Conservation of Mechanical energy

Introduction :
Conservation of mechanical energy states that the whole mechanical power (sum of kinetic energy+ potential energy) of a structure remains stable provided the forces acting on the body are conservative. The standard of conservation of mechanical energy states, the whole mechanical energy of a conservation system leftovers constant.

Proof of Conservation of Mechanical energy

Suppose a particle of mass m moves from position x₁ to another position x₂ under the action of conservation force. As a result its velocity changes from v₁ to v₂. By work-energy theorem, we have

If V₁ and V₂ are the potential energies at positions x₁ and x₂, we have

If effort done by a conservative force is positive, the kinetic energy of body increases and potential energy of body decrease to stay total mechanical energy conserved.

Example of Freely Falling Body

Gravitational force is conservation force. When a body of mass m, initially at rest at height H, above the ground, falls under gravity, then its total mechanical energy remains constant.

At heighest point A:
The body is at rest, therefore kinetic energy,

Potential energy, V=mgh
Total mechanical energy at A=K+V=0+mgH=mgH

At point B:
Let the body reach at intermediate point B at distance x below point.

Potential energy=mg(H-x)

At point C:
Let C be the point on the ground. If v_C is the velocity at point C, then relation
V²=u²+2gs gives v²_C =0 +2gH=2gH
Kinetic energy,

Potential energy = 0
Total mechanical energy at C=mgH=0=mgH
Thus it is clear that for a freely falling body, the total mechanical energy remains constant.

Example of Mass-Spring System in Conservation of Mechanical Energy

Consider a spring-block system located on a horizontal frictionless table. Mass of block is M and force stable of spring is K. when mass is in position A, spring is in its usual length. So that elastic PE of spring is zero. When mass is taken to position B, the spring is stretched by an amount x₀, so elastic potential energy of spring (= (1/2 )kx₀²) and kinetic energy in this position = zero.