Physics · 12th Grade · Energy and Momentum Systems · Weeks 10-18

Simple Harmonic Motion: Springs and Pendulums

Students will analyze simple harmonic motion (SHM) in spring-mass systems and pendulums.

• Common Core State Standards
• HS-PS4-1

About This Topic

Simple harmonic motion (SHM) describes the oscillation of objects subject to a restoring force proportional to their displacement from equilibrium. The two most common systems studied in US 12th grade physics are the mass-spring system and the simple pendulum. Both exhibit SHM under specific conditions: the spring must follow Hooke's law and the pendulum must swing through small angles (less than about 15 degrees). Understanding when these conditions hold, and what breaks them, develops critical scientific thinking.

The period of a spring-mass system depends only on mass and spring constant, independent of amplitude, a non-obvious result that students often find surprising. The simple pendulum's period depends only on length and gravitational field strength, not on the mass of the bob or the amplitude (for small angles). These independence results have practical significance: they are why pendulum clocks keep consistent time regardless of the weight of the pendulum bob.

Physical pendulum and spring labs are ideal for HS-PS4-1, which asks students to use mathematical representations to support claims about relationships between variables. When students experimentally verify that doubling mass does not change pendulum period, or that tripling amplitude does not change spring period, they are collecting authentic evidence for a powerful physical principle.

Key Questions

1. Explain the conditions necessary for an object to undergo simple harmonic motion.
2. Analyze how the period of a spring-mass system depends on mass and spring constant.
3. Predict the period of a simple pendulum given its length and the acceleration due to gravity.

Learning Objectives

• Calculate the period and frequency of a mass-spring system given its mass and spring constant.
• Analyze the relationship between the length of a simple pendulum and its period for small angular displacements.
• Compare and contrast the factors affecting the period of a mass-spring system versus a simple pendulum.
• Identify the conditions under which a system exhibits simple harmonic motion, distinguishing it from other types of oscillation.
• Predict the effect of changing mass, spring constant, length, or gravitational acceleration on the period of SHM systems.

Before You Start

Newton's Laws of Motion

Why: Understanding forces, acceleration, and inertia is fundamental to grasping the concept of a restoring force and its effect on motion.

Circular Motion and Gravitation

Why: Knowledge of gravitational acceleration is necessary for calculating the period of a simple pendulum.

Hooke's Law

Why: Students must understand the relationship between force and displacement for a spring to analyze mass-spring systems.

Key Vocabulary

Simple Harmonic Motion (SHM)	A type of periodic motion where the restoring force is directly proportional to the displacement and acts in the direction opposite to that of displacement.
Period (T)	The time it takes for one complete cycle of oscillation to occur in a system undergoing simple harmonic motion.
Frequency (f)	The number of complete cycles of oscillation that occur per unit of time, typically measured in Hertz (Hz).
Spring Constant (k)	A measure of the stiffness of a spring; a higher spring constant indicates a stiffer spring that requires more force to stretch or compress.
Restoring Force	The force that acts to bring an oscillating object back to its equilibrium position.

Watch Out for These Misconceptions

Common Misconception

A heavier mass on a pendulum swings faster because it has more momentum.

What to Teach Instead

Pendulum period is independent of mass because both the restoring force (gravity component) and the inertia (resistance to acceleration) increase proportionally with mass, canceling out. Only length and g affect the period. Students who experimentally vary mass while holding length constant produce compelling data to counter this misconception.

Common Misconception

Larger amplitude means faster oscillation because the object has more distance to cover and must move faster.

What to Teach Instead

For SHM within the small-angle or Hooke's law regime, period is independent of amplitude. Larger amplitude means the restoring force and therefore maximum speed both increase proportionally, so the oscillation takes the same time. Students can test this directly with spring-mass systems, measuring period at small and large amplitudes.

Active Learning Ideas

Inquiry Circle: What Affects Period?

Groups systematically vary one factor at a time (mass on spring, spring constant, pendulum length, pendulum mass, amplitude) while measuring period with a stopwatch or motion sensor. Students record results in a structured data table and identify which variables affect period and which do not, supporting claims with evidence from their own measurements.

65 min·Small Groups

Think-Pair-Share: The Pendulum Clock Problem

Ask students to predict how a grandfather clock with a 1-meter pendulum should be adjusted if it runs slow. Pairs work through the period formula to determine the required length change, then discuss what would happen on the Moon. Whole-class sharing connects the formula to real-world mechanical clock design.

20 min·Pairs

Gallery Walk: SHM in Context

Stations present oscillating systems with given parameters (spring constant, mass, pendulum length) and ask groups to calculate period, frequency, and angular frequency, then identify what would change the oscillation rate. A final synthesis station asks groups to design a spring-mass system with a specified period.

35 min·Small Groups

Real-World Connections

• Engineers designing shock absorbers for vehicles analyze mass-spring systems to control oscillations and provide a smooth ride, ensuring passenger comfort and vehicle stability.
• Watchmakers and clock manufacturers historically relied on the predictable period of pendulums to create accurate timekeeping devices, with pendulum length being a critical adjustment for precision.
• Seismologists study the oscillations of the Earth's crust, which can exhibit characteristics of simple harmonic motion, to understand earthquake wave propagation and Earth's internal structure.

Assessment Ideas

Quick Check

Present students with scenarios: a mass on a spring, a swinging pendulum, a bouncing ball, a vibrating guitar string. Ask them to identify which systems exhibit SHM and briefly explain why or why not, referencing the conditions for SHM.

Discussion Prompt

Pose the question: 'Imagine you have a pendulum clock that is running too fast. Based on your understanding of SHM, what specific adjustment would you make to the pendulum to correct its timekeeping, and why?'

Exit Ticket

Provide students with the formula for the period of a spring-mass system (T = 2π√(m/k)). Ask them to calculate the new period if the mass is quadrupled, keeping the spring constant the same, and explain the result in one sentence.

Frequently Asked Questions

What conditions are needed for simple harmonic motion?

A system undergoes SHM when the restoring force is proportional to displacement and directed toward equilibrium. For a spring, this requires Hooke's law to hold. For a pendulum, this is approximately satisfied for small angles where the arc can be approximated as a straight line. Outside these conditions, the motion becomes anharmonic.

How does the period of a spring-mass system depend on mass and spring constant?

The period T equals 2 pi times the square root of mass divided by the spring constant. Increasing mass increases the period (slower oscillation). Increasing the spring constant decreases the period (stiffer spring oscillates faster). Neither amplitude nor maximum speed appears in this formula, which is one of the non-obvious results of SHM analysis.

Why does a pendulum clock need to be adjusted if moved to a higher altitude?

At higher altitude, gravitational acceleration g is slightly smaller. The pendulum period T equals 2 pi times the square root of length divided by g, so a smaller g increases the period. The clock runs slow. To compensate, the pendulum length must be shortened to restore the original period.

How does hands-on investigation improve understanding of SHM variables?

The independence of period from amplitude and (for pendulums) mass are genuinely counterintuitive. Students who measure this themselves and compare data across conditions develop real conviction about the result. Collecting contradictory evidence that matches theory is far more convincing than being told the formula, and it builds the experimental reasoning skills that AP Physics exams assess.