A small object of mass 'm' is moved over a rough, curved surface as shown. The force which moves it along the dotted path shown is always tangential to the curved path. If the object moves with negligible speed, find the work done by the force F in moving it from A to B. The coefficient of friction between the object and the curved surface is `mu.`

Explanation

Consider a small section 'ds' of the curced path followed by the particle. In this small section, the vartious forces acting on the object are shown.

As the object moves with negligible speed, the change in kinetic energy is zero
`thereforedW_(F)+dW_(mg)+dW_(N)+dW_("friction")=0`
`dW_(F)+(-mgdy)+0-muNds=0`
Now, `N=mgcos thetaand cos theta=(dx)/(ds)`
`impliesdW_(f)=mg dy +mumg dx `
`thereforeW_(F)=underset(0)overset(H)intmg dy +underset(0)overset(L)intmu mg dx`
`W_(F)=mgH+mu mgL.`