a small sphere of mass `m` susopended by a thread is first taken aside so that the thread forms the right angle with the verticle and then realease. (a) find the total acceleration of the sphere and the thread tension as a function of `theta`, the angle of deflection of the thread from the vertical. (b) Find the angle `theta` between the thread and the vertical. at the moment when the total acceleration vector of the sphere is directed (i) horizontally (ii)vertically upward and (iii) vertically downward.

Explanation

(a) `v^(2)=u^(2)+2gh=0+2gL cos theta`
`a_(c)=(v^(2))/(L)=2g cos theta`
`a_(t)=g sin theta`
`a=sqrt(a_(c)^(2)+a_(t)^(2))=gsqrt(1+3 cos^(2)theta)`
Total acceleration: `a=gsqrt(1+3 cos^(2)theta)`
`T-mg cos theta=(mv^(2))/(L)=2mg cos theta`
`T=3mg cos theta`
`a_(c) cos theta=a_(t) sin theta`
`2g cos^(2)theta=g sin^(2)theta`
implies `tan^(2)theta=2`
`tan theta=sqrt(2)`
`cos theta=(1)/sqrt(3)`
`theta=cos^(-1)((1)/sqrt(3))`
(ii) when `theta=0^^(@)`, i.e., when the sphere is at the lowest postion, `a_(t)=0`, (since `a_(t)=g sin theta)`
`a=a_(c)`, directed vertically upwards
(iii) when `theta=90^(@)`, i.e., when the string is horizontal,
`a_(c)=0(since a_(c)=(u^(2))/(L))`
`a=a_(t)=g sin 90^(@)=g`, directed vertically downwards