classical mechanics - Finding the steady state solution of a damped oscillator

A damped harmonic oscillator with $m = 10$kg, $k = 250$N/m, and $c = 60$kg/s is subject to a driving force given by $F_0\cos(\omega t)$, where $F_0 = 48$N. (a) What value of $\omega$ results in steady-state oscillations with maximum amplitude? (b) What is the maximum amplitude? What is the phase shift at the resonance

Studying for my exam tomorrow, I am kinda of confused to how to even begin this problem. What does it mean by steady-state solution?

I know maximum amplitude occurs when velocity is zero. The diff equation looks as follows:

$$a + \frac{cv}{m} + \frac{kx}{m}$$

and thats all I know. Please do not give me the answer, but please tell me how I can approach this.

Answer

As Mark Eichenlaub said in his comment, start by writing down $F=ma$, and including all the forces (damping force, spring, driving force) on the left.

The result is a differential equation for the unknown function $x(t)$. You're looking for a "steady state" solution, which just means a solution in which $x(t)$ is a sinusoidal oscillation: $x(t)=A\cos(\omega t+\phi)$.

There are such steady solutions for any $\omega$ you care to name. Find the value of $\omega$ that leads to a solution with the biggest amplitude.

And, sorry if this sounds rude, but you should probably also look up "resonance" and "driven harmonic oscillator" in your textbook. There's bound to be a bunch of useful stuff there.