Simple harmonic motion (SHM) is everywhere around you - from...
Physics148 views·Updated 27 Jul 2026·4 pagesAdvanced Higher Physics: Simple Harmonic Motion Notes
1 / 4
1
of 4
Understanding Simple Harmonic Motion
The key to SHM is surprisingly simple: the restoring force is always proportional to how far you are from the centre, but it always pulls you back towards equilibrium. Think of a mass on a spring - the further you stretch it, the harder it pulls back.
The mathematical relationship is F = -ky, where k is the spring constant and the negative sign shows the force opposes displacement. This connects directly to Newton's second law, giving us a = -ω²y - acceleration is proportional to displacement but in the opposite direction.
The angular frequency ω links everything together through ω = 2πf = 2π/T. This determines how quickly your system oscillates, whether it's a pendulum or a vibrating string.
Key insight: The graphs of displacement, velocity, and acceleration are all sinusoidal but shifted relative to each other - when displacement is maximum, velocity is zero!
2
of 4
SHM Equations and Graphs
The beauty of SHM lies in its predictable equations. For displacement, you'll use y = A sin ωt (starting from equilibrium) or y = A cos ωt (starting from maximum displacement), where A is the amplitude.
Velocity follows as v = -Aω cos ωt or v = -Aω sin ωt depending on your starting conditions. Notice how velocity is maximum when displacement is zero - that's when the object flies through the equilibrium position.
Acceleration is always a = -ω²y, directly linking it to displacement. The graphs show these three quantities as sine and cosine waves, each phase-shifted by 90° from the next.
Remember: When displacement is at its peak, velocity is zero (the object momentarily stops before changing direction), and acceleration is at maximum (pulling hardest towards centre).
3
of 4
Deriving Key SHM Relationships
One of the most useful equations in SHM is v = ±ω√, which connects velocity to position at any moment. The derivation uses the trigonometric identity sin²θ + cos²θ = 1, transforming the basic equations into this practical form.
This equation tells you something fascinating: as displacement increases, velocity must decrease. At the extreme positions , velocity becomes zero, whilst at equilibrium , velocity reaches its maximum value of ±ωA.
For pendulums, the period formula T = 2π√(l/g) only works for small angles. This is crucial for A-level problems - large amplitude pendulum swings don't follow simple harmonic motion.
Pro tip: The ± sign in the velocity equation accounts for direction - the object can be moving towards or away from equilibrium at any given displacement.
4
of 4
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Physics148 views·Updated 27 Jul 2026·4 pagesAdvanced Higher Physics: Simple Harmonic Motion Notes
Simple harmonic motion (SHM) is everywhere around you - from the pendulum in a grandfather clock to the vibrations in your phone. It's a special type of motion where objects oscillate back and forth with a predictable pattern, making it...
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Understanding Simple Harmonic Motion
The key to SHM is surprisingly simple: the restoring force is always proportional to how far you are from the centre, but it always pulls you back towards equilibrium. Think of a mass on a spring - the further you stretch it, the harder it pulls back.
The mathematical relationship is F = -ky, where k is the spring constant and the negative sign shows the force opposes displacement. This connects directly to Newton's second law, giving us a = -ω²y - acceleration is proportional to displacement but in the opposite direction.
The angular frequency ω links everything together through ω = 2πf = 2π/T. This determines how quickly your system oscillates, whether it's a pendulum or a vibrating string.
Key insight: The graphs of displacement, velocity, and acceleration are all sinusoidal but shifted relative to each other - when displacement is maximum, velocity is zero!
2
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SHM Equations and Graphs
The beauty of SHM lies in its predictable equations. For displacement, you'll use y = A sin ωt (starting from equilibrium) or y = A cos ωt (starting from maximum displacement), where A is the amplitude.
Velocity follows as v = -Aω cos ωt or v = -Aω sin ωt depending on your starting conditions. Notice how velocity is maximum when displacement is zero - that's when the object flies through the equilibrium position.
Acceleration is always a = -ω²y, directly linking it to displacement. The graphs show these three quantities as sine and cosine waves, each phase-shifted by 90° from the next.
Remember: When displacement is at its peak, velocity is zero (the object momentarily stops before changing direction), and acceleration is at maximum (pulling hardest towards centre).
3
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Deriving Key SHM Relationships
One of the most useful equations in SHM is v = ±ω√, which connects velocity to position at any moment. The derivation uses the trigonometric identity sin²θ + cos²θ = 1, transforming the basic equations into this practical form.
This equation tells you something fascinating: as displacement increases, velocity must decrease. At the extreme positions , velocity becomes zero, whilst at equilibrium , velocity reaches its maximum value of ±ωA.
For pendulums, the period formula T = 2π√(l/g) only works for small angles. This is crucial for A-level problems - large amplitude pendulum swings don't follow simple harmonic motion.
Pro tip: The ± sign in the velocity equation accounts for direction - the object can be moving towards or away from equilibrium at any given displacement.
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