Animated Solution for Physics - Oscillations: A particle of mass $m$ is attached to a spring (of spring constant $k$) and has a natural angular frequency ${\omega }_0$. An external force $F(t)$ proportional to $\cos \omega t(\omega \ne {\omega }_0)$ is applied to the oscillator. The time displacement of the oscillator will be proportional to

Imagine a simple pendulum or a mass on a spring. If you give it a little nudge, it oscillates at its own preferred rhythm. We call this the natural frequency, denoted by ${\omega }_0$. But what happens if you don't just let it be? What if you continuously push and pull it with a rhythmic force of your own?

This is the classic setup of a forced harmonic oscillator. In this problem, we have a mass $m$ attached to a spring with constant $k$. We are applying an external force $F(t)$ that varies as $\cos \omega t$. Notice that the driving frequency $\omega$ is different from the natural frequency ${\omega }_0$. Our goal is to find out how the amplitude of the resulting motion depends on these parameters.

To understand the motion, we must turn to the ultimate rulebook of classical mechanics: Newton's Second Law. The net force on the mass dictates its acceleration.

There are two forces at play here: 1. The spring's restoring force, which always tries to pull the mass back to equilibrium: $-kx$. 2. The external driving force: $F_0\cos \omega t$.

Putting these together, the equation of motion becomes:

We also know that the natural angular frequency ${\omega }_0$ is related to the spring constant and mass by the equation ${\omega }_0^2=\frac{k}{m}$. This means we can replace $k$ with $m{\omega }_0^2$.

Substituting this into our differential equation and rearranging, we get:

This is a non-homogeneous second-order linear differential equation. It might look intimidating, but physics gives us a beautiful shortcut to solve it!

When you force an oscillator, it initially throws a tantrum. It tries to oscillate at its natural frequency while you force it at your driving frequency. This chaotic phase is called the transient state. However, due to inevitable tiny damping in the universe, this tantrum dies out.

Eventually, the mass surrenders and oscillates purely at the driving frequency $\omega$. This is the steady-state. Because the driving force is a cosine function, it is a brilliant and physically sound guess to assume the steady-state displacement is also a cosine function:

Here, $A$ is the maximum displacement, or the amplitude, which we want to find.

To plug this guess into our differential equation, we need the acceleration. We differentiate $x(t)$ twice with respect to time:

Now, let's substitute our expressions for $x$ and its second derivative back into the master equation:

Look at the elegance of this! Every single term contains the factor $\cos \omega t$. We can factor out $A\cos \omega t$ on the left side:

Since this equation must hold true for any time $t$, we can safely divide both sides by $\cos \omega t$:

Finally, solving for the amplitude $A$, we get:

The question asks what the time displacement (the amplitude $A$) is proportional to. Since $F_0$ is just the proportionality constant of the external force, we can clearly see that:

Resonance

Before we wrap up, let's look at the denominator of our result: $m({\omega }_0^2-{\omega }^2)$. What happens if you tune your driving frequency $\omega$ to exactly match the natural frequency ${\omega }_0$?

The term $({\omega }_0^2-{\omega }^2)$ becomes zero. This means the amplitude $A$ would theoretically shoot up to infinity! This spectacular phenomenon is known as resonance. It's the same physics that allows a singer to shatter a wine glass by hitting exactly the right note. In reality, infinite amplitude is prevented by damping (friction or air resistance), but the amplitude still becomes dangerously large.

By understanding this simple equation, you've just unlocked the secret behind everything from tuning a radio to designing earthquake-resistant skyscrapers!