Figure shows, a system of four pulley with two masses A and B. Find at an instant.

(a) Speed of block A when block B is going up at `1ms^(-1)` and pulley Y is going up at `2 ms^(-1)`.
(b) Acceleration of block A if block B is going up `3 ms^(-2)` and pulley Y is going down at `1 ms^(-2)`.

Explanation

Let in a given time displacement of A is `x_(1)`, of B is `x_(2)`, and that of yis `x_(3)` in downward direction.
As the length of string will remain constant, so we can write.
`x_(1)+(x_(1)-x_(3))+x_(3)+(x_(2)-x_(3))+x_(2)=0`
`implies 2x_(1_+2x_(2)-x_(3)=0`
(a) Differentiating, we get `2v_(1)+2v_(2)-v_(3)=0`
`implies 2v_(1)+2(-1)-(-2)=0 implies v_(1)=0`
(b) Again differentiating, we get `2a_(1)+2a_(2)-a_(3)=0`
`implies 2a_(1)+2(-3)-1=0 implies a_(1)`
`=7//2 ms^(-2)`.