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A constant horizontal force F of magnitu...

A constant horizontal force F of magnitude equal to `(mg)/(2)` begins to act on the bob of pendulum shown when the bob was at rest. What maximum angle the string makes with the vertical ?

Text Solution

Verified by Experts

Let the string makes angle `theta` with the vertical when it comes to rest (momentarily) as shown
`therefore x=lsintheta`
and `h=l-lcostheta=l(1-costheta)`
Here work done by gravity
`W_(g)=-mgh=-mgl(1-costheta)`
Work done by tension, `W_(T)=0[vecT_|_vecv]`
Work done by force F,
`W_(F)=Fxx` (displacement in the direction of F) = Fx
`=(mg)/(2)lsintheta`
`therefore` Applying work-energy theorem

`W_(g)+W_(F)+W_(T)=DeltaK.E=0`
`therefore (1)/(2)sintheta=1-costheta`
`therefore (1)/(2)sintheta=1-costheta`
`therefore tan((theta)/(2))=(1)/(2)`
`therefore theta=2tan^(-1)((1)/(2))`
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