Simple harmonic motion (SHM) is everywhere around you - from...
Physics149 views·Updated 6 Sept 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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Physics149 views·Updated 6 Sept 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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We thought you’d never ask...
Our AI Companion is a student-focused AI tool that offers more than just answers. Built on millions of Knowunity resources, it provides relevant information, personalised study plans, quizzes, and content directly in the chat, adapting to your individual learning journey.
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