Frequency and Period | AP Physics C: Mech Unit 7 Study Guide

Getting Started

Many systems in nature, from vibrating atoms to orbiting planets, exhibit periodic motion. A special and fundamental type of this motion is Simple Harmonic Motion (SHM), which occurs whenever an object is subject to a restoring force that is directly proportional to its displacement from an equilibrium position. Our central question is: what physical properties of an oscillating system determine its period and frequency—the very tempo of its motion?

What You Should Be Able to Do

After studying this section, you will be able to:

Derive the second-order differential equation of motion for a mass-spring oscillator using Newton's second law.
Derive the second-order differential equation of motion for a simple pendulum using Newton's second law for rotation and the small-angle approximation.
Identify the angular frequency, $ω$ , of an oscillating system by comparing its equation of motion to the canonical form for SHM.
Calculate the period, $T$ , and frequency, $f$ , for both mass-spring and simple pendulum systems from their physical parameters.

Key Concepts & Mechanisms

System & Preconditions: The Setup for SHM

The defining characteristic of any system that undergoes SHM is a linear restoring force. This means the net force acting on the object is directed toward its equilibrium position and its magnitude is directly proportional to the displacement, $x$ , from that position: $F_{n e t} = - C x$ , where $C$ is a positive constant. This relationship is the physical cause of the sinusoidal motion we observe. We will analyze two canonical systems under ideal conditions.

Mass-Spring Oscillator: A block of mass $m$ (in kg) is attached to an ideal (massless) spring with spring constant $k$ (in N/m). The block slides on a frictionless horizontal surface. The equilibrium position is where the spring is neither stretched nor compressed.
Simple Pendulum: A point mass (or "bob") of mass $m$ is suspended from a pivot by a massless, inextensible string of length $ℓ$ (in m). The system is in a uniform gravitational field of strength $g$ (in m/s²). We specifically assume the motion is restricted to small angular displacements from the vertical equilibrium position.

Key Steps / Relations: From Force to Period

The period of an oscillator is not an arbitrary value; it is a direct consequence of the system's dynamics. We can derive it by applying Newton's second law.

Represent the Forces: Draw a free-body diagram for the object when it is displaced from equilibrium.
Apply Newton's Second Law: Write the equation of motion using $Σ F = m a$ for linear motion or $Σ τ = I α$ for rotational motion.
Formulate the Differential Equation: Substitute the calculus definitions for acceleration ( $a = \frac{d ^{2} x}{d t ^{2}}$ or $α = \frac{d ^{2} θ}{d t ^{2}}$ ) and any necessary geometric or algebraic relations to express the equation of motion in terms of a single position variable (e.g., $x$ or $θ$ ).
Identify the SHM Form: Manipulate the equation into the canonical second-order linear differential equation for SHM:
$\frac{d ^{2} x}{d t ^{2}} + ω^{2} x = 0$
Here, $x$ represents the displacement variable, and $ω$ is the angular frequency (in rad/s), a constant determined by the physical properties of the system.
Extract the Period: By comparing the system's specific differential equation to the canonical form, we can identify $ω^{2}$ . The period $T$ (in s), the time for one complete cycle, is then found using the fundamental relationship:
$T = \frac{2 π}{ω}$
The frequency $f$ (in Hz), the number of cycles per second, is the reciprocal of the period: $f = 1/ T = ω /2 π$ .

