Simple Harmonic Motion: Pendulum and Spring Period

The two period formulas

A mass on a spring oscillates with period $T_s = 2\pi\sqrt{m/k}$, where $m$ is the attached mass in kilograms and $k$ is the spring constant in newtons per meter. A simple pendulum swings with period $T_p = 2\pi\sqrt{\ell/g}$, where $\ell$ is the string length in meters and $g = 9.8 \text{ m/s}^2$ near Earth's surface. The period $T$ is the time for one complete cycle, in seconds, and frequency is its reciprocal: $T = 1/f$.

All three equations are printed on the AP Physics 1 equation sheet, so you never have to memorize them. What the exam actually tests is whether you know which variables matter, which do not, and how the period responds when one of them changes. That is where the rest of this guide goes.

What counts as simple harmonic motion

Simple harmonic motion happens when the net force on an object is a restoring force whose strength is proportional to the displacement from equilibrium. A spring does this exactly: Hooke's law gives $F_s = -k\Delta x$, so doubling the stretch doubles the pull back toward center. The negative sign means the force always points opposite the displacement.

A pendulum is only approximately simple harmonic. The restoring force component along the swing is $mg\sin\theta$, and only for small angles (roughly under 15 degrees) is $\sin\theta$ close enough to $\theta$ for the motion to count as SHM. That is why the pendulum formula carries a small-angle assumption while the spring formula has none.

In both cases the object speeds up toward equilibrium, overshoots, slows, stops momentarily at the far side, and repeats. Maximum speed occurs at equilibrium. Maximum acceleration occurs at the endpoints, where displacement and restoring force are largest.

What does not affect each period

Amplitude appears in neither formula. Pull the spring mass twice as far before releasing it, or start the pendulum from a slightly wider angle, and the period stays the same: the object travels farther each cycle but also moves faster, and the two effects cancel exactly. This amplitude independence is the signature of simple harmonic motion and one of the most tested facts in Unit 7.

Mass does not affect the pendulum. A heavier bob feels a stronger gravitational pull, but it also has proportionally more inertia, so $m$ cancels out of the motion. Swap a steel bob for a wooden one on the same string and the period is unchanged.

Gravity does not affect the spring-mass period. Since $g$ never enters $T_s = 2\pi\sqrt{m/k}$, a horizontal spring oscillator on Earth and the identical oscillator on the Moon share one period. Hanging the mass vertically shifts the equilibrium position but leaves the period alone.

Energy trades back and forth

Total mechanical energy stays constant in ideal SHM; it just changes form twice per cycle. At maximum displacement the object is momentarily at rest, so kinetic energy is zero and everything is potential: $U_s = \frac{1}{2}k(\Delta x)^2$ for a spring, or gravitational $\Delta U_g = mg\Delta y$ for a pendulum bob at the top of its swing. Passing through equilibrium, potential energy hits its minimum and the energy is all kinetic, $K = \frac{1}{2}mv^2$, which is where speed peaks.

For a spring oscillator with amplitude $A$, the total energy is $\frac{1}{2}kA^2$, set entirely by amplitude and spring constant. Doubling the amplitude quadruples the stored energy but, as always, leaves the period untouched.

Every energy question in topic 7.4 is really a conservation of energy problem: write the total energy at one position, set it equal to the total at another, and solve. The third worked example below does exactly that.

Proportional reasoning beats plugging in

Multiple-choice questions rarely hand you clean numbers. They ask what happens to $T$ when a variable changes, and the square root is the whole game: period scales with the square root of the numerator and the inverse square root of the denominator.

• Quadruple the mass on a spring: $T_s$ doubles, since $\sqrt{4} = 2$.
• Quadruple the pendulum length: $T_p$ doubles for the same reason.
• Swap in a spring four times stiffer: $T_s$ halves.
• Move a pendulum to a planet where $g$ is four times larger: $T_p$ halves.

Handle one factor at a time and most Unit 7 ranking questions take under a minute.

Where SHM sits in AP Physics 1

Oscillations is Unit 7 of AP Physics 1, worth 5 to 8 percent of the multiple-choice section. This guide covers topic 7.1 (defining SHM), topic 7.2 (frequency and period), and topic 7.4 (energy of simple harmonic oscillators); topic 7.3 adds the position, velocity, and acceleration graphs. A calculator is allowed on both sections of the exam, so arithmetic is never the obstacle in a period problem. The traps are conceptual, like assuming a heavier bob swings slower.

Keep the equation sheet open while you practice so you know exactly where $T_s = 2\pi\sqrt{m/k}$, $T_p = 2\pi\sqrt{\ell/g}$, and $T = 1/f$ sit on the page. For the full topic list, see the Unit 7 overview. The same oscillation is what launches a periodic wave, so the frequency here carries straight into wave speed, frequency, and wavelength. If you are in the calculus-based course instead, AP Physics C: Mechanics tests the same two formulas in its own Unit 7 and adds simple and physical pendulums as topic 7.5.