Frequency and Period | AP Physics 1 Unit 7 Study Guide

Getting Started

This chapter explores simple harmonic motion (SHM), the periodic back-and-forth movement of an object, such as a mass on a spring or a swinging pendulum. We will focus on the timing of this motion, specifically what physical properties of a system determine how long it takes to complete one full cycle. The core question is: How can we predict the timing of an oscillation based on the object's physical characteristics?

What You Should Be Able to Do

After completing this section, you will be able to:

Define period and frequency, and relate them mathematically.
Identify the physical properties that determine the period of a mass-spring system.
Identify the physical properties that determine the period of a simple pendulum.
Calculate the period and frequency for both mass-spring and simple pendulum systems.
Predict how changes to a system's properties (e.g., mass, spring constant, length) will affect its period of oscillation.

Key Concepts & Mechanisms

We will compare the two primary models of simple harmonic motion you will encounter: the ideal mass-spring oscillator and the simple pendulum. Understanding their similarities and differences is key to predicting their behavior.

Feature	Model A: Mass-Spring Oscillator	Model B: Simple Pendulum	Why It Matters
System Components	An object of mass, m, attached to an ideal (massless) spring with spring constant, k.	A point mass (bob), m, suspended from a massless string of length, L, in a uniform gravitational field, g.	The components define the system and are the only variables that can potentially affect the period.
Source of Restoring Force	The spring itself. The force is described by Hooke's Law, $F_{s} = - k x$ , where x is the displacement from equilibrium.	The component of the gravitational force tangent to the arc of motion, $F_{g} sin θ$ .	The restoring force is what drives the object back to equilibrium. Its relationship with displacement determines if the motion is SHM.
Key Parameters Affecting Period	Mass (m) and spring constant (k).	String length (L) and local gravitational field strength (g).	This is the central concept. Notice that mass affects the spring system's period but not the pendulum's, and gravity affects the pendulum's period but not the spring's.
Equation for Period	$T_{s} = 2 π \frac{m}{k}$	$T_{p} = 2 π \frac{L}{g}$	These equations are your predictive tools. They encapsulate the relationship between the system's physical properties and its timing.
Assumptions & Limitations	The spring is massless and obeys Hooke's Law. There is no friction or air resistance.	The angle of displacement is small (typically < 15°). The string is massless and does not stretch. There is no air resistance.	These idealizations allow for the simple period equations. In real-world scenarios, factors like friction and large angles can alter the period.

Key Models & Diagrams

This matrix summarizes how to connect the physical properties of an oscillating system to its period, the key observable quantity.

System Model	Key Physical Parameters	Governing Equation for Period (T)	Predicted Observable
Mass-Spring Oscillator	Mass (m), Spring Constant (k)	$T_{s} = 2 π \frac{m}{k}$	Increasing mass increases the period (slower oscillation). Increasing spring stiffness decreases the period (faster oscillation).
Simple Pendulum	Length (L), Gravitational Field (g)	$T_{p} = 2 π \frac{L}{g}$	Increasing length increases the period (slower oscillation). Increasing gravity (e.g., moving from the Moon to Earth) decreases the period (faster oscillation).

Key Components & Evidence

Period (T): The time required to complete one full cycle of motion. It is measured in seconds (s). A longer period means a slower oscillation.
Frequency (f): The number of complete cycles of motion that occur per unit time. It is measured in Hertz (Hz), where 1 Hz = 1 cycle/second. A higher frequency means a faster oscillation.
Period-Frequency Relationship: Period and frequency are reciprocals: $T = \frac{1}{f}$ . This means an object with a high frequency has a short period, and vice-versa.
Simple Harmonic Motion (SHM): A specific type of periodic motion where the restoring force is directly proportional to the displacement from an equilibrium position. This condition leads to a constant period, independent of the amplitude.
Mass (m): A measure of an object's inertia. In a mass-spring system, more mass (more inertia) makes it harder for the spring's force to change the object's direction, resulting in 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) exerts a stronger restoring force for a given displacement, causing the object to accelerate more quickly and have a shorter period. Its SI unit is newtons per meter (N/m).
Length (L): For a simple pendulum, this is the distance from the pivot point to the center of mass of the bob. A longer pendulum has a longer path and experiences a less effective restoring force, resulting in a longer period. Its SI unit is the meter (m).
Gravitational Field Strength (g): The acceleration due to gravity at a specific location. A stronger gravitational field provides a stronger restoring force for a pendulum, resulting in a shorter period. Its SI unit is meters per second squared (m/s²) or newtons per kilogram (N/kg).

Skill Snapshots

Causation

Increasing the mass on a spring causes the system's inertia to increase, which results in a longer, slower period of oscillation.
Making a pendulum string longer causes the restoring force to be a smaller component of gravity for a given angle, which results in a longer period.
Using a stiffer spring (higher k) causes a greater restoring force for any given displacement, which results in a shorter, faster period of oscillation.

Comparison

The period of a mass-spring system depends on mass, whereas the period of a simple pendulum is independent of mass.
Unlike a mass-spring system, a simple pendulum's period is dependent on its location in the universe (i.e., the local value of g).
For both a mass-spring system and a simple pendulum in ideal SHM, the period is independent of the amplitude of the oscillation.

Change Over Time

Baseline: An ideal mass-spring system with mass m and spring constant k oscillates with a constant period $T_{s}$ .
Change 1: If the mass is replaced with a new mass of 4m, the new period will be $4$ or 2 times the original period.
Change 2: If a simple pendulum of length L is taken from Earth to the Moon (where g is about 1/6th of Earth's), its period will increase by a factor of $6$ .
Continuity: For an ideal oscillating system without friction, the period remains constant throughout the motion, regardless of whether the object is at its maximum speed at equilibrium or momentarily at rest at its maximum displacement.

Common Misconceptions & Clarifications

Misconception: A larger amplitude (pulling a spring back farther) makes the period shorter because the object travels faster.
- Clarification: For ideal SHM, the period is independent of amplitude. While a larger amplitude means the object has a greater distance to travel, the restoring force is also proportionally larger, causing a greater average speed. These two effects exactly cancel, leaving the period unchanged.
Misconception: A heavier pendulum bob will swing with a shorter period.
- Clarification: The period of a simple pendulum is independent of its mass ( $T_{p} = 2 π L / g$ ). While a heavier bob has a greater gravitational force acting on it, it also has proportionally greater inertia. These two effects cancel out, so mass does not affect the period.
Misconception: The equation for a pendulum's period works for any starting angle.
- Clarification: The equation $T_{p} = 2 π L / g$ is derived using the "small angle approximation," where we assume $sin θ \approx θ$ . This approximation is only valid for small angles (typically less than 15°). At larger angles, the restoring force is no longer directly proportional to the displacement, and the actual period becomes slightly longer than the one predicted by the formula.

One-Paragraph Summary

The timing of simple harmonic motion is characterized by its period (T), the time for one cycle, and its frequency (f), the cycles per second. These two quantities are inversely related by $T = 1/ f$ . The period of an oscillator is not determined by the amplitude of its motion but by the physical properties of the system itself, which define the relationship between inertia and the restoring force. For a mass-spring system, the period is determined by the mass and the spring constant ( $T_{s} = 2 π m / k$ ). For a simple pendulum oscillating at small angles, the period is determined by its length and the local gravitational field strength ( $T_{p} = 2 π L / g$ ). These equations provide a powerful way to predict the oscillatory behavior of these fundamental physical systems.

Frequency and Period of SHM - AP Physics 1: Algebra-Based Study Guide