Classical Mechanics (Kibble & Berkshire) · Problem 6.5

Orbits in a central field — T.W.B. Kibble, F.H. Berkshire

Physics — Mechanics · rectilinear-motion-energy

Problem

A particle PP of unit mass moves on the positive xx-axis under the force field F=36x3−9x2(x>0).F = \frac{36}{x^3} - \frac{9}{x^2} \qquad (x > 0). Show that each motion of PP consists of either (i) a periodic oscillation between two extreme points, or (ii) an unbounded motion with one extreme point, depending upon the value of the total energy. Initially PP is projected from the point x=4x = 4 with speed 0.5. Show that PP oscillates between two extreme points and find the period of the motion. [You may make use of the formula ∫abx dx[(x−a)(b−x)]1/2=π(a+b)2. ]\int_a^b \frac{x\,dx}{[(x-a)(b-x)]^{1/2}} = \frac{\pi(a+b)}{2}.\,] Show that there is a single equilibrium position for PP and that it is stable. Find the period of small oscillations about this point.

Related theory

Browse the problem library