Toy cart tied to the end of an unstretched string of length a, when revolved moves in a horizontal circle of radius 2a with a time period T. Now the toy cart is speeded up until it moves in a horizontal circle of radius 3a with a period T. If Hooke's law (F=kx) holds, then

Options

A. `T_(1)=(2)/(sqrt3)T`
B. `T_(1)=sqrt((3)/(2))T`
C. `T_(1)=sqrt((2)/(3))T`
D. `T_(1)=((sqrt3)/(2))T`

Answer

D

Explanation

When toy -cat moves in a circle of radius `2l` the string streches by `'l'`
The restoring force = centripetal force
`Kl=mr omega ^(2)=m2l omega ^(2)=m2lxx(4pi^(2))/(T_(1)^(2))`
`Kl=mxx2l(4pi^(2))/T_(1)^(2)" "......(i)`
`K2l=m3lxx(4pi^(2))/(T_(2)^(2))" "......(ii)`Dividing (i) by (ii),
`(1)/(2)=(2)/(3)xx(T_(2)^(2))/(T_(1)^(2))`
`therefore T_(2)^(2)=(3)/(4)T_(1)^(2)`
`T_(1)=(sqrt(3))/(2)T`