State and establish principle of conservation of mechanical energy .

Explanation

Let this principle will be obtain in one dimension .
If a body undergoes distance `Deltax` by conservative force F then according to work energy (WE) theorem ,
`DeltaK = F (x) Deltax " "….(1)`
but according to definition of potential energy for conservative force ,
`-DeltaV = F(x)Deltax " "....(2)`
From equation (1) and (2) ,
`DeltaK = - DeltaV `
` :. DeltaK + DeltaV = 0 `
` :. Delta(K+V) =0`
` :. ` The change of kinetic energy and potential energy is zero . That means sum of kinetic energy for conservative force ,
` -DeltaV = F(x) Deltax " "....(2)`
From equation (1) and (2) ,
`DeltaK+DeltaV = 0 `
` :. Delta(K+V) = 0 `
` :. ` The chance of kinetic energy and potential energy is zero . That means sum of kinetic energy and potential energy is a constant .
` :. K+V ` = constant
Over the whole path , `x_(i) ` to `x_(f)` this means that
`K_(i) +V(x_(i))=K_(f) +V(x_(f))`
` :. ` The quantity `K + V(x) ` is called the total mechanical energy of the system .
The kinetic energy K and potential energy V(x) is called the total mechanical energy of the system .
The kinetic energy K and potential energy V(x) may vary from point to point but the sum remains constant .
Mechanical energy is conserved in conservative field . That means as we go above and above form earth.s surface the kinetic energy and potential energy increases .
The principle of total mechanical energy is written as under :
"The total mechanical energy of a system is conserved if the forces doing work on it are conservative ".
"Under influence of conservative forces the mechanical energy of a mechanically isolated " system remain constant.