction , which is proportional to the product of their masses and inversely proportional to the square of distance between the exert force on each other is given by
F = Gm1m2 / r²
Where G is a constant , Called the universal gravitation constant . It’s value in SI units is G = 6.674×10-11 N.m²/kg². ## Kepler’s laws of planetary motion | Definition , Diagrams , Equations ,Facts and Solved Examples .

Johannes Kepler used empirical observations, and sophisticated calculations to arrive at the famous Kepler’s laws of planetary motion, published in 1609 and 1619. What is particularly significant about Kepler’s laws is that they challenged the prevailing world’s view, with the Earth in the center of the universe (a geocentric theory) and the Sun and all the planets and stars orbiting around it, just as the Moon does . ## Equation of continuity in fluid mechanics – Explanation , Derivation and Solved examples

The continuity equation is a mathematical expression of the law of conservative of mass in fluid dynamics .
This equation defines the steady flow of a fluid in a tube. It states that if flow of a fluid is steady then the mass of the fluid entering  per second at one end is equal to the mass of fluid leaving per second at the other end .