Outputs & Effects: The Derived Periods

For the Mass-Spring Oscillator:

The only horizontal force is the spring force, given by Hooke's Law: $F_{s} = - k x$ .
Applying Newton's second law: $Σ F_{x} = - k x = m a_{x}$ .
Substituting $a_{x} = \frac{d ^{2} x}{d t ^{2}}$ : $- k x = m \frac{d ^{2} x}{d t ^{2}}$ .
Rearranging into the canonical form: $\frac{d ^{2} x}{d t ^{2}} + \frac{k}{m} x = 0$ .
By comparison, we see that $ω^{2} = k / m$ . Therefore, $ω = k / m$ .
The period of the mass-spring system is:
$T_{s} = 2 π \frac{m}{k}$

For the Simple Pendulum:

The restoring force is the tangential component of gravity: $F_{res t ore} = - m g sin θ$ .
Using the rotational form of Newton's second law, the restoring torque is $τ = r F_{⊥} = - ℓ (m g sin θ)$ . The moment of inertia of a point mass is $I = m ℓ^{2}$ .
So, $Σ τ = I α$ becomes $- m g ℓ sin θ = m ℓ^{2} \frac{d ^{2} θ}{d t ^{2}}$ .
Here we apply the small-angle approximation: for small angles $θ$ (in radians), $sin θ \approx θ$ . This linearizes the restoring torque.
The equation becomes: $- m g ℓ θ \approx m ℓ^{2} \frac{d ^{2} θ}{d t ^{2}}$ .
Rearranging into the canonical form: $\frac{d ^{2} θ}{d t ^{2}} + \frac{g}{ℓ} θ = 0$ .
By comparison, we see that $ω^{2} = g / ℓ$ . Therefore, $ω = g / ℓ$ .
The period of the simple pendulum (for small angles) is:
$T_{p} = 2 π \frac{ℓ}{g}$

Regulation & Limits: The Domain of Validity

The mass-spring period formula is valid as long as the spring is "ideal"—it obeys Hooke's law, is massless, and there is no damping (e.g., friction or air resistance). Within this domain, the period is independent of the amplitude of oscillation.
The simple pendulum period formula is an approximation. Its validity is strictly limited to small angular displacements (typically < 15°). Beyond this limit, the restoring torque is no longer proportional to $θ$ , the motion is not SHM, and the period becomes dependent on the amplitude (a larger amplitude results in a longer period).

Key Models & Diagrams

The process of deriving the period of an SHM system can be visualized as a consistent workflow from physical principles to a predictive equation.

System Representation	Governing Law (Differential Form)	Key Parameter ( $ω^{2}$ )	Predicted Observable (Period, T)
Mass-Spring: Block on a frictionless surface attached to an ideal spring.	$\frac{d ^{2} x}{d t ^{2}} + (\frac{k}{m}) x = 0$	$ω^{2} = \frac{k}{m}$	$T_{s} = 2 π \frac{m}{k}$
Simple Pendulum: Point mass on a massless string, small angle displacement.	$\frac{d ^{2} θ}{d t ^{2}} + (\frac{g}{ℓ}) θ = 0$	$ω^{2} = \frac{g}{ℓ}$	$T_{p} = 2 π \frac{ℓ}{g}$

Key Components & Evidence

Period (T): The time required to complete one full cycle of motion. Its SI unit is the second (s).
Frequency (f): The number of cycles completed per unit time. Its SI unit is the Hertz (Hz), where 1 Hz = 1 s⁻¹.
Angular Frequency (ω): The rate of change of the phase of the sinusoidal motion, related to frequency by $ω = 2 π f$ . Its SI unit is radians per second (rad/s).
Mass (m): A measure of an object's inertia. In a mass-spring system, greater mass leads to a longer period. Its SI unit is the kilogram (kg).
Spring Constant (k): A measure of a spring's stiffness. A stiffer spring (larger k) produces a larger restoring force, leading to a shorter period. Its SI unit is Newtons per meter (N/m).
Length (ℓ): For a simple pendulum, the distance from the pivot to the center of mass of the bob. A longer pendulum has a longer period. Its SI unit is the meter (m).
Gravitational Acceleration (g): The strength of the local gravitational field. A stronger gravitational field increases the restoring force on a pendulum, leading to a shorter period. Its SI unit is meters per second squared (m/s²).
Small-Angle Approximation: The mathematical condition ( $sin θ \approx θ$ ) that linearizes the pendulum's equation of motion, making it a model for SHM.

