In simple terms
A friendly intro before the formal notes — no formulas yet.
Energy in simple harmonic motion
Cambridge 9702 Paper 4 — Energy in simple harmonic motion (17.2). Senpai Corner diagram-backed pilot with premium structure and live visuals.
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17.2 Energy in simple harmonic motion.
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During SHM, energy is constant exchanged between KE and PE.
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When one goes up, the other goes down and vice versa.
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E.g., the PE of a pendulum swing is maximum when it is at the top of the swing whereby it momentarily stops (KE=0) and reverse direction.
What this topic covers
The official Cambridge syllabus points this lesson works through.
- 17.2.1
Describe the interchange between kinetic and potential energy during simple harmonic motion
- 17.2.2
Recall and use for the total energy of a system undergoing simple harmonic motion
Explore the concept
Use the live diagram, PhET or GeoGebra sim, and synced steps — play it, drag controls, or tap a step.
Step-synced diagram — highlights what to look for in the simulation above.
Step 1
17.2 Energy in simple harmonic motion.
13 more simulations for this topic — run them in the Simulations section below
Simulations
Every simulation here runs the real model — try the steps on a card, then check what you see against the notes.
13 simulations · 4 to start with
Start herein this order — each one shows a different piece of the topic
- 3JCN PhysicsStart here · 19702 17.1 · 9702 17.2 · IB C.1
Simple Harmonic Motion (SHM)
Set mass, spring constant and amplitude; watch x, v, a graphs and the energy exchange of a block on a spring
Why this one: Match the energy exchange to the x and v graphs: KE peaks at the centre, PE at the ends.
Try this
- Set mass, spring constant and amplitude and read the period from the x graph.
- Double the amplitude and compare the period and the peak velocity.
- Compare the x, v and a graphs: note where each crosses zero and peaks.
- Watch the KE and PE exchange across one cycle.
Look for a is proportional to minus x, v peaks at x = 0, and the period is independent of amplitude.
3JCN Physics Simulation by Thomas Nguyen · CC BY 4.0
- oPhysicsStart here · 29702 5.1 · 9702 5.2 · 9702 17.2
Conservation of Mechanical Energy: Mass on a Vertical Spring
Mass oscillating on a vertical spring with live KE / GPE / EPE bar graphs; adjust mass and spring constant
Why this one: Watch KE, gravitational PE and elastic PE bars add to a constant total as the mass bounces.
Try this
- Watch the three bars through one full oscillation.
- Increase the mass and compare the bar heights.
- Increase the spring constant and compare the period.
Look for KE, GPE and EPE trade against each other while their total stays constant.
Simulation by Tom Walsh, oPhysics.com — made with GeoGebra · Licensed to MarkScheme (site: free for non-profit educational use; applets made with GeoGebra)
- PhysicsHubStart here · 39702 17.1 · 9702 17.2 · 9702 17.3
Spring Mass System Simulator
Vertical mass on a spring; student sets bob mass, spring constant, damping, rest length and gravity and reads period/frequency and decay in real time
Why this one: Add damping and watch the total energy drain away as the amplitude decays.
Try this
- Set damping to zero and read the period.
- Double the bob mass and read it again.
- Add damping and watch the decay.
Look for Period grows with mass and falls with spring constant; damping shrinks the amplitude while leaving the period almost unchanged.
PhysicsHub (@mattqdev) · MIT
- SimuPhysicsStart here · 49702 17.1 · 9702 17.2 · IB C.1
Mass-Spring Simple Harmonic Motion
A mass on a spring bounces while displacement is plotted against time and energy shifts between kinetic and potential
Why this one: Follow displacement against time while energy flows between kinetic and potential.
Try this
- Read the period from the displacement graph.
- Watch the energy at the ends of the motion.
- Watch the energy at the centre.
Look for Energy is all potential at the extremes and all kinetic at the centre.
Open on SimuPhysicsRuns on their siteSimuPhysics by Mohamed Abdelsalam · Licensed to MarkScheme (site publishes no licence; served with frame-ancestors self)
More simulations9 more on this topic — core ones first
- oPhysicsCore9702 17.1 · 9702 17.2 · IB C.1
Simple Harmonic Motion: Mass on a Spring
Mass on a horizontal spring: set initial displacement, mass and k; step through motion with x, v, a graphs
Try this
- Set an initial displacement and step through one period.
