# Work, Energy and Power VI

PROVE THAT WORK DONE AGAINST NON–CONSERVATIVE FORCES IS A PATH FUNCTION
Consider a body of mass (m) which is required to be lifted to height ‘h’ by pushing it up the rough incline of inclination.
Minimum force required to slide the body up the rough inclined plane having
coefficient of kinetic friction µ with the body is
F = mgSinθ + fk
or,                                       F = mgSinθ + µN
or,                                       F = mgSinθ + µmgCosθ
Displacement of the body over the incline in moving through height h is
s = h/Sinθ
Hence work done in moving the body up the incline is
W = F.s.Cos0 (since force and displacement are in opposite direction)
or,                                     W = (mgSinθ + µmgCosθ). h/Sinθ.1
or,                                     W = mgh + µmgh/Tanθ
Similarly if we change the angle of inclination from θ to θ1, then work done will be
W1 = mgh + µmgh/Tanθ1
This clearly shows that work done in both the cases is different & hence work done against non-conservative force in a path function and not a state function i.e. it not only depends upon initial & final states of body but also depends upon the path through which process has been carried out.

To Prove That Work Done Against Non-conservative Forces In A Complete Cycle Is Not Zero
Consider a body displaced slowly on a rough horizontal plane through displacement
s from A to B.
Minimum force required to move the body is
F = fk = µN = µmg
Work done by the body in displacement s is
W = F.s.Cos0(since force and displacement are in same direction)
or,                                                     W = µmgs
Now if the same body is returned back from B to A
Minimum force required to move the body is
F = f= µN = µmg
Work done by the body in displacement s is
W = F.s.Cos0(since force and displacement are in same direction)
or,                                                   W = µmgs
Hence total work done in the complete process
W = W1 + W2 = 2µmgs
Note:- When body is returned from B to A friction reverse its direction.

POWER

Rate of doing work by a body with respect to time is known as power.

Average Power
It is defined as the ratio of total work done by the body to total time taken.
Pavg = Total work done / Total time taken = ∆W / ∆t

Instantaneous Power
Power developed within the body at any particular instant of time is known as instantaneous power.
Or
Average power evaluated for very short duration of time is known as instantaneous power.
EFFICIENCY
It is defined as the ratio of power output to power input.
Or
It is defined as the ratio of energy output to energy input.
Or
It is defined as the ratio of work output to work input.

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