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Simple harmonic motion (SHM) is one of the most important types of periodic motion in physics. It appears everywhere — from the back-and-forth swing of a pendulum to the vibration of atoms in a crystal lattice. Understanding SHM thoroughly is essential for Edexcel A-Level Physics, because it connects to waves, resonance, and even quantum mechanics.
Simple harmonic motion is defined as oscillatory motion in which the acceleration of the object is:
This is expressed mathematically as:
a = −ω²x
where:
The negative sign is crucial — it tells us that the acceleration always acts in the opposite direction to the displacement. When the object is displaced to the right of equilibrium (positive x), the acceleration points to the left (negative). When displaced to the left (negative x), the acceleration points to the right (positive). This restoring behaviour is what causes the object to oscillate rather than simply move away.
flowchart LR
A["Object displaced\nright (+x)"] -->|"Restoring force\npoints left (−a)"| B["Object passes\nthrough equilibrium\n(x = 0, a = 0)"]
B -->|"Momentum carries\nobject left (−x)"| C["Object displaced\nleft (−x)"]
C -->|"Restoring force\npoints right (+a)"| B
B -->|"Momentum carries\nobject right (+x)"| A
For a system to undergo SHM, two conditions must be satisfied:
If the restoring force is proportional to displacement but not directed towards equilibrium, the motion is not SHM. If the restoring force is directed towards equilibrium but not proportional to displacement, the motion is oscillatory but not simple harmonic.
Common exam mistake: Students often state only one condition — either that acceleration is proportional to displacement, or that it is directed towards equilibrium. Both conditions must be stated for full marks. The word "proportional" alone is not enough; you must also specify the direction.
Several quantities describe the motion:
| Quantity | Symbol | Unit | Meaning |
|---|---|---|---|
| Displacement | x | m | Distance from equilibrium at any instant |
| Amplitude | A | m | Maximum displacement from equilibrium |
| Period | T | s | Time for one complete oscillation |
| Frequency | f | Hz | Number of complete oscillations per second |
| Angular frequency | ω | rad s⁻¹ | Rate of change of phase angle |
| Phase | φ | rad | Position in the cycle at a given time |
These are connected by the relationships:
f = 1/T
ω = 2πf = 2π/T
Angular frequency ω is not the same as the angular velocity of a rotating object, although the mathematics is analogous. In SHM, ω tells us how rapidly the phase of the oscillation advances. A higher ω means more oscillations per second.
The defining equation a = −ω²x is not an assumption — it arises directly from the physics of the restoring force. Consider a mass m on a spring with spring constant k:
This derivation shows that the SHM equation is a consequence of a linear restoring force. Any system where F ∝ −x will produce SHM.
When a mass attached to a spring is displaced from its natural length and released, the spring exerts a restoring force given by Hooke's law: F = −kx. Since the restoring force is proportional to displacement and directed towards equilibrium, this system performs SHM (provided the spring obeys Hooke's law and the oscillations are not so large that the spring deforms permanently).
A pendulum bob displaced through a small angle θ from vertical experiences a restoring component of gravitational force proportional to sin θ. For small angles (typically less than about 10°), sin θ ≈ θ in radians, making the restoring force approximately proportional to displacement. Under this small-angle approximation, the pendulum performs SHM. At larger angles, the motion is periodic but not truly simple harmonic.
In every case, the restoring mechanism is different, but the mathematical description is the same: a = −ω²x.
The angular frequency ω determines how quickly the system oscillates. It depends on the physical properties of the system, not on the amplitude. For example:
This is a remarkable result: the angular frequency (and therefore the period and frequency) of SHM does not depend on the amplitude. Whether you pull a spring a little or a lot, the time for one complete oscillation is the same (provided the motion remains simple harmonic). This property is called isochronicity and was first observed by Galileo watching a swinging chandelier in Pisa Cathedral.
A mass of 0.4 kg on a spring is observed to oscillate 15 times in 12 seconds. Calculate: (a) the frequency (b) the period (c) the angular frequency (d) the spring constant
Solution:
(a) f = 15/12 = 1.25 Hz
(b) T = 1/f = 1/1.25 = 0.80 s
(c) ω = 2πf = 2π × 1.25 = 2.5π ≈ 7.85 rad s⁻¹
(d) Since ω² = k/m: k = mω² = 0.4 × (2.5π)² = 0.4 × 6.25π² = 0.4 × 61.69 ≈ 24.7 N m⁻¹
The acceleration of an oscillating object is measured at two displacements:
| Displacement x (m) | Acceleration a (m s⁻²) |
|---|---|
| 0.020 | −3.2 |
| 0.050 | −8.0 |
Show that this data is consistent with SHM and find the frequency.
Solution:
For SHM, a/x must be constant (= −ω²):
At x = 0.020: a/x = −3.2/0.020 = −160 s⁻² At x = 0.050: a/x = −8.0/0.050 = −160 s⁻²
The ratio is constant and negative, confirming SHM with ω² = 160 s⁻².
ω = √160 = 4√10 ≈ 12.65 rad s⁻¹
f = ω/(2π) = 12.65/6.283 ≈ 2.0 Hz
A mass on a spring is pulled down 8 cm and released. It completes 5 full oscillations in 4 seconds. Calculate the magnitude of the acceleration at (a) maximum displacement and (b) half the amplitude.
Solution:
T = 4/5 = 0.8 s, so ω = 2π/T = 2π/0.8 = 2.5π rad s⁻¹
A = 0.08 m
(a) At x = A: |a| = ω²A = (2.5π)² × 0.08 = 6.25π² × 0.08 ≈ 4.93 m s⁻²
(b) At x = A/2 = 0.04 m: |a| = ω²x = 6.25π² × 0.04 ≈ 2.47 m s⁻²
Note that halving the displacement halves the acceleration — this proportionality is the hallmark of SHM.
| Formula | Use |
|---|---|
| a = −ω²x | Defining equation of SHM |
| ω = 2πf = 2π/T | Relating angular frequency to f or T |
| ω = √(k/m) | Angular frequency for mass-spring |
| ω = √(g/L) | Angular frequency for pendulum |
| f = 1/T | Frequency-period relationship |
Seismometers detect ground motion during earthquakes using a mass on a spring inside a housing. When the ground shakes, the housing moves but the suspended mass (due to its inertia) tends to stay still. The relative motion between the mass and housing is recorded. The natural frequency of the seismometer's mass-spring system determines which earthquake frequencies it is most sensitive to. Low-frequency seismometers (long period, large mass, soft spring) detect distant earthquakes; high-frequency ones detect nearby events.
SHM is a model — an idealisation. Real oscillating systems experience friction, air resistance, and other complications. But SHM provides the starting point for understanding all oscillatory behaviour. Once you understand the simple harmonic case, you can add damping, driving forces, and resonance as modifications.
In the next lesson, we will develop the mathematical equations that describe how displacement, velocity, and acceleration vary with time during SHM.