7.2 Frequency and Period of SHM

Frequency (f) and period (T) describe how fast something oscillates in simple harmonic motion, and they are inverses: T = 1/f. For SHM systems, two core results are the mass-spring period T = 2π√(m/k) and the small-angle pendulum period T = 2π√(ℓ/g).

Why This Matters for the AP Physics 1 Exam

SHM problems test whether you can connect a system's physical properties to how it oscillates over time. On the exam you may need to calculate period or frequency, predict how changing mass, spring constant, length, or gravity affects the motion, and explain why amplitude does not change the period. These skills support both multiple-choice reasoning and free-response questions where you translate between equations, graphs, and verbal descriptions of oscillating systems.

A common free-response task asks you to create and compare representations, such as sketching free-body diagrams and energy bar charts for a block on a spring at maximum displacement and at equilibrium, then explaining how those representations agree. Knowing the period and frequency relationships gives you a foundation for that kind of reasoning.

Key Takeaways

• Period T is the time for one full cycle (seconds); frequency f is cycles per second (hertz, Hz = s⁻¹). They are inverses: T = 1/f.
• A mass-spring oscillator has period T = 2π√(m/k). Stiffer spring (larger k) means shorter period; larger mass means longer period.
• A small-angle simple pendulum has period T = 2π√(ℓ/g). Longer length or weaker gravity means longer period.
• Pendulum period does not depend on the bob's mass.
• Amplitude does not affect the period of an ideal SHM system.
• The pendulum formula only works for small angles (roughly under about 15°); larger swings increase the period.

Frequency and Period in SHM

When objects oscillate in SHM, they repeat the same motion over and over. Two properties describe this repeating behavior.

Period is the time required for one complete oscillation. For a swinging pendulum, the period is how long it takes to swing back and forth once and return to its starting position and direction.

• Symbol $T$, measured in seconds (s)
• It tells you the time per cycle
• A longer period means each oscillation takes more time

Frequency is how many oscillations happen per unit time.

• Symbol $f$, measured in hertz (Hz), which equals $s^{-1}$
• It tells you cycles per second
• A higher frequency means more oscillations in the same amount of time

Period and frequency are inverses:

• $f = \frac{1}{T}$ and $T = \frac{1}{f}$
• If frequency increases, period decreases
• Example: a system oscillating at 4 Hz has a period of 0.25 seconds per cycle

Object-Spring Oscillator Systems

When a mass attached to a spring is displaced and released, it exhibits SHM. The period depends on two things: the mass and the spring constant.

• Period of a mass-spring system: $T = 2\pi\sqrt{\frac{m}{k}}$
• $m$ is the mass in kilograms, $k$ is the spring constant in N/m

The spring constant $k$ measures how stiff the spring is:

• A higher $k$ means a stiffer spring that exerts more force per unit displacement
• As $k$ increases, the period decreases (oscillation speeds up)
• Quadruple the spring constant and the period is cut in half

The mass $m$ affects how quickly the system responds to the spring force:

• As mass increases, the period increases (oscillation slows down)
• Double the mass and the period increases by a factor of $\sqrt{2}$ (about 1.41)
• A larger mass has more inertia and accelerates more slowly under the same force

The amplitude (how far the spring stretches) does not affect the period in an ideal system. That amplitude independence is a defining feature of SHM.

Simple Pendulum Systems

A simple pendulum is a small mass (the bob) hanging from a lightweight string or rod. When displaced and released, it swings back and forth in motion that approximates SHM for small angles.

• Period of a simple pendulum: $T = 2\pi\sqrt{\frac{\ell}{g}}$
• $\ell$ is the length in meters, $g$ is the gravitational acceleration in $m/s^2$

The length directly affects the period:

• Longer pendulums swing more slowly (longer periods)
• Double the length and the period increases by a factor of $\sqrt{2}$
• This is why tall grandfather clocks use long pendulums for a slow, steady swing

The gravitational acceleration affects how quickly the pendulum accelerates toward equilibrium:

• On Earth's surface, $g$ is about $9.8 \, m/s^2$
• In lower gravity (like the Moon), a pendulum would swing more slowly
• In stronger gravity, it would swing more quickly

The mass of the bob does not affect the period. A greater mass increases inertia, but it also increases the gravitational force proportionally, so there is no net effect on the acceleration.