Skill Snapshots

Causation

Driver: A restoring force directly proportional to displacement ( $F \propto - x$ ). → Change: The system exhibits simple harmonic motion with a period determined by the system's physical constants.
Driver: Increasing the mass m on a spring of constant k. → Change: The system's inertia increases, causing it to accelerate more slowly and increasing its period of oscillation according to $T_{s} \propto m$ .
Driver: Increasing the length ℓ of a simple pendulum. → Change: The lever arm for the gravitational torque increases, but the moment of inertia increases more significantly ( $I \propto ℓ^{2}$ ), resulting in a slower angular acceleration and a longer period according to $T_{p} \propto ℓ$ .

Comparison

A mass-spring system's period is determined by intrinsic properties (mass, spring constant), while a simple pendulum's period is determined by a mix of an intrinsic property (length) and an external field (gravity).
The restoring force in a mass-spring system is fundamentally linear (Hooke's Law), making it a true SHM oscillator (barring material limits). In contrast, a simple pendulum is an approximate SHM oscillator that behaves as such only in the limit of small angles.
For a mass-spring system, the period is independent of the local gravitational field. For a simple pendulum, the period is fundamentally dependent on g; the same pendulum will oscillate more slowly on the Moon than on Earth.

Change and Continuity

Baseline: An oscillator rests at its stable equilibrium position ( $x = 0$ ) with zero net force and zero velocity.
Change 1: When the object is displaced to its maximum amplitude ( $x = A$ ), its velocity is momentarily zero, but the restoring force and acceleration are at their maximum magnitude, directed back toward equilibrium.
Change 2: As the object passes through equilibrium ( $x = 0$ ), the restoring force and acceleration become zero, but its speed is at its maximum, causing it to overshoot.
Continuity: In an ideal (undamped) system, the total mechanical energy (the sum of kinetic and potential energy) is conserved throughout the entire oscillation cycle.

Common Misconceptions & Clarifications

Misconception: The period of a pendulum depends on its mass or the amplitude of its swing.
Clarification: The period of a simple pendulum is independent of its mass because mass cancels from both sides of the equation of motion ( $τ = I α$ ). It is also independent of amplitude only because we assume the small-angle approximation holds. For large angles, the period does increase with amplitude.
Misconception: Angular frequency ( $ω$ ) is the same as angular velocity ( $ω$ ).
Clarification: This is a classic notational confusion. For SHM, angular frequency $ω$ (a scalar constant in rad/s) describes how rapidly the oscillations occur. For circular motion, angular velocity $ω$ (a vector) describes the rate of rotation. While related through projection, they are not the same concept.
Misconception: A stronger spring (larger k) or a stronger gravitational field (g) will make an oscillator move more slowly.
Clarification: A larger k or g value corresponds to a stronger restoring force for a given displacement. A stronger force produces a greater acceleration, causing the object to complete a cycle more quickly. Therefore, the period is shorter ( $T_{s} \propto 1/ k$ and $T_{p} \propto 1/ g$ ).

One-Paragraph Summary

The period and frequency of a system in Simple Harmonic Motion are not arbitrary but are determined entirely by the physical characteristics of the system. By applying Newton's second law, we can derive a second-order differential equation of motion whose form, $\overset{x}{¨} + ω^{2} x = 0$ , is the mathematical signature of SHM. The term $ω^{2}$ encapsulates the system's physical properties, allowing us to derive the period using $T = 2 π / ω$ . For a mass-spring system, this yields $T_{s} = 2 π m / k$ , dependent on inertia and spring stiffness. For a simple pendulum, the small-angle approximation is crucial to linearize the dynamics, resulting in a period $T_{p} = 2 π ℓ / g$ that is independent of mass and amplitude. These equations provide powerful predictive tools for analyzing the timing of ideal oscillatory systems.

Frequency and Period of SHM - AP Physics C: Mechanics Study Guide