Classical Mechanics (Kibble & Berkshire) · Problem 5.5

Energy conservation — T.W.B. Kibble, F.H. Berkshire

Physics — Mechanics · free-linear-oscillations

Problem

A damped oscillator satisfies the equation

\ddot{x} + 2K\dot{x} + \Omega^2 x = 0

where

K

and

\Omega

are positive constants with

K < \Omega

(under-damping). At time

t = 0

the particle is released from rest at the point

x = a

. Show that the subsequent motion is given by

x = ae^{-Kt}\left(\cos\Omega_D t + \frac{K}{\Omega_D}\sin\Omega_D t\right),

where

\Omega_D = (\Omega^2 - K^2)^{1/2}

. Find all the turning points of the function

x(t)

and show that the ratio of successive maximum values of

x

e^{-2\pi K/\Omega_D}

. A certain damped oscillator has mass 10 kg, period 5 s and successive maximum values of its displacement are in the ratio

3:1

. Find the values of the spring and damping constants

\alpha

and

\beta