Question

What is meant by damping or dissipative force? Establish the differential equation of a damped harmonic oscillator and solve it to obtain an expression for the displacement of the oscillator.

Answer 1

Dissipative force The dissipative force is a force in which work done by the force is negative, and in this energy always be losses.

$F_d=-bv$

$F_{net}=-kx(t)-b\frac{dx(t)}{dt}$

$\Rightarrow m\frac{d^2x(t)}{dt^2}= -kx(t)-b\frac{dx(t)}{dt}$

$\Rightarrow m\frac{d^2x(t)}{dt^2}+b\frac{dx(t)}{dt}=-kx(t)$

$\Rightarrow \frac{d^2x(t)}{dt^2}+\frac{b}{m}\frac{dx(t)}{dt}=-\frac{kx(t)}{m}$

$\frac{d^2x}{dt^2}+\gamma\frac{dx}{dt}+\omega_o^2 x=0$

Now, comparing with the damped oscillation,

$\omega_o^2=\frac{k}{m}$

$\omega_o=\sqrt{\frac{k}{m}}$

Hence, we can conclude that $\frac{2\pi }{T}=\sqrt{\frac{k}{m}}$

$\Rightarrow T = 2\pi \sqrt{\frac{m}{k}}$