A particle moves in a circle through the origin under the influence of a force a/rk towards the origin (where r is its distance from the origin). Find k.
The equations of motion are r (θ ')2 - r'' = a/rk, r2θ ' = A (conservation of angular momentum).
If the particle moves in a circle as described, then we can write its orbit as r = B cos θ. Differentiating, r' = - B θ ' sin θ = -AB/r2 sin θ. Differentiating again, r'' = -AB cos θ A/r2 + 2AB/r3 sin θ r' = -A2/r3 - 2A2B2/r5 sin2θ = -A2/r3 - 2A2B2/r5 (1 - r2B2) = A2/r3 - 2A2B2/r5. So substituting back in the equation of motion we get: 2A2B2/r5 = a/rk. Hence k = 5.
Note that this is unphysical, since we require infinite velocity as we reach the origin.
5th Putnam 1942
© John Scholes
5 Mar 2002