A toy cart is tied to the end of an unstrectched string of length 'l' when revolved , the toy cart moves in horizontal circle with radius '2l' and time period T. IF it is speeded until it moves in horizontal circle of radius '3l' with period `T_(1) ,` relation between `T and T_(1) ` is (Hooke 's law is obeyed )

Options

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

Answer

D

Explanation

By defination `K= ("force (F )")/( "Extension "(Delta l)) "" therefore F= K (Delta l) `
in this problem , ` Delta l_(1) =2 l- l=l and Delta l_(2) =3 l-l =2l`
the centrifugi force acting on the toy cart is `mr omega ^(2) ,` where `r_(1) = 2l and r_(2) = 3l `
Thus `kl = m omega ^(2) (2l) `
` and K (2l ) = m omega_(1) ^(2) (3l) `
`therefore ` Dividing (1) by (2) , we get
`(1)/(2) =((omega )/(omega_(1)))^(2) -(2)/(3) `
` therefore (omega )/(omega _(1)) = sqrt((3)/(4))`
` therefore ((2pi)/(T ))/((2pi ) /(T_(1)))=(T_(1))/(T) = (sqrt(3))/(2) `
` therefore T_(1) =(sqrt(3))/(2) T `