A block A of mass m is tied to a fixed point C on a horizontal table through a string passing round a massless smooth pulley B. A force F is applied by the experimenter to the pulley. Show that if the pulley is displaced by a distance x the block will be displaced by 2x. Find the acceleration of the block and the pulley.

Explanation

Suppose the pulley is displaced to B and the block to A. The length of the string is `CB+BA` and is also equal to `CB+BB'+B'B+BA'`.
Hence,`CB+BA'+A'A=CB+BB'+B'B+BA'` or, A'A=2BB`.


The displacement of A is therefore, twice the displacement of B in any given time interval. Differentiating twice, we find that the acceleration of A is twice the acceleration of B.

To find the acceleration of the block we will need the tension in the string. That can be obtained by considering the pulley as the system.
The forces acting on the pulley are
i. F towards right by the experimenter,
ii T towards let by the portion BC of the string and
iii. T towards left by the portion BA of the string.
The vertical forces, if any, add to zero as there is no vertical motion.
As the mass of the pulley is zero. the equation of motion is
`F - 2T = 0` giving `T =F / 2`
Now consider the block as the system. The only horizontal force acting on the block is the tension T towards right. The acceleration of the block is therefore, `a=T/m=F/(2m)`. The acceleration of the pulley is `a/2=F/(4m)`