A small stone of mass `m` is attached to a light string which passes through a hollow tube. The tube is held by one hand and the free end of the string by the other hand. The mass is set into revolution in a horizontal circle of radius `r_(1)` with a speed `v_(1)` . The string is pulled down shortening the radius of the circular path to `r_(2)`. Which of the following is not correct? (`omega_(1)` and `T` have usual meanings)

Options

A. `(omega_(2))/(omega_(1))=(r_(1)^(2))/(r_(2)^(2)`
B. `(E_(k1))/(E_(k2))=(r_(2)^(2))/(r_(1)^(2))`
C. `(E_(k1))/(E_(k2))=(r_(2)^(2))/(r_(1)^(2))`
D. `(T_(2))/(T_(1))=(r_(1)^(3))/(r_(2)^(3))`

Answer

D

Explanation

Using conservation of angular momentum
`mr_(2)^(2)omega_(2)=mr_(1)^(2)omega_(1)` or `(omega_(2))/(omega_(1))=(r_(1)^(2))/(r_(2)^(2)`
Again `E_(k_(1))/E_(k_(2))=(1/2mv_(1)^(2))/(1/2mv_(2)^(2))=(r_(1)^(2)omega_(1)^(2))/(r_(2)^(2)omega_(2)^(2))=(r_(1)^(2))/(r_(2)^(2))xx[(r_(2)^(2))/(r_(1)^(2))]^(2)=(r_(2)^(2))/(r_(1)^(2)`
Again `(T_(2))/(T_(1))=(mr_(2)omega_(2)^(2))/(mr_(1)omega_(1)^(2))=(r_(2))/(r_(1)=[(r_(1)^(2))/(r_(2)^(2))]=(r_(1)^(3))/(r_(2)^(3))`