Classical Mechanics (Kibble & Berkshire) · Problem 4.15

Linear oscillations — T.W.B. Kibble, F.H. Berkshire

Physics — Mechanics · constrained-motion-rotating-wire

Problem

A particle

P

of mass

m

can slide along a smooth rigid straight wire. The wire has one of its points fixed at the origin

O

, and is made to rotate in the

(x,y)

-plane with angular speed

\Omega

. By using the vector equation of motion of

P

in polar co-ordinates, show that

r

, the distance of

P

from

O

, satisfies the equation

\ddot{r} - \Omega^2 r = 0,

and find a second equation involving

N

, where

N\hat{\boldsymbol{\theta}}

is the force the wire exerts on

P

. [Ignore gravity in this question.] Initially,

P

is at rest (relative to the wire) at a distance

a

from

O

. Find

r

as a function of

t

in the subsequent motion, and deduce the corresponding formula for

N