recall and use E = 21mω2x02 for the total energy of a system undergoing simple harmonic motion

What is SHM? 🤔

Imagine a playground swing. When you push it, it moves back and forth with a smooth, repeating motion. That’s a simple harmonic oscillator! Common examples: a mass on a spring, a pendulum (for small angles), or even a vibrating guitar string. The key idea is that the restoring force is proportional to the displacement: $$F = -kx \quad \text{or} \quad F = -m\omega^2x$$

Total Energy of a SHM System 🧪

In SHM, the total mechanical energy (kinetic + potential) stays constant. It’s given by:

$$E = \frac{1}{2}\,m\,\omega^2\,x_0^2$$

Where $m$ is mass, $\omega$ is angular frequency, and $x_0$ is the amplitude (maximum displacement). Notice that energy depends only on amplitude, not on time.

Quick Derivation 📐

1. Start with Newton’s 2nd law: $m\ddot{x} = -k x$.
2. Rewrite as $\ddot{x} + \omega^2 x = 0$, where $\omega = \sqrt{k/m}$.
3. Integrate the kinetic energy $K = \frac{1}{2}m\dot{x}^2$ and potential energy $U = \frac{1}{2}k x^2$.
4. Show that $K + U = \frac{1}{2}m\omega^2x_0^2$ is constant.

Example Problem 🎯

A 0.5 kg mass is attached to a spring with $k = 200\;\text{N/m}$ and oscillates with an amplitude of $0.02\;\text{m}$. Find the total mechanical energy.

1. Compute $\omega$: $$\omega = \sqrt{\frac{k}{m}} = \sqrt{\frac{200}{0.5}} = \sqrt{400} = 20\;\text{rad/s}$$

2. Plug into the energy formula: $$E = \frac{1}{2}\,m\,\omega^2\,x_0^2 = \frac{1}{2}\,(0.5)\,(20)^2\,(0.02)^2$$

3. Calculate: $$E = 0.25 \times 400 \times 0.0004 = 0.04\;\text{J}$$

So the system carries 0.04 J of energy, which stays the same at all times.

Energy Distribution Table 📊