Consider the situation as shown in the figure. If `mu_(1)` is the friction coefficient between block `A` and incline and `mu_(2)` is the friction coefficient between block `B` and incline. Find the acceleration of block `A` if (a) `mu_(1) = (1)/(5) , mu_(2) =(2)/(5)` and (b) `mu_(1) = (2)/(5) , mu_(2) = (1)/(5)`.

Explanation

Considering the independent motion of the blocks
(a) `a_(B) = g sin 37^(@) - mu_(2) g cos 37^(@)`
` = 10 xx (3)/(5) - (2)/(5) xx 10 xx (4)/(5) = 6 - 3.2 = 2.8 m//s^(2)`
`a_(A) = g sin 37^(@) - mu_(1) g cos 37^(@)`
` = 10 xx (3)/(5) - (1)/(5) xx 10 xx (4)/(5) = 6 - 1.6 = 4.4 m//s^(2)`
Since `a_(B) lt a_(A)` , both the blocks will move in contact,
`( 2m + m) g sin 37^(@) - mu_(2) xx 2 mg cos 37^(@) - mu_(1) xx mg cos 37^(@) = ( 2m + m)a`
`3 m xx 10 xx (3)/(5) - (2)/(5) xx 2m xx 10 xx (4)/(5) - (1)/(5) xx m xx 10 xx (4)/(5) = 3 ma`
`18 m - 6.4 m - 1.6 m = 3ma`
`a = (10)/(3) = 3.3 m//s^(2)`
Acceleration of `A = 3.3 m//s^(2)`
(b) `mu_(1) = (2)/(5) , mu_(2) = (1)/(5)`
`a_(B) = g sin 37^(@) - mu_(2) g cos 37^(@) = 4.4 m//s^(2)`
`a_(A) = g sin 37^(@) - mu_(1) g cos 37^(@) = 2.8 m//s^(2)`
The blocks will move independently.
Acceleration of `A = 2.8 m//s^(2)`