One end of a light string of length L is connected to a ball and the other end is connected to a fixed point O. the ball is released from rest at `t=0` with string horizontal and just taut. The ball then moves in vertical circular path as shown. The time taken by ball to go from position A to B is `t_(1)` and from B to lowest position C is `t_(2)` let the velocity of ball at B is `vecv_(B)` and at C is `vecv_(C)` respectively.
Q. If `|vecv_(C)|=2|vecv_(B)|` then the value of `theta `as shown is

Options

A. `cos^(-1)((1)/(4))^(1//3)`
B. `sin^(-1)((1)/(4))^(1//3)`
C. `cos^(-1)((1)/(2))^(1//3)`
D. `sin^(-1)((1)/(2))^(1//3)`

Answer

B

Explanation

`V_(B)=sqrt(2gLsintheta)` and `V_(C)=sqrt(2gL)`
`V_(C)=2V_(B)`
Then `2gL=4(2gL sin theta )`
or `sin theta =(1)/(4) ` or `theta=sin ^(-1)((1)/(4))`
`|vec(v)_(B)-vec(v)_(C)|=sqrt(v_(B)^(2)+v_(C)^(2)-2v_(B)v_(C)sin theta )=v_(B)`
`rArr v_(B)^(2)+v_(C)^(2)-2v_(B)v_(C)sin theta -v_(B)^(2)`
`v_(C)=2v_(B) sin theta`
`rArr sqrt(2gl)=2sqrt(2gl sin theta ) sin theta`

`:. sin ^(3) theta =(1)/(4) rArr sin theta =((1)/(4))^(1//3)`
`theta = sin ^(-1)((1)/(4))^(1//3)`