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Question:

The potential energy of a particle of mass m at a distance r from a fixed point O is given by V(r) = kr²/2, where k is a positive constant of appropriate dimensions. This particle is moving in a circular orbit of radius R about the point O. If v is the speed of the particle and L is the magnitude of its angular momentum about O, which of the following statements is (are) true?

v=√(kR/m)

v=√(kmR)

L=√(mkR²)

L=√(mk²R²)

Solution:

Given potential energy V = kr²/2
F = -kr (towards centre) [F = -dV/dr]
At r = R, kR = mv²/R [Centripetal force]
where v is the speed of the particle
⇒ velocity v = √(kR²/m) = √(kR/m)
Angular momentum L = mvr = mR√(kR/m) = √(mkR³)
Therefore, v = √(kR/m) and L = √(mkR³)