PHYS381 Assignment
A mass $m$ is attached to a spring with spring constant $s$ and is subjected to an external oscillatory force $F_0 \cos(\omega t)$. The resulting motion of the mass can be described by the equation
$$m\ddot{x} + sx = F_0 \cos(\omega t).$$
Determine the steady-state response of the system by solving this equation, and show that it can be expressed as
$$x = \frac{F_0 \cos(\omega t)}{m(\omega_0^2 - \omega^2)},$$
where $\omega_0^2 = \frac{s}{m}$ represents the natural frequency of the system.
If the initial conditions are $x = \dot{x} = 0$ at $t = 0$, show that the solution can be rewritten as
$$x = \frac{F_0}{m} \frac{1}{\omega_0^2 - \omega^2} (\cos(\omega t) - \cos(\omega_0 t)).$$
Now, by expressing $\omega = \omega_0 + \Delta \omega$ with $\frac{\Delta \omega}{\omega_0} \ll 1$ and $\Delta \omega t \ll 1$, show that near resonance, the displacement $x$ can be approximated by
$$x(t) \approx \frac{F_0}{m \omega_0 \Delta \omega} \sin(\omega_0 t) \sin\left(\frac{\Delta \omega t}{2}\right).$$
Sketch the behavior of $x$ as a function of time, noting how the second term grows with time, allowing the oscillations to increase due to resonance. Observe that the condition $\Delta \omega t \ll 1$ focuses attention on the transient behavior of the system.