The work delivered by force equals `W = f*x` , when x is the displacement. However, when F is not constant with respect to x, the work cannot be calculated as the product of the force and displacement. However, this relationship is still true for a work `DeltaW` done on a very small displacement `Deltax` , because for a very small change in x, the force is approximately constant:
`DeltaW = fDeltax` , or in differential notation, `dW = Fdx`
The total work would then be the sum of the work done on small displacements, or the integral:
`W = int fdx` .
The given force is `f = kx^(-2/3)` , so the work would be
`W = int_0 ^x kx^(-2/3) dx = 3kx^(1/3)`
The power delivered is the rate at which the work is done:
`P = (dW)/(dt) = (d(dkx^(-1/3)))/(dt) = kx^(-2/3) (dx)/(dt)` , where dx/dt is the velocity.
The power will be proportional to x^(-2/3) and v = dx/dt.
The power delivered by the force can be calculated as the ratio of work done per unit time. That is:
power = work done/time
Work done by the force is the product of force and the displacement. That is,
Work done = force x displacement.
Hence, power = force x displacement/time
The force is a function of displacement (x) and is given as:
`f = kx^(-2/3)`
Thus, the power generated by the force will be given as:
`power = fx/t = (kx^(-2/3)x)/t = (kx^(1/3))/t`
Thus, the power generated by the force will be directly proportional to the displacement (raised to the power 1/3) and is inversely proportional to time t. In other words,
`Power alpha x^(1/3)`
`Power alpha 1/t`
Thus, given the displacement and the time taken, we can calculate the power generated by a certain force on a body.
Hope this helps.