- Double the mass and compare the period.
- Double k instead and compare again.
Look for Acceleration is always opposite to displacement, and the period grows with mass and shrinks with spring constant.
Simulation by Tom Walsh, oPhysics.com — made with GeoGebra · Licensed to MarkScheme (site: free for non-profit educational use; applets made with GeoGebra)
- The Physics ClassroomCore9702 17.1 · 9702 17.2 · 9702 6.2
Vibrating Mass on a Spring
Place a mass on a spring, or different masses on two springs, press Start and measure the height over time
Try this
- Place a mass on the spring and press Start.
- Swap for a heavier mass and compare the period.
- Run both springs at once with different masses.
Look for A larger mass gives a longer period on the same spring.
Physics Interactives by The Physics Classroom · Licensed to MarkScheme (site terms otherwise permit linking only)
- 3JCN PhysicsCore9702 17.1 · 9702 17.2 · 9702 12.1
Circular Motion & SHM Relationship
Run uniform circular motion beside its projection to see SHM as a shadow of circular motion
Try this
- Run the circular motion and watch its projection move alongside.
- Compare the projection's displacement with the angle turned.
Look for The projection of uniform circular motion is SHM with x = r cos ωt, so ω of the SHM equals the angular speed.
3JCN Physics Simulation by Thomas Nguyen · CC BY 4.0
- 3JCN PhysicsCore9702 17.1 · 9702 17.2 · 9702 1.3
SHM for Lab
Timed mass-spring oscillator for lab: change mass and k and measure the period
Try this
- Set a mass and k, time ten oscillations and divide for the period.
- Double the mass and measure the period again.
- Double k instead and measure the period again.
Look for T = 2π√(m/k): quadrupling the mass doubles the period.
3JCN Physics Simulation by Thomas Nguyen · CC BY 4.0
- 3JCN PhysicsCore9702 17.1 · 9702 17.2 · IB C.1
Simple Pendulum
Set length and amplitude of a simple pendulum; read the period and compare with the formula
Try this
- Set a length and amplitude and read the period.
- Quadruple the length and compare the period with the formula.
- Change the amplitude only and compare.
Look for T = 2π√(L/g): the period doubles when the length quadruples and is independent of small amplitudes.
3JCN Physics Simulation by Thomas Nguyen · CC BY 4.0
- 3JCN PhysicsCore9702 17.1 · 9702 17.2 · 9702 5.1
Simple Pendulum Problem
Set length, mass and release angle of a pendulum; step through a worked period and speed problem
Try this
- Set length, mass and release angle and step through the worked period.
- Change the mass only and compare the period.
- Change the length only and compare the period and the speed at the bottom.
Look for Period depends on length, not mass; the speed at the lowest point follows from the height lost, mgh = ½mv².
3JCN Physics Simulation by Thomas Nguyen · CC BY 4.0
Key formulas
Tap any symbol to reveal exactly what it means and its units.
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Full topic notes
Formal explanation with the rigour you need for the exam.
Energy Interconversion in SHM
Imagine a mass-spring system: as the mass moves, its energy continuously transforms. When it's fastest at the equilibrium position, all energy is kinetic. As it slows down moving towards maximum displacement, kinetic energy converts into potential energy, stored in the stretched or compressed spring. At the very ends of its motion, it momentarily stops, meaning all its energy is now potential, ready to pull or push it back.
17.2 Energy in simple harmonic motion.
During SHM, energy is constant exchanged between KE and PE.
When one goes up, the other goes down and vice versa.
E.g., the PE of a pendulum swing is maximum when it is at the top of the swing whereby it momentarily stops (KE=0) and reverse direction.
The KE is maximum at the point of equilibrium (bottom PE=0).
Speed (v) is max when displacement x = 0. Hence KE is maximum.
Key Energy Formulas
The beauty of SHM lies in its predictable energy variations. We have specific formulas to quantify the kinetic and potential energy at any given displacement, . These equations are derived from the velocity-displacement relationship in SHM and are crucial for understanding how energy distribution shifts throughout an oscillation.
Kinetic Energy (KE):
Potential Energy (PE):
Total Energy (E):
Remember that total energy is constant in ideal SHM. If you know the amplitude, you can calculate the total energy, which simplifies finding KE or PE at any point!
