A block of mass `m` is slowly pulled along a curved surface from position 1 and 2. If the coefficient of kinetic friction between the block and surface is `mu`, find the work done by the applied force.

Explanation

The forces acting on the block are gravity,
`mgdarr`, `Nuarr`, `f_kdarr`, and `Fuarr`.
Using W-E theorem: `W=DeltaK`

`W_N+W_(gr)+W_N=W_f=DeltaK` (i)
For slowly shifting the body,
`DeltaK=0` (ii)
Work done normal reaction
`W_N=vecN*dvecs=0` `( :. N _|_ds)` (iii)
Work done by the gravity
`W_(gr)=-mgy` ( `:.` the block moves up) (iv)
Using equations(i), (ii), (iii), and (iv),
`W_F+W_f=mgy` (v)
`W_f=intvecf_k*vec(ds)`
`=intf_kdscos180^@`
`=-intf_kds=-intmuNds`
`=-intmu(mgcostheta)ds`
`=-mumgintdscostheta(dscostheta)=dx)`
`=-mumgint_0^xdx=-mumgx` (vi)
Using equations (v) and (vi), `W_F=mg(mux+y)`