A small sphere of mass `m` suspended by a thread is first taken a side so that the thread forms the right angle with the vertical and then released, then

Options

A. Total acceleration of sphere as a function of `theta` measured from the vertical is `g sqrt(1 + 3 cos^(3) theta)`
B. Thread tension as a function of `theta` measured from the vertical is `T = 3 mg cos theta`
C. The angle `theta` between the thread and the vertical at the moment when the total acceleration vector of the sphere is directed horizontally is `cos^(-1) 1//sqrt(3)`.
D. The thread tension at the moment when the vertical component of the sphere's velocity is maximum will be `mg`.

Answer

A, B, C

Explanation

Between `A` and `B`
`mgl cos theta = (1)/(2) mv_(B)^(2) , V_(B) = sqrt(2gL cos theta)`
`a_(r) = 2g cos theta , a_(t) = g sin theta`
Now at `B , T_(B) - mg cos theta = (mv_(B)^(2))/(L)`
Put `V_(B) rArr T_(B) = 3mg cos theta`
`tan (90-theta)=(a_(t))/(a_(r)) =(1)/(2) tan thetarArr tan theta = sqrt(2)`
`rArr cos theta = (1)/(sqrt(3))`.