5th Putnam 1942 Problem B4 A particle moves in a circle through the origin under the influence of a force a/r k towards the origin (where r is its distance from the origin). Find k.
The equations of motion are r (θ ') 2 - r'' = a/r k , r 2 θ ' = 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/r 2 sin θ. Differentiating again, r'' = -AB cos θ A/r 2 + 2AB/r 3 sin θ r' = -A 2 /r 3 - 2A 2 B 2 /r 5 sin 2 θ = -A 2 /r 3 - 2A 2 B 2 /r 5 (1 - r 2 B 2 ) = A 2 /r 3 - 2A 2 B 2 /r 5 . So substituting back in the equation of motion we get: 2A 2 B 2 /r 5 = a/r k . Hence k = 5. Note that this is unphysical, since we require infinite velocity as we reach the origin. 5th Putnam 1942 © John Scholes jscholes@kalva.demon.co.uk 5 Mar 2002