A smooth horizontal table rotates with angular speed ω. A radial groove contains a particle initially at distance a from the axis, placed gently with no radial speed. Find its speed relative to the groove when its distance becomes L.
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Question

A smooth horizontal table rotates with angular speed ω. A radial groove contains a particle initially at distance a from the axis, placed gently with no radial speed. Find its speed relative to the groove when its distance becomes L.

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G1 Physics · Accepted Answer

In the rotating groove, radial motion obeys r¨=ω²r. Multiplying by ṙ and integrating from r=a, ṙ=0 gives ṙ²=ω²(r²-a²). Hence the speed relative to the groove at r=L is ω√(L²-a²).