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

Energy in Simple Harmonic Motion

Students will explore energy transformations within systems undergoing simple harmonic motion.

• Common Core State Standards
• HS-PS3-1
• HS-PS4-1

About This Topic

Energy in simple harmonic motion continuously transforms between kinetic and potential forms while total mechanical energy remains constant. In a mass-spring system, maximum elastic potential energy occurs at maximum displacement (amplitude) where velocity is zero, and maximum kinetic energy occurs at equilibrium where potential energy is zero. At all intermediate positions, energy is split between the two forms according to the displacement from equilibrium.

This energy perspective offers an alternative route to solving SHM problems without needing to know the exact time. Students can calculate maximum velocity from amplitude and the energy equation, connect amplitude to total energy, and understand why a spring with more energy oscillates faster at equilibrium but with the same period. HS-PS3-1 asks students to create computational models relating energy to motion, and HS-PS4-1 asks for mathematical representations of oscillatory phenomena.

Energy diagrams and graphs of KE and PE versus time or position are powerful visualization tools for this topic. When students graph both energies on the same axes and see that their sum is a horizontal line, the conservation principle becomes visually concrete. Active group work on constructing and interpreting these graphs before deriving the algebra makes the mathematics much more meaningful.

Key Questions

1. Explain how energy is conserved and transformed between kinetic and potential forms in SHM.
2. Analyze the relationship between amplitude, total energy, and maximum velocity in an SHM system.
3. Construct an energy diagram to represent the energy changes in a mass-spring system over time.

Learning Objectives

• Calculate the total mechanical energy of a mass-spring system given its amplitude and spring constant.
• Analyze the instantaneous kinetic and potential energy of an oscillating mass at various points in its motion.
• Compare the energy distribution in a simple harmonic oscillator at maximum displacement versus at equilibrium.
• Construct a graphical representation of kinetic and potential energy over one period of oscillation for a mass-spring system.
• Explain the relationship between the amplitude of oscillation and the total energy stored in a simple harmonic system.

Before You Start

Introduction to Energy: Kinetic and Potential Energy

Why: Students need a foundational understanding of what kinetic and potential energy are and how they are calculated before exploring their transformations in SHM.

Introduction to Oscillatory Motion and Waves

Why: Students should have a basic grasp of periodic motion and concepts like displacement and equilibrium before analyzing the energy within SHM.

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.
Kinetic Energy (KE)	The energy an object possesses due to its motion, calculated as 1/2 * mass * velocity^2.
Potential Energy (PE)	Stored energy in a system due to its position or configuration. In SHM, this is often elastic potential energy (1/2 * k * x^2) for a spring.
Amplitude	The maximum displacement or distance moved by a point on a vibrating body or wave measured from its equilibrium position.
Conservation of Mechanical Energy	In an ideal system (no friction or air resistance), the total mechanical energy (KE + PE) remains constant throughout the motion.

Watch Out for These Misconceptions

Common Misconception

At the equilibrium position in SHM, energy is zero because the object has no potential energy.

What to Teach Instead

At equilibrium, potential energy is zero but kinetic energy is at its maximum. Total energy is conserved and equals the system's total mechanical energy throughout the oscillation. An energy bar chart at equilibrium showing a full KE bar and empty PE bar makes this explicit.

Common Misconception

A larger amplitude means more energy, so the object oscillates faster (shorter period).

What to Teach Instead

A larger amplitude does mean more total energy, but it does not change the period. A larger amplitude spring has both more PE at the extremes and more KE at equilibrium (higher max speed), but these scale together so the time to complete one cycle is unchanged. This is a key result that distinguishes SHM from other oscillatory motion.

Active Learning Ideas

Inquiry Circle: Spring Energy Audit with Sensors

Groups use a motion sensor and force sensor to measure position and velocity of an oscillating mass-spring system in real time. Using the measured spring constant and mass, they calculate KE and PE at every moment and plot both on the same graph. Groups identify the equilibrium crossing, amplitude points, and verify total energy stays constant across the full oscillation.

60 min·Small Groups

Think-Pair-Share: Maximum Velocity from Energy

Students are given a spring constant, mass, and amplitude, and asked to find the maximum velocity of the oscillating mass without using time-based equations. Pairs use energy conservation to set maximum KE equal to total energy and solve for v. Class comparison highlights why the energy method is faster than the phase equation for this type of question.

20 min·Pairs

Gallery Walk: Interpreting SHM Energy Diagrams

Stations show a variety of KE-PE versus time or position graphs from different SHM systems and ask groups to identify amplitude, maximum speed, period, total energy, and the system's current energy state from a marked point. A synthesis station asks groups to sketch the energy diagram for a pendulum swinging between two heights.

35 min·Small Groups

Real-World Connections

• Mechanical engineers designing shock absorbers for vehicles use principles of SHM and energy transformation to absorb impacts and provide a smooth ride, ensuring passenger comfort and vehicle stability.
• Physicists studying seismic waves analyze the energy transfer in Earth's crust, which can exhibit characteristics of SHM, to understand earthquake magnitudes and predict potential damage zones.
• Instrument makers creating musical instruments like guitars or pianos rely on the predictable energy oscillations of strings and springs to produce specific musical pitches and tones.

Assessment Ideas

Quick Check

Present students with a diagram of a mass-spring system at various positions (e.g., amplitude, equilibrium, halfway point). Ask them to label each position with the relative amounts of KE and PE (e.g., 'Max KE, Zero PE', 'Half KE, Half PE', 'Zero KE, Max PE') and indicate if total energy is conserved.

Discussion Prompt

Pose the question: 'If you double the amplitude of a mass-spring system, how does the total energy change? Explain your reasoning using the energy equation and what you observe in energy diagrams.' Facilitate a class discussion where students share their calculations and interpretations.

Exit Ticket

Provide students with a graph showing KE and PE versus time for a mass-spring system. Ask them to identify the point on the graph where the velocity is maximum and the point where the potential energy is maximum. They should also write one sentence explaining why the sum of KE and PE is constant.

Frequently Asked Questions

How is energy conserved in simple harmonic motion?

Total mechanical energy in SHM equals one-half times the spring constant times the square of the amplitude, and this value is constant throughout the oscillation. As the object moves from amplitude to equilibrium, potential energy converts to kinetic energy. As it moves back to amplitude, kinetic energy converts back to potential energy, in an endless cycle.

How do you find maximum velocity in simple harmonic motion using energy?

At the equilibrium position, all energy is kinetic. Set the total energy (0.5kA squared) equal to kinetic energy (0.5mv squared) and solve for v. This gives maximum velocity as A times the square root of k over m, which equals A times omega, the angular frequency. This approach requires no knowledge of phase angle or time.

What does an energy diagram of SHM look like?

If you plot KE and PE versus position, KE is maximum at the center (equilibrium) and zero at the extremes (amplitude), while PE is zero at the center and maximum at the extremes. The two curves are mirror images that add to a constant total. If plotted versus time, both are sinusoidal curves out of phase by 90 degrees, and their sum is always constant.

How does graphical analysis support learning energy in SHM?

Energy graphs give students a visual handle on a concept that is otherwise purely algebraic. When students plot KE and PE from their own sensor data and see the sum flatten to a horizontal line, energy conservation becomes an observed fact rather than a stated principle. Group interpretation of these graphs, especially at unusual positions, builds the qualitative reasoning that multiple-choice and free-response questions test.