The pendulum approximates SHM only for small angles (under about 15°). For larger angles, the period increases slightly with amplitude.

How to Use This on the AP Physics 1 Exam

Problem Solving

• Start by identifying the system. Is it a mass-spring oscillator or a pendulum? That tells you which period formula to use.
• Watch the inverse relationship. If a problem gives frequency, use $T = 1/f$ before plugging into a period equation.
• Track the square root. Both period formulas have a square root, so changing a variable by a factor of 4 changes the period by a factor of 2.
• To solve for a buried variable like $k$, rearrange first. From $T = 2\pi\sqrt{m/k}$ you get $k = \frac{4\pi^2 m}{T^2}$.

Free Response

• Be ready to explain why amplitude does not change the period, using the period formulas as support.
• When asked to compare representations, connect the equation to graphs or energy bar charts for the same system.
• State assumptions clearly, especially the small-angle approximation for pendulums.

Common Trap

• Do not confuse stiffer with slower. A larger spring constant gives a shorter period, not a longer one.
• Do not include the bob's mass in pendulum period reasoning. Mass cancels out.

Practice Problem 1: Period-Frequency Relationship

Use the inverse relationship between frequency and period:

$T = \frac{1}{f}$

Given:

• Frequency $f = 440 \text{ Hz}$

Substituting: $T = \frac{1}{440 \text{ Hz}} = 0.00227 \text{ seconds}$

Each complete vibration of the tuning fork takes about 2.27 milliseconds.

Practice Problem 2: Object-Spring Oscillator

Use the period formula for a mass-spring system and solve for $k$:

$T = 2\pi\sqrt{\frac{m}{k}}$

Rearranging: $k = \frac{4\pi^2 m}{T^2}$

Given:

• Mass $m = 0.5 \text{ kg}$
• Period $T = 0.4 \text{ seconds}$

Substituting: $k = \frac{4\pi^2 \times 0.5 \text{ kg}}{(0.4 \text{ s})^2}$ $k = \frac{4\pi^2 \times 0.5 \text{ kg}}{0.16 \text{ s}^2}$ $k = \frac{2\pi^2 \text{ kg}}{0.16 \text{ s}^2}$ $k \approx 123 \text{ N/m}$

The spring constant is approximately 123 N/m.

Practice Problem 3: Simple Pendulum

Use the pendulum period formula:

$T = 2\pi\sqrt{\frac{\ell}{g}}$

Given:

• Length $\ell = 2 \text{ meters}$
• Acceleration due to gravity $g = 9.8 \text{ m/s}^2$

Substituting: $T = 2\pi\sqrt{\frac{2 \text{ m}}{9.8 \text{ m/s}^2}}$ $T = 2\pi\sqrt{\frac{2}{9.8}}$ $T = 2\pi\sqrt{0.204}$ $T = 2\pi \times 0.452$ $T \approx 2.84 \text{ seconds}$

The pendulum takes about 2.84 seconds to complete one full swing back and forth.

Common Misconceptions

• A stiffer spring means a longer period. It is the opposite. A larger spring constant $k$ gives a shorter period because the system oscillates faster.
• Heavier pendulum bobs swing slower. The bob's mass does not appear in the pendulum period formula, so it has no effect on the period.
• Bigger amplitude means a longer period. For ideal SHM, amplitude does not change the period at all.
• The pendulum formula always works. It only holds for small angles (roughly under about 15°). For larger swings the period grows with amplitude, so the simple formula no longer applies.
• Frequency and period are the same thing. They are inverses. High frequency means short period, and low frequency means long period.

Related AP Physics 1 Guides

• 7.1 Defining Simple Harmonic Motion (SHM)
• 7.3 Representing and Analyzing SHM
• 7.4 Energy of Simple Harmonic Oscillators