A block of mass m rests on a rough horizontl surface as shown in figure (a) and (b). Coefficient of friciton between block and surface is `mu`. A force F-mg acting at an angle `theta` with the vertical side of the block. Find the condition for which block will move along the surface.

Explanation

For(a): normal reaction `N=mg-mg cos theta` fricitional force `=muN=mu(mg-mg cos theta)`
Now block can be pulled when : Horizontal component of force `ge` fricitional force i.e. `mg sin theta gemu(mg-mg cos theta)`
or `2 sin (theta)/(2)cos (theta)/(2)gemu(1-cos theta)`
or `2sin(theta)/(2)cos (theta)/(2)ge2musin^(2)(theta)/(2)` or `cot(theta)/(2)gemu`
For (b): Normal reaction `N=mg+mg cos theta=mg(1+cos theta)` Hence, block can be pushed along the horizontal surface when horizontal component of force `ge` fricitional force
i.e. `mg sin thetagemu mg(1+cos theta)` or
`2 sin(theta)/(2)cos (theta)/(2gemuxx2 cos^(2)(theta)/(2)rArrtan(theta)/(2)gemu`