A ball suspended by a thread swing in a vertical plane that its acceleration values in the lowest possition and the extreme postition are equal . Find the thread deffection angle in the extreme possition.

Explanation

In the extreme position, `v=0`
At `A:a_(c)=(u^(2))/(L),a_(t)=g sin theta=0`
`a_(A)=a_(c)=(u^(2))/(L)`
At `B:_(c)=(v^(2))/(L)=0,a_(t)=gsintheta`
`a-(B)=g sin theta`
Given: `a_(A)=a_(B)`
`(u^(2))/(L)=g sin thetaimpliesu^(2)=gL sin theta` ...`(i)`
`v^(2)=u^(2)-2gh`
`0=u^(2)-2gL(1-cos theta)`
`implies u^(2)=2gL(1-cos theta)` ....`(ii)`
Equating (i) and (ii), we get
`gL sin theta=2gL(1-cos theta)`
`sin theta=2-2cos theta`
Squaring both sides, we have
`sin^(2)theta=4+4 cos^(2)theta-8cos theta`
`1-cos^(2)theta=4+4 cos^(2) theta-8cos theta`
`5 cos^(2)theta-8 cos theta+3=0`
`cos theta=-(-8)+-sqrt((-8)^(2)-4xx5xx3)/(2xx5)=(+-2)/(10)`
`cos theta=1impliestheta=0^(@),` not possible
`cos theta=(6)/(10)=(3)/(5)impliestheta=53^(@)`