Simple harmonic motion (SHM) is an oscillation in which acceleration is directly proportional to displacement from equilibrium and is always directed back toward equilibrium. Its defining equation is:
a = −ω²x
In plain English, the farther the object moves from its central resting position, the more strongly it accelerates back. A mass on an ideal spring is the clearest example, while a pendulum approximates SHM only when its angular displacement is small.
For IB Physics, you must understand the restoring force, equilibrium position, amplitude, period, frequency, and angular frequency. You should also be able to recognize SHM from equations and graphs, calculate the periods of springs and pendulums, and explain how kinetic and potential energy change during an oscillation.
What makes an oscillation simple harmonic?
An oscillation is repeated motion backward and forward about an equilibrium position. However, not every oscillation is simple harmonic.
For motion to be SHM, two conditions must be satisfied:
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The acceleration or resultant restoring force must be directly proportional to displacement from equilibrium.
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The acceleration or resultant restoring force must act in the opposite direction to the displacement.
These conditions are summarized by:
a ∝ −x
Introducing the constant ω² gives the defining IB equation:
a = −ω²x
Here, a is acceleration in m s⁻², x is displacement from equilibrium in metres, and ω is angular frequency in rad s⁻¹. The square on ω ensures that ω² is positive, so the negative sign alone describes the direction of acceleration.
This definition matters because simply moving back and forth is not sufficient. A bouncing ball, for example, repeats its motion but does not experience an acceleration proportional to its displacement from a central equilibrium position, so it is not undergoing SHM.
For wider syllabus coverage and linked practice, use RevisionDojo’s IB Physics C.1 Simple Harmonic Motion explained resources. This article focuses specifically on the meaning and physical behavior of SHM rather than duplicating the complete topic page.
Restoring force and the negative sign
A restoring force is a force that acts toward a stable equilibrium position. If an object is displaced to the right, the restoring force acts left; if it is displaced left, the force acts right.
For an ideal spring, Hooke’s law states:
F = −kx
The spring constant k, measured in N m⁻¹, indicates the stiffness of the spring. A larger value of k means that a greater force is required to produce the same extension or compression.
Using Newton’s second law, F = ma:
ma = −kx
Therefore:
a = −(k/m)x
Comparing this with a = −ω²x gives:
ω² = k/m
This demonstrates why an ideal mass-spring system performs SHM. Its acceleration is proportional to displacement and directed toward equilibrium.
A frequent exam mistake is to say that the restoring force always opposes the object’s motion. That is not the defining condition. The force opposes the object’s displacement from equilibrium; while the object moves back toward equilibrium, the force and velocity are in the same direction.
The key quantities used to describe SHM
QuantityMeaningUnitEquilibrium positionPosition where the resultant force is zerom or another position unitDisplacement, xSigned distance from equilibrium at a particular instantmAmplitude, AMaximum magnitude of displacementmPeriod, TTime for one complete oscillationsFrequency, fNumber of complete oscillations per secondHzAngular frequency, ωRate at which the phase of the oscillation changesrad s⁻¹PhaseStage reached within an oscillationrad
Period, frequency, and angular frequency are related by:
T = 1/f = 2π/ω
Therefore:
f = 1/T
and
ω = 2πf
Angular frequency is not the same as ordinary frequency. Frequency counts cycles per second, whereas angular frequency represents the corresponding phase change in radians per second. One complete oscillation corresponds to a phase change of 2π radians.
What happens during one complete oscillation?
Imagine a mass oscillating horizontally between x = −A and x = +A. At the positive extreme, its velocity is zero but its acceleration has maximum magnitude toward the negative direction.
The mass then moves toward equilibrium and speeds up. At equilibrium, displacement and acceleration are zero, but speed is maximum because the mass has converted its stored potential energy into kinetic energy.
After passing equilibrium, the mass continues because of its inertia. The restoring force now acts opposite to its velocity, so the mass slows until it reaches the negative extreme, where its velocity is again zero.
PositionDisplacementSpeedAccelerationPositive extreme+AZeroMaximum, negativeEquilibrium0MaximumZeroNegative extreme−AZeroMaximum, positive
Notice that zero acceleration does not mean zero velocity. At equilibrium, acceleration is instantaneously zero because the restoring force is zero, but the oscillator is moving at its greatest speed.
Why SHM produces sine and cosine graphs
The equation a = −ω²x has sinusoidal solutions. A general displacement equation can be written as:
x = A sin(ωt + φ)
where φ is the initial phase angle. A cosine equation is equally valid:
x = A cos(ωt + φ)
The choice between sine and cosine depends on the oscillator’s initial position and direction. If it starts at maximum positive displacement and is released from rest, x = A cos(ωt) is convenient. If it passes through equilibrium in the positive direction at t = 0, x = A sin(ωt) is convenient.
Differentiating displacement gives velocity, and differentiating velocity gives acceleration. For x = A sin(ωt + φ):
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x = A sin(ωt + φ)
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v = ωA cos(ωt + φ)
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a = −ω²A sin(ωt + φ) = −ω²x
The three graphs are shifted relative to one another:
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Velocity is π/2 radians, or one-quarter of a cycle, out of phase with displacement.
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Acceleration is π radians, or half a cycle, out of phase with displacement.
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Acceleration and displacement always have opposite signs except when both are zero.
An acceleration-displacement graph for SHM is a straight line through the origin with gradient −ω². This is one of the most direct ways to identify SHM from experimental data.
Under the current IB Physics course, qualitative SHM behavior and period relationships are studied at both SL and HL. Detailed use of phase angle and the full displacement, velocity, and energy equations forms part of the additional higher-level treatment. The RevisionDojo notes on defining and analyzing SHM provide focused equation and graph review.
