A small block of mass `m` on a horizontal smooth table is attached to a light string passing through a hole as shown in the figure. Initially, the block moves in a circle of radius `r` with speed `v` and the string is held by a person. The person pulls the strin slowly to decrease the radius of circle to `r//2`. (a) Find the tension in the string when the block moves in a circle of radius `r//2`. (b) Calculate the change in the kinetic energy of the block.

Explanation

The torque acting on the block about vertical axis through the hole is zero hence the angular momentum about this axis is constant.
By the conservation of angular momentum
`mvr=mv'(r )/(2) implies v'=2v`
Tension provides necessary centripetal force
`T'=(mv'^(2))/(r//2)=(m(2v)^(2))/(r//2)=(8mv^(2))/(r)`
`DeltaK=(1)/(2)mv'^(2)-(1)/(2)mv^(2)`
`=(1)/(2)m(2v)^(2)-(1)/(2)mv^(2)`
`=(3)/(2)mv^(2)`