Work, Energy and Power V

TO PROVE WORK DONE AGAINST CONSERVATIVE FORCES IS A STATE FUNCTION Consider a body of man m which is required to be lifted up to height h. This can be done in 2 ways. (i) By directly lifting the body against gravity

(ii) By pushing the body up a smooth inclined plane.

Min force required to lift the body of mass m vertically is

                                                               F = mg

And displacement of body in lifting is s

                                                              s = h

Hence work done in lifting is            

                                                             W₁ = FsCos0

                                                             W₁ = mgh

Now we consider the same body lifted through height h by pushing it up a smooth inclined plane

Min force required to push the body is 

                                                             F = mgSinθ 

And displacement of body in lifting is 

                                                            s = h/Sinθ 

Hence work done in pushing is W₂ = FsCos

or,                                               W₂ = mgSinθ. h/Sinθ. 1

or,                                               W₂ = mgh 

From above W₁ = W₂ we can say that in both the cases work done in lifting the body through height ‘h’ is same. 

To Prove That Work Done Against Conservative Forces Is Zero In A Complete Cycle

Consider a body of man m which is lifted slowly through height h & then allowed to come back to the ground slowly through height h.

For work done is slowly lifting the body up, Minimum force required in vertically upward direction is

F = mg

Vertical up displacement of the body is

                                     s = h

Hence work done is

                                  W = FsCosθ

or,                              W₁ = FsCos0

or,                              W₁ = mgh (since force and displacement are in same direction)

For work done is slowly bringing the body down,

Minimum force required in vertically upward direction is

                                   F = mg

Vertical down displacement of the body is

                                   s = h

Hence work done is

or,                            W₂ = Fs Cos180(since force and displacement are in opposite direction)

or,                            W₂ = - mgh

Hence total work done against conservative forces in a complete cycle is

                                 W = W₁ + W₂ 

or,                             W = (mgh) + (-mgh)

or,                             W = 0

NON-CONSERVATIVE FORCES

Non conservative forces are the forces, work done against which does not get conserved in the body in the form of potential energy.

PROPERTIES OF NON-CONSERVATIVE FORCES

1. Work done against these forces does not get conserved in the body in the form of P.E.

2. Work done against these forces is always dissipated by being converted into non usable forms of energy like heat, light, sound etc.

3. Work done against non-conservative force is a path function and not a state function.

4. Work done against non-conservative force in a complete cycle is not zero.

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