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\$
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.
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.
| Time | Displacement \$x\$ | Velocity \$v\$ | Potential Energy \$U\$ | Kinetic Energy \$K\$ | Total Energy \$E\$ |
|---|---|---|---|---|---|
| 0 s | \$x_0\$ | 0 | \$\tfrac{1}{2}k x_0^2\$ | 0 | \$E\$ |
| \$t = \tfrac{T}{4}\$ | 0 | \$\pm \omega x_0\$ | 0 | \$\tfrac{1}{2}m(\omega x_0)^2\$ | \$E\$ |
Exam Tip:
• If the problem gives you amplitude \$x_0\$, you can skip calculating \$\omega\$ if \$k\$ and \$m\$ are not needed.
• Always check units: energy should be in joules (J).
• Remember that \$E\$ is the same at every point in the motion – it’s a constant of motion.
• When in doubt, write down \$E = K + U\$ and substitute \$K = \tfrac{1}{2}m\dot{x}^2\$ and \$U = \tfrac{1}{2}k x^2\$ to see the cancellation.