Find the acceleration `a_(1) , a_(2) and a_(3)` of the three blocks shown in Fig if a horizontal force of `10 N` is applied on

(a) `2 kg` blocks, (b) `3 kg` blocks and ( c) `7 kg` blocks (Take `g = 10 ms^(-2))`

Answer

(a) `3.0 m // s^(2) , 0.4 m//s^(2) , 0.4 m//s^(2)` (b) 5/6 `m//s^(2)` , (c) 5/6 `m//s^(2) , 5//6 m//s^(2) , 5//6 m//s^(2)`

Explanation

(a) When force of 10 N is applied on 2 kg block . The limiting frictional force between 2 kg and 3 kg blocks
`f_(1) = 0.2 xx 2g = 0.2 xx 2 xx 10 = 4 N `
The limiting frictional force between 3 kg and 7 kg blocks that can be
`f_(2) = 0.3 xx 5 g = 0.3 xx 5 xx 10 = 15 N`

As applied force 10 N is greater than `f_(1)` but less than `f_(2)` so 2 kg block will slide over 3 kg but 3 kg will move together with 7 kg .
Thus we have ,
`a_(1) = (10-4)/(2) = 3 m//s^(2)`
`a_(2) = a_(3) = (4)/(3 + 7) = 0.4 m//s^(2)`
(b) When force of 10 N is applied on 3 kg block . As the applied force is less than the friction between 3 kg and 7 kg blocks (that can be 15 N) so the blocks will move together as one unit with an acceleration
`a= (10)/( 2 + 3 + 7) = (5)/(6) m//s^(2)`

Now find pseudo force on 2 kg block because of acceleration of 3 kg block
`F_("pseudo") = 2 xx (5)/(6) xx (5)/(3) N`
As the `F_("pseudo")` is smaller than the friction between 2 kg and 3 kg (that can be 4 N) .So , 2 kg will move together with other blocks .
(c) When force of 10 N is applied on 7 kg block . Supposing 3 kg and 2 kg blocks move togther with 7 kg block .The acceleration of whole system , a = `(10)/(2 + 3 +7) = (5)/(6) m//s^(2)`

The pseudo force on 2 kg block = `(2 xx 5)/(6) = (5)/(3) N` which is less than the frictional force between 2 kg and 3 kg block , so they move together .
Now check whether 2kg and 3 kg move together over 7 kg block . The pseudo force on ( 2+ 3) kg is = ` 5 xx (5)/(6) = (25)/(6)` N , which is also less than frictional force between 3 kg and 7 kg blocks , so all the blocks move together with a common acceleration of `(5)/(6) m//s^(2)`.
Therefore `a = a_(1) =a_(2) = a_(3) = (10)/(2 + 3 + 7) = (5)/(6) m//s^(2)`