Proof:
Let a mass (m) to move along a curve in space from initial position P1 with co-ordinates ( x1 , y1 , z1 ) to final position P2 is given by ( x2 , y2 , z2 ) as shown in Figure, where XOY is a horizontal plane and Z-axis is pointing vertically upward.
Let z1 > z2 .
Gravitational force, m × g, does work on the body along vertical downward distance of
(z1 - z2 ) .
If the velocity (V1) of body at P1 increases to (V2) as it comes to point P2, by using work energy equations, we can write down :
WD on the body = Increase in its KE
∴ mg ( z1 - z2 ) = (m (V2)2) /2 - (m (V1)2)/2
∴ mg ( z1 ) + (m (V1)2) /2 = mg z2 + (m (V2)2) /2
or,
(Wz1 + (W/g)((V1 )2 /2)= (Wz2 + (W/g)((V2 )2 /2)=
i.e., (PE + KE) at position P1 = (PE + KE) at position P2.
The above equation shows that in a conservative field, the total mechanical energy of a body at any point remains constant. In engineering mechanics of rigid bodies, this mechanical energy is partly potential energy and partly kinetic energy.
In case, a body subjected to a spring force, is undergoing a motion, the W. D. on the body again depend only on the component of displacement only along the axis of the spring and shall be again independent of the path of motion of the body. Thus, this spring force along a given axis is also a conservative force. Therefore, both gravitational field and the spring force field are conservative as these are unidirectional forces.