3.6.1.2ThermalStretch

Simple Harmonic Motion

A pendulum, a mass bouncing on a spring, a plucked guitar string — wildly different systems, but all obeying exactly the same mathematical rule. Whenever the restoring force pulling something back to equilibrium is proportional to how far it has been displaced, you get $simple harmonic motion (SHM)$ , and a single equation describes the position, velocity and acceleration of every single example.

45 min Video by Science Shorts 3.6.1 Periodic motion

SHM (Simple Harmonic Motion) — A-level PhysicsWatch the full walkthrough before the notes below.

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What you'll be able to do

State the defining equation of SHM, $a = - ω^{2} x$
Use $x = A cos (ω t)$ to describe the displacement of an oscillator
Calculate velocity and maximum speed using $v = \pm ω A^{2} - x^{2}$
Use the time period equations for a mass-spring system and a simple pendulum
Describe how kinetic and potential energy exchange during SHM, with the total energy constant

The defining equation of SHM

A system undergoes $simple harmonic motion$ if its acceleration is always directed toward a fixed equilibrium point and directly proportional to its displacement from that point. The minus sign is essential: it shows the acceleration (and hence the restoring force) always acts to pull the object back toward equilibrium, opposing the displacement.

From this single defining relationship, the whole mathematical description of SHM follows: a sinusoidal variation of displacement with time, at a single, fixed angular frequency $ω$ that depends only on the physical properties of the system (never on the amplitude).

a = - ω^{2} x

Defining equation of SHM — acceleration proportional to (and opposing) displacement.

Tip — The angular frequency ω in SHM is a genuinely fixed property of the system — for a mass on a spring or a pendulum swinging with small amplitude, ω does not depend on how far you pull it back before releasing it.

Displacement, velocity and maximum values

If an oscillator is released from its maximum displacement (amplitude $A$ ) at $t = 0$ , its displacement follows $x = A cos (ω t)$ . Its velocity is greatest as it passes through the equilibrium position ( $x = 0$ ) and exactly zero at the extremes of its motion ( $x = \pm A$ ), where it momentarily stops before reversing direction.

x = A cos (ω t) v = \pm ω A^{2} - x^{2} v_{m a x} = ω A

Displacement, velocity, and maximum speed (at x = 0).

1Maximum speed:

v_{m a x} = ω A = 4.0 \times 0.050 = 0.20

m s⁻¹.

2At

x = 0.030

v = ω A^{2} - x^{2} = 4.0 0.05 0^{2} - 0.03 0^{2}

v = 4.0 0.0025 - 0.0009 = 4.0 0.0016 = 4.0 \times 0.040

Answer

v_{m a x} = 0.20

m s⁻¹;

v = 0.16

m s⁻¹ at

x = 0.030

Tip — Maximum acceleration occurs at maximum displacement (a_max = ω²A), exactly where speed is zero — and maximum speed occurs at zero displacement, exactly where acceleration is zero. The two are never both large at once.

Time period: springs and pendulums

Two classic SHM systems have their own time period formulas, both derivable from Newton’s second law and Hooke’s law (or the small-angle approximation for a pendulum): a mass on a spring, and a simple pendulum swinging through small angles.

T = 2 π \frac{m}{k}

Time period of a mass m on a spring of stiffness k.

T = 2 π \frac{l}{g}

Time period of a simple pendulum of length l (small-angle oscillations only).

1Use

T = 2 π \frac{m}{k}

T = 2 π \frac{0.40}{25} = 2 π 0.016

Answer

T \approx 0.79

Tip — Neither time period formula depends on amplitude — a pendulum swinging through a larger (small) angle takes the same time per swing as one swinging through a smaller angle, which is precisely why pendulum clocks work reliably.

Energy in SHM

As an oscillator moves, energy continuously exchanges between kinetic and potential forms, but (ignoring damping) the $total$ mechanical energy stays constant. At maximum displacement, all the energy is potential (the oscillator is momentarily at rest); at zero displacement, all the energy is kinetic (moving at maximum speed).

E_{total} = \frac{1}{2} m ω^{2} A^{2}

Total mechanical energy of an SHM oscillator (constant throughout the motion).

1Total energy:

E = \frac{1}{2} m ω^{2} A^{2} = 0.5 \times 0.20 \times 5. 0^{2} \times 0.1 0^{2} = 0.5 \times 0.20 \times 25 \times 0.01

E = 0.025

3Potential energy at

x = 0.060

m scales as

x^{2}

E_{p} = E \times (\frac{x}{A})^{2} = 0.025 \times (\frac{0.060}{0.10})^{2} = 0.025 \times 0.36 = 0.009

4Kinetic energy:

E_{k} = E - E_{p} = 0.025 - 0.009

AnswerTotal energy = 0.025 J; kinetic energy at x = 0.060 m is 0.016 J.

Tip — On a graph of energy against displacement, the potential energy curve is a parabola (∝ x²) and the kinetic energy curve is an inverted parabola — their sum is a constant horizontal line, the total energy.

Equation recap

a = - ω^{2} x

Defining equation of SHM.

x = A cos (ω t)

Displacement, released from maximum amplitude.

v = \pm ω A^{2} - x^{2}, v_{m a x} = ω A

Velocity and maximum speed.

T = 2 π \frac{m}{k} (spring), T = 2 π \frac{l}{g} (pendulum)

Time period formulas.

E_{total} = \frac{1}{2} m ω^{2} A^{2}

Total energy of an SHM oscillator.

Common mistakes to avoid

Forgetting the minus sign in a = −ω²x, or thinking acceleration and displacement point the same way.

The acceleration always opposes the displacement (it’s a RESTORING force/acceleration) — the minus sign is not optional decoration.

Assuming maximum speed and maximum acceleration occur at the same point in the motion.

Maximum speed occurs at x = 0 (equilibrium); maximum acceleration occurs at x = ±A (the extremes) — they occur at opposite ends of the motion.

Thinking a larger amplitude changes the time period of an SHM oscillator.

Time period depends only on the physical properties of the system (m and k, or l and g) — never on amplitude, for genuine SHM.

Key takeaways

SHM is defined by a = −ω²x: acceleration proportional to displacement, always directed back toward equilibrium.
x = A cos(ωt); v = ±ω√(A²−x²), with maximum speed ωA at x = 0.
Time period: T = 2π√(m/k) for a spring, T = 2π√(l/g) for a pendulum — independent of amplitude.
Total energy (½mω²A²) stays constant, continuously exchanging between kinetic and potential forms.

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Simple Harmonic Motion

simple harmonic motion (SHM)

, and a single equation describes the position, velocity and acceleration of every single example.

45 min Video by Science Shorts 3.6.1 Periodic motion