Two pucks on a level air table

PHYSICS · Multi-part · Fermi University Physics · Mechanics & Dynamical Systems · Gravitation and central-force motion · Gravitation and central-force motion

Action and Symmetry, Ch The Kepler problem redone: reduction, quadrature and the Runge-Lenz vector

Problem

Two pucks on a level air table, of masses

m_1=0.12\,\mathrm{kg}

and

m_2=0.16\,\mathrm{kg}

, are joined by a light elastic cord whose natural length is negligible next to the stretches involved, so that the pair interacts through

V(r)=\tfrac12k_{\rm s}r^2

, with

r

their separation. Friction is negligible. ((a)) Find the reduced mass

\mu

of the pair, in kilograms. ((b)) Write down (do not evaluate) the quadrature integral

\varphi(r)

for this potential, with

V_{\rm eff}(r) = \tfrac12 k_{\rm s}r^2 + \ell^2/2\mu r^2

substituted, in terms of

E

\ell

\mu

and

k_{\rm s}

. ((c)) The force between the pucks is

F = -k_{\rm s}r

, a power law with

\beta = 1

. Find the apsidal angle, in radians and in degrees.

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