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

Fro (a): normal reaction N = mg - mg `cos theta,` frictionsl force `=muN=mu(mg-mgcostheta)` Now block can be pulled when : Horizontal component of force `ge` frictional force i.e., mg `sinthetagemu(mg-mgcostheta)`
or `2sin""(theta)/(2)cos""(theta)/(2)gemu(1-costheta)`
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 horizonal surface when horizontal component of force `ge` frictional force i.e., `mssinthetagemumg(1+costheta)`
`or 2sin""(theta)/(2)cos""(theta)/(2)gemuxx2cos^(2)""(theta)/(2)impliestan""(theta)/(2)gemu`