This principle state that if only the conservative force are doing work on a body, then its total mechanical energy (kinetic energy + potential energy) remains constant.
Let a body undergoes displacement ∆x under the action of conservative force F(x), then from work energy theorem, the change in Kinetic Energy is
∆K = F(x).∆x (I)
As the force is conservative the change in Potential energy is given by
∆U = Negative of work done = – F(x).∆x (II)
Combining the above two equations, We get
Although, individually the Kinetic Energy K and Potential Energy U may change from one state of the system to another, but their sum of the total Mechanical Energy of the system remains constant under the conservative force.
Mechanical Energy Of A Stretched Spring
If the block of mass m is extended to position Xm and released from rest, then it’s total Mechanical Energy at any arbitrary point X, where X is lies between – Xm and Xm.
At any intermediate position X between -Xm and Xm, the energy is partially Kinetic Energy and Potential Energy.