A ball suspended by a thread swings in a vertical plane so that its acceleration in the extreme position and lowest position are equal in magnitude. Angle `theta` of thread deflection in the extreme position will be `:`

Options

A. `30^(@)`
B. `60^(@)`
C. `53^(@)`
D. `37^(@)`

Answer

C

Explanation

Acceleration at lowest position
`a_(L)=(v^(2))/R`
from energy conservation
`mgR(1- cos theta)=(mv^(2))/2`
`(v^(2))/R=2g(1-cos theta)`
`a_(L)=2g(1-cos theta)`
acceleration at highest position
`a_(H)=g sin theta`
according to problem
`a_(L)=a_(H)`
`2g(1-cos theta)=g sin theta`
`2(1- cos theta)=sin theta`
`rArr 2(1-1+2 sin^(2)theta//2)=2 sintheta//2 cos theta//2`
`tan .(theta)/(2)=1/2`
`tan theta=(2 tan .(theta)/(2))/(1-tan^(2).(theta)/(2))=(2(1/2))/(1-1/4)=4/3`
`rArr theta=53^(@)`