(a) A block slides down an incline of angle `45^(@)` with an acceleration `(g)/(2 sqrt(2))`. Find the friction coefficient.
(b) A block slides down at `45^(@)` inclined plane in twice the time it takes to slide down a smooth `45^(@)` incline. What is friction co-efficient between the block and the incline?
( c) A block slides down on a rough inclined plane of inclination `30^(@)` with constant velocity. If this block is projected up the plane with a velocity of `10 m//s`, then at what distance along the inclined plane , the block will come to rest?
(d) A body is launched up on a rough inclined plane of inclination `theta`. If time of ascent is half of time of descent, find the friction coefficient.

Explanation

(a) Acceleration in downward direction :
`a = g sin theta - mu g cos theta = g sin 45^(@) - mug cos 45^(@)`
`(g)/(2 sqrt(2)) = (g)/(sqrt(2)) - (mu g)/(sqrt(2)) rArr (1)/(2) = 1 - mu rArrmu = (1)/(2)`
(b) Smooth inclined plane :
`a = g sin 45^(@)`
`d = (1)/(2) at^(2)` (i)
Rough inclined plane ,
`a' = g sin 45^(@) - mu g cos 45^(@)`
` d = (1)/(2) a'(2t)^(2)` (ii)
Equating (i) and (ii) , we get
`(1)/(2) at^(2) = (1)/(2) a'(4t)^(2)`
`a = 4a'`
`g = sin 45^(@) = 4( g sin 45^(@) - mug cos 45^(@))`
` 1 = 4(1 - mu) rArr 1 - mu = (1)/(4) rArr mu = (3)/(4)`
(c) When the block moves with constant velocity ,
Acceleration = 0
`a = g sin theta - mu g cos theta`
`0 = g sin theta - mug cos theta rArr mug cos theta = g sin theta`
Retardation of the block when it moves up
`a' = g sin theta + mug cos theta = g sin theta + g sin theta = 2 g sin theta`
`= 2g sin 30^(@) = g`
`v^(2) = u^(2) - 2a' s`
`0 = (10)^(2) - 2 g d rArr d = ((10)^(2))/( 2 g) = (100)/(20) = 5 m`
(d) Motion up the plane :
Retardation `a_(1) = g sin theta + mu g cos theta , time = t_(0)`
`s = (1)/(2) a_(1) t^(2)`
`d = (1)/(2) a_(1) t_(0)^(2)` (i)
Motion down the plane :
Acceleration `a_(2) = g sin theta - mu g cos theta , time = 2t_(0)`
`s = (1)/(2) a_(2) ( 2t_(0))^(2)`
`d = (1)/(2) a_(2) ( 2t_(0))^(2)`
Equating (i) and (ii) , we get
`(1)/(2) a_(1) t_(0)^(2) = (1)/(2) a_(2) (2t_(0))^(2)`
`a_(1) = 4 a_(2)`
`g sin theta + mu g cos theta = 4(g sin theta - mu g cos theta)`
`sin theta + mu cos theta = 4 sin theta - 4 mu cos theta`
` 5 mu cos theta = 3 sin theta`
`mu = (3)/(5) tan theta`
Recall : If a particle is moving under constant acceleration/deceleration in a straight line and its velocity at one end (initial or final) is zero , displacement in time `t , s = (1)/(2) at^(2)`.