Find the friction on the block in the following cases :
(a) (b)
(c ) If (i) `m = 0.1 kg` and (ii) `m = 1 kg`
(d)
(e)
(f)
`If (i) `t = 4 s` and (ii) `t = 9 s`

Explanation

Friction is a self - adjusting force. It develops according to the need from zero to `(f_(s))_(max) = mu N or muN`. To dertime friction force (a) calculate normal reaction `N` , (b) then `(f_(s))_(max) = mu_(s) N or mu N` , (c ) search for that which is trying to move the body ,let us say it movable force (it may be applied force , or weight or component of weight , etc.)
(i) If movable force `lt (f_(s))_(max)` , the block is at rest static friction `f_(s)` = movable force
(ii) If movable force `gt (f_(s))_(max)` , the block is in motion , kinetic friction `f_(k) = mu_(k) N or mu N`
(iii) If movable force `= (f_(s))_(max)`, motion is about to start
`N = 100`
`(f_(s))_(max) = mu N = (1)/(2) xx 100 = 50 N`
Applied force `= 30 N`
Applied force `lt (f_(s))_(max)` , the block is at rest
Friction force = `30 N`
(b) `N = 100 cos 37^(@) = 0.8 xx 80 = 64 N`
The force which can move the block is
`100 sin37^(@) = 100 xx (3)/(5) = 60 N`
`60 lt 64` , the block is at rest
Friction force `= 60 N`
( c ) `N = 5`
`(f_(s))_(max) = mu N = (1)/(2) xx 5 = 2.5 N`
The force that can move the block `W = mg`
(i) Movable force `W = mg = 0.1 xx 10 = 1 N`
`1 N lt 2.5 N` , the block is at rest
(ii) Movable force `W = mg = 1 xx 10 = 10 N`
`10 N gt 2.5 N` , the block is in motion
Friction force `= 2.5 N`
(d) `uarr : N + 50 sin 37^(@) = 100`
`N + 50 xx (3)/(5) = 100 rArr N = 70`
`(f_(s))_(max) = mu N = 0.8 xx 70 = 56 N`
Movable charges ` = 50 cos 37^(@) = 50 xx (4)/(5) = 40 N`
`40 N lt 56 N` , the block is at rest
Friction force `= 40 N`
(e) `uarr N = 50 sin 37^(@) + 100 = 50 xx (3)/(5) + 100 = 130` `(f_(s))_(max) = muN = (1)/(2) xx130 = 65 N`
Movable force ` = 50 cos 37^(@) = 50 xx (4)/(5) = 40 N`
`40 N lt 65 N` , the block is at rest
Friction force `= 40 N`
(f) `N = 100`
`(f_(s))_(max) = muN = 0.8 xx 100 = 80 N`
(i) At `t = 4 s`, movable force `F = 10 t = 40 N`
`40 lt 80` , the block is at rest
Friction force `= 40 N`
`(ii) At `t = 9 s`, movable force `F = 10 t = 90 N`
`90 gt 80` , the block is in motion
Friction force =` 80 N`