A block of mass 2 kg rests on an inclined plane which makes an angle of `30^(@)` with the horizontal. The coefficient of friction between the block and the surface is `sqrt(3//2)`. (i) What force should be applied on the block so that it moves down without any acceleration ? (ii) What force should be applied on the block so that it moves up without any acceleration ? (iii) Calculate the ratio of the powers in the above two cases if the block moves with uniform speed in both the cases.

Explanation

Make a 'free-body' diagram of the block. The the force of friction opposite of the direction of motion.
(i) Project forces along and perpendicular to the plane
`{:("perpendicular to plane",N=mg cos theta),("along the plane",F+mg sintheta-f=0):}`
(`because` there is no acceleration along the plane)
`F+mg sintheta-muN=0impliesFmgsintheta=mumgcostheta`
`F=mg(mucostheta-sintheta)=2xx9.8"("(sqrt3)/(2)cos30^(@)-sin30^(@)")"`
`=19.6"("(sqrt3)/(2)xx(sqrt3)/(2)-1/2")"=19.6"("(3)/(4)-(1)/(2)")"=4.9N`

(ii) This time the direction of F is reversed and that of the frictional force is also reversed.
`thereforeN=mgcostheta,F=mgsintheta+f`
`impliesF=mg(mucostheta+sintheta)=19.6"("3/4+1/2")"=24.5N`