A block of mass m is pulled slowly by a minimum constant force on a rough (`mu`) horizontal surface through a distance x. Find work done by force F.

Explanation

The work done by the force F is

`W=int vecF_(min)*dvecs=F_(min)xcostheta` (i)
For slowly pulling the block.
`Fcos theta=muN` (ii)
In vertical direction
`Fsintheta+N-mg=0`
`N=Fsintheta-mg` (iii)

Using Eqs. (ii) and (iii),
`F=(mumg)/(sintheta+mucostheta)`
F is minimum when `tan theta=mu`
`F_(min)=(mumg)/(sqrt(1+mu^2))` (iv)
Using Eqs. (i) and (iv),
`W=(mumgx)/(sqrt(1+mu^2))*(1)/(sqrt(1+mu^2))=(mumg)/(1+mu^2)`