Graphical Representation of Energy in SHM
The relationship between energy and displacement in SHM can be visualized with a graph. The potential energy () is a parabola opening upwards, zero at equilibrium () and maximum at the amplitudes (). The kinetic energy () is an inverted parabola, maximum at equilibrium and zero at the amplitudes. The total energy () is a constant horizontal line, representing the conservation of energy. The points where the KE and PE graphs intersect are where the energy is split equally between kinetic and potential.
Damping: Energy Loss
In the real world, no oscillation lasts forever. Damping is the process where energy is gradually lost from an oscillating system, typically converted into heat or sound. This energy dissipation causes the amplitude of the oscillations to decrease over time. The rate at which energy is removed determines the type of damping, influencing how quickly the system returns to equilibrium.
The physical cause of damping is any force that opposes motion and is non-conservative, such as air resistance or friction within the material of a spring. In a car's suspension system, shock absorbers are designed to provide critical damping, preventing the car from bouncing excessively after hitting a bump. This ensures a smooth ride and maintains tyre contact with the road.
Damping dissipates energy from the system, usually as heat or sound.
It causes the amplitude of oscillations to decrease over time.
Underdamping shows slow amplitude decay over many cycles.
Critical damping returns to equilibrium fastest without oscillation.
Overdamping returns to equilibrium slowly, also without oscillation.
Resonance: Energy Gain
While damping removes energy, forced oscillations involve an external force continuously supplying it. When this external driving force's frequency exactly matches the system's natural frequency, a phenomenon called resonance occurs. This leads to highly efficient energy transfer, causing a dramatic increase in the amplitude of oscillations. Think of pushing a child on a swing at just the right time!
Resonance has many important applications. In radio receivers, tuning circuits are adjusted to resonate at the frequency of a specific radio station, amplifying its signal while ignoring others. Magnetic Resonance Imaging (MRI) uses resonance of atomic nuclei in a magnetic field to create detailed images of body tissues. However, resonance can also be destructive, as famously demonstrated by the collapse of the Tacoma Narrows Bridge in 1940, where wind-induced oscillations matched the bridge's natural frequency.
Forced oscillations involve an external driving force.
Resonance occurs when driving frequency equals natural frequency.
Efficient energy transfer causes a large amplitude increase.
Damping limits the maximum amplitude at resonance.
Increased damping makes the resonance curve broader and lower.
Resonance is often misunderstood! Remember it's about efficient energy transfer, not just any large amplitude. Damping always works against resonance, reducing its peak effect.
Worked examples
See the formulas applied — reveal one step at a time, like the exam.
A 0.2 kg mass oscillates with SHM on a spring. Its angular frequency is 5 rad s.05 m. Calculate the kinetic energy when the displacement from equilibrium is 0.03 m.
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Identify given values: , , , .
A simple pendulum has a bob of mass 0.50 kg and oscillates with a period of 2.0 s. The amplitude of the oscillation is 8.0 cm. Calculate: (a) the total energy of the oscillation, and (b) the displacement at which the kinetic energy is equal to the potential energy.
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Identify given values and convert units: , , .
How it all connects
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Glossary
Key terms for this topic — skim now; the Check step will test them.
- Damping
In the real world, no oscillation lasts forever. Damping is the process where energy is gradually lost from an oscillating system, typically converted into heat or sound.
- natural frequency
When this external driving force's frequency exactly matches the system's natural frequency, a phenomenon called resonance occurs.
- Critical damping
Critical damping is the most efficient form of damping, allowing a system to return to equilibrium in the shortest time without oscillating.
- increased damping affect
Increased damping reduces the maximum amplitude at resonance and broadens the resonance curve.
Quick check
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Revision flashcards
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Key takeaways
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17.2 Energy in simple harmonic motion.
During SHM, energy is constant exchanged between KE and PE.
When one goes up, the other goes down and vice versa.
E.g., the PE of a pendulum swing is maximum when it is at the top of the swing whereby it momentarily stops (KE=0) and reverse direction.
The KE is maximum at the point of equilibrium (bottom PE=0).
Speed (v) is max when displacement x = 0. Hence KE is maximum.
Practice — then mark it
The whole point: a real Cambridge question, marked mark-by-mark.
the energy E of the oscillations.
Calculate the total energy of the oscillations.
Extra simulations & links
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Frequently asked
Checkpoint
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