Classical Mechanics (Kibble & Berkshire) · Problem 7.26

Non-linear oscillations and phase space — T.W.B. Kibble, F.H. Berkshire

Physics — Mechanics · effect-of-resistance

Problem

A satellite of mass

m

moves under the attractive inverse square field

-(m\gamma/r^2)\hat{\boldsymbol{r}}

and is also subject to the linear resistance force

-mK\boldsymbol{v}

, where

K

is a positive constant. Show that the governing equations of motion can be reduced to the form

\ddot{r} + K\dot{r} + \frac{\gamma}{r^2} - \frac{L_0^2 e^{-2Kt}}{r^3} = 0, \qquad r^2\dot{\theta} = L_0 e^{-Kt},

where

L_0

is a constant which will be assumed to be positive. Suppose now that the effect of resistance is slight and that the satellite is executing a 'circular' orbit of slowly changing radius. By neglecting the terms in

\dot{r}

and

\ddot{r}

, find an approximate solution for the time variation of

r

and

\theta

in such an orbit. Deduce that small resistance causes the circular orbit to contract slowly, but that the satellite speeds up!

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