Q.342. Pulley and blocks - mechanics

Question 342.
The figure shows a light inextensible cord,
light frictionless pulleys, and blocks A and B 
having equal mass.

It is initially held at rest so that the blocks are

at the same height above the ground. The blocks

are then released.

Show that the speed of block A at the moment

when the vertical separation of the blocks is h is:

v = √ [8gh/15].

Answer 342.

Let T = tension in the cord

If a = acceleration downwards of block A,
then a/2 = acceleration upwards of block B
 For the motion of block A,
mg - T = ma ... ( 1 )
For the motion of block B,
2T - mg = m * (a/2)
=> T - mg/2 = ma/4 ... ( 2 )
Adding ( 1 ) and ( 2 ),
mg/2 = 5ma/4
=> a = 2g/5

For separation of h,
block A moves down by 2h/3 with acceleration a = 2g/5
=> v^2 = 2a * (2h/3) = (4g/5) * (2h/3) = (8gh)/(15)
=> v = √[(8gh)/(15)].

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