Period of a mass-spring system
For a mass m attached to an ideal spring of constant k:
T = 2π√(m/k)
This relationship shows that:
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Increasing the mass increases the period, so the oscillation becomes slower.
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Increasing the spring constant decreases the period, so a stiffer spring oscillates more quickly.
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The ideal period does not depend on amplitude, provided the spring remains within the region where Hooke’s law applies.
For example, a 0.50 kg mass attached to a spring with k = 200 N m⁻¹ has period:
T = 2π√(0.50/200) ≈ 0.31 s
In a vertical system, gravity changes the equilibrium position by stretching the spring. It does not change the ideal period because displacement in the SHM equation is measured from the new equilibrium position, not from the spring’s natural length.
For derivations and additional examples, see RevisionDojo’s time period of oscillatory systems notes.
When does a pendulum perform SHM?
A simple pendulum consists of a small bob suspended from a fixed point by a light string. Its restoring force is the tangential component of weight:
F = −mg sin θ
This is not exactly proportional to angular displacement because sin θ is not exactly equal to θ. For sufficiently small angles measured in radians, however:
sin θ ≈ θ
The restoring force then becomes approximately proportional to displacement, so the pendulum behaves approximately as a simple harmonic oscillator. Its period is:
T = 2π√(L/g)
where L is the distance from the pivot to the bob’s center of mass and g is gravitational field strength.
The formula shows that the ideal small-angle period:
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increases when the pendulum is longer;
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decreases when gravitational field strength is greater;
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is independent of the bob’s mass;
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is approximately independent of amplitude only in the small-angle limit.
Textbooks sometimes use about 10° to 15° as a practical small-angle range, but the underlying requirement is mathematical rather than a universal cutoff. As amplitude increases, the small-angle approximation becomes less accurate and the true period becomes slightly longer than the SHM prediction.
Energy changes in SHM
In an ideal undamped oscillator, total mechanical energy remains constant. Energy repeatedly changes between kinetic energy and potential energy.
For an ideal spring oscillator:
Eₜ = ½kA² = ½mω²A²
At displacement x, the elastic potential energy is:
Eₚ = ½kx² = ½mω²x²
The kinetic energy is therefore:
Eₖ = Eₜ − Eₚ = ½mω²(A² − x²)
At the extreme positions, potential energy is maximum and kinetic energy is zero. At equilibrium, kinetic energy is maximum and the displacement-dependent potential energy is minimum.
Real oscillators experience friction, air resistance, or internal energy losses. Their amplitude gradually decreases, which is called damping. Such motion can remain approximately sinusoidal, but it is not ideal undamped SHM because mechanical energy and amplitude are no longer constant.
IB students at both levels should be able to describe the qualitative energy exchange. HL students should also be prepared to apply the quantitative energy relationships included in the current course. RevisionDojo’s SHM energy transformation notes develop this higher-level treatment.
Real-world examples and their limitations
Many systems can be modeled using SHM, at least for small displacements:
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A mass on a spring: close to ideal SHM while Hooke’s law applies and damping is small.
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A simple pendulum: approximate SHM for small angular displacements.
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A vibrating tuning fork: each prong can be modeled approximately as an oscillator near equilibrium.
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Atoms in a solid: atoms vibrate around equilibrium positions and can often be treated as harmonic oscillators for small vibrations.
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A floating object displaced vertically: approximate SHM can occur when the change in upthrust is proportional to displacement.
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Electrical LC circuits: charge and current oscillate sinusoidally in the ideal model, although the moving quantity is charge rather than a mechanical object.
SHM is therefore best understood as a model. A real system does not need to behave perfectly for the model to be useful, but you must identify the assumptions under which the restoring relationship is approximately linear.
A reliable IB exam method
When an exam question asks whether motion is simple harmonic, avoid relying only on the appearance of a repeating graph. Use the defining relationship.
A strong method is:
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Identify the equilibrium position.
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Define displacement from that position, including a positive direction.
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Determine the resultant force or acceleration.
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Show that it has the form F = −Cx or a = −Cx, where C is a positive constant.
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Compare with a = −ω²x to identify angular frequency or period.
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Check that the approximation used, such as Hooke’s law or the small-angle approximation, is valid.
For graph questions, remember that an a-x graph must be linear with a negative gradient. For calculation questions, convert masses to kilograms, lengths to metres, and distinguish frequency in hertz from angular frequency in rad s⁻¹.
Common mistakes include:
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defining amplitude as the total distance between the two extremes rather than the maximum displacement from equilibrium;
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treating every periodic motion as SHM;
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forgetting the negative sign in a = −ω²x;
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stating that acceleration is maximum at equilibrium;
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using the pendulum period formula without mentioning the small-angle condition;
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assuming that a vertical spring’s equilibrium is at its natural length;
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confusing one trip from one extreme to the other with a complete oscillation.
After learning the model, apply it using the C.1 SHM Questionbank and reinforce definitions with C.1 SHM Flashcards. Jojo AI can help explain errors, but always reconstruct the force relationship and sign convention independently before checking feedback.
Conclusion
Simple harmonic motion is oscillation governed by the defining relationship a = −ω²x. The negative sign shows that acceleration points toward equilibrium, while the proportionality shows that its magnitude increases with displacement.
An ideal spring provides the clearest example, and a pendulum approximates SHM at small angles. During each cycle, speed is greatest at equilibrium, acceleration magnitude is greatest at the extremes, and energy moves between kinetic and potential forms.
For effective revision, learn the definition precisely, connect it to force diagrams, and practise recognizing the model in unfamiliar situations. RevisionDojo’s C.1 Study Notes, Flashcards, Questionbank, and Jojo AI are most useful when combined in that order: understand the model, retrieve it from memory, and then apply it under exam conditions.