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A smooth spehre of radius R is made to t...

A smooth spehre of radius R is made to translate oin a straight line with a constant acceleration a. A particle kept on the top of the sphere is released rom there at zero velocity with respect to the sphere. Find the speed of the particle with respect to the sphere as a functon of the angle `theta` it slides.

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The correct Answer is:
A, B, C
Let the spehere moves towards left with an accelertion a.
Let m=mass of the particle
The particle m will also experience the inertia due to acceleratioiN/A as it is in the sphere. It will also experience the tangential inertia force `[m((dv)/(dt)] and centrifugal force ((mv^2)/R)`.
for the free body diagram
`m(dv)/(dt)=macostheta+mgsintheta`

`rarr mv(dv)/(dt)`
`=ma.costheta(R(dtheta)/(dt)+mgsintheta(R(dtheta)/(dt))` (beause v=R (dtheta)/(dt)`
`v dv =a R costheta dtheta+gRsinthetadtheta`
`v^2/2=aRsintheta-gRcostheta+C`
givin that at `theta=0, v=0`

So, C=gR
`rarr v^2/2=aRsintheta-gRcostheta+gR`
`rarr v^2=2R(asintheta)+g-gcostheta)`
`rarr v=[2R(asintheta+g-gcostheta)]^(1/2)`
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