A particle of mass moving under an attractive inverse-square force with fixed angular…

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Mechanics & Dynamical Systems · Calculus of change and accumulation

Mathematical Methods for Physics, Ch The derivative as a physical instrument

Problem

A particle of mass

m

moving under an attractive inverse-square force with fixed angular momentum

L

has an effective potential energy

U_{\text{eff}}(r) = \frac{L^2}{2mr^2} - \frac{k}{r}, \qquad r > 0,

where

k > 0

is the force constant. The distance

r

from the centre then changes as if the particle moved in one dimension with energy

E = \tfrac12 m\dot r^2 + U_{\text{eff}}(r)

, and

E

stays constant. Take

L = 4.0\,\mathrm{kg\,m^2\,s^{-1}}

m = 2.0\,\mathrm{kg}

and

k = 8.0\,\mathrm{J\,m}

. ((a)) Find the stationary point

r_0

U_{\text{eff}}(r)

. ((b)) Evaluate

U_{\text{eff}}''(r_0)

and state whether

r_0

is a minimum or a maximum. ((c)) Given that

U_{\text{eff}}\to+\infty

r\to0^+

and

U_{\text{eff}}\to0^-

r\to\infty

, sketch in words the shape of the curve. A particle placed at

r_0

with

\dot r = 0

keeps a constant

r

, so it moves in a circle. Use your result from (b) to say whether a small change in

r

from

r_0

meets a force back towards

r_0

or away from it.

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