Think of a playground swing or a mass attached to a spring. When you push it and let go, it moves back and forth in a regular pattern. That regular back‑and‑forth motion is called simple harmonic motion (SHM).
In SHM we use a few handy equations. They let us predict where the object will be and how fast it’s moving.
| Quantity | Equation |
|---|---|
| Displacement | \$x(t)=A\cos(\omega t+\phi)\$ |
| Velocity | \$v(t)=-A\omega\sin(\omega t+\phi)\$ |
| Acceleration | \$a(t)=-\omega^2x(t)\$ |
| Energy | \$E=\tfrac{1}{2}kA^2\$ |
Here, \$A\$ is the maximum displacement (amplitude), \$\omega\$ is the angular frequency, \$k\$ is the spring constant, and \$\phi\$ is the phase shift.
Two common ways to write the velocity:
\$v(t)=v_0\cos(\omega t)\$
This form is handy when you know the speed at the start of the motion.
\$v=\pm\omega\sqrt{A^2-x^2}\$
The “±” shows that the speed is the same whether the object is moving right or left.
Both equations give the same result; they’re just different ways of looking at the same physics.
Imagine a mass \$m=0.5\,\text{kg}\$ attached to a spring with \$k=20\,\text{N/m}\$. Pull the mass 0.1 m to the right and let go.
At the instant the mass passes the equilibrium point (where \$x=0\$), its speed is at the maximum, \$v_{\text{max}}\$.
Picture a child on a swing. When the child pushes off the ground, the swing goes forward and then swings back. The motion is almost perfectly SHM if the swing is not too damped.
Just like a spring, the swing’s motion can be described by the same equations!
Try solving them before checking the solutions in the next lesson!