PHYC10003 Chap.5 Simple Harmonic Motion
Simple Harmonic Motion
Simple harmonic motion is defined by a restoring force proportional and opposite to displacement. The resulting position is sinusoidal, with amplitude, angular frequency and phase constant playing distinct roles. Differentiation links displacement, velocity and acceleration, while initial position together with velocity direction selects the correct phase rather than leaving two trigonometric candidates.
Periodicity alone is insufficient: the model requires acceleration proportