Classical Mechanics (Kibble & Berkshire) · Problem 7.7

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

Physics — Mechanics · path-equation

Problem

A particle

P

of mass

m

moves under the attractive inverse cube field

\boldsymbol{F} = -(m\gamma^2/r^3)\hat{\boldsymbol{r}}

, where

\gamma

is a positive constant. Initially

P

is at a great distance from

O

and is projected towards

O

with speed

V

along a line whose perpendicular distance from

O

p

. Obtain the path equation for

P

. For the case in which

V = \frac{15\gamma}{\sqrt{209}\, p},

find the polar equation of the path of

P

and make a sketch of it. Deduce the distance of closest approach to

O

, and the final direction of departure.