AP® Physics C: Mechanics · Unit 7 · Topic 7.2
The Simple Pendulum
A simple pendulum swings with period T = 2 π √(L/g), set only by its length and local gravity. Mass cancels out of the equation of motion, so heavier bobs swing in the same time as light ones.
Picture a point mass on a massless string, displaced by angle theta. Gravity's restoring torque is -mgL sin(theta), and Newton's second law for rotation gives mL2 (d2 theta/dt2) = -mgL sin(theta). The mass m appears on both sides and cancels, which is exactly why a bowling ball and a golf ball on identical strings swing in step. What's left is d2 theta/dt2 = -(g/L) sin(theta), a differential equation that is not yet the clean harmonic oscillator you want, because of that sin(theta).
For small angles (roughly under 15 degrees), sin(theta) is almost exactly theta in radians. Swap it in and the equation becomes d2 theta/dt2 = -(g/L) theta, the same shape as F = -kx for a spring. That makes the angular frequency omega = √(g/L) and the period T = 2 π √(L/g). Same recipe as every oscillator in this unit: find the restoring force, linearize it, read off omega from the coefficient.
This result sits right next to the spring equation T = 2 π √(m/k): both come from forcing a messy restoring force into the form -(something)x. The pendulum trades spring stiffness k for gravity's pull g/L, and trades mass for geometry. Push the angle too large, though, and sin(theta) is no longer close to theta, the restoring torque gets weaker than the linear approximation predicts, and the period stretches out, growing with amplitude instead of staying constant.
Key terms
- Simple pendulum
- An idealized point mass swinging on a massless, inextensible string under gravity.
- Small-angle approximation
- Treating sin(theta) as equal to theta in radians, valid for angles roughly under 15 degrees.
- Restoring torque
- The torque from gravity that pulls the pendulum back toward its lowest point.
- Angular frequency (omega)
- The constant √(g/L) that sets how fast the pendulum oscillates, in radians per second.
Watch it explained
- How it worksHow does the small-angle approximation turn a swinging pendulum into a harmonic oscillator?
- Worked exampleFind the length of a pendulum whose period is exactly 2.0 seconds on Earth.
- Worked exampleFind the period of a 0.8 m pendulum taken to the Moon, where g is 1.6 m/s2.
- Going furtherWhat happens to the period once you swing a pendulum through a really large angle?
Check yourself
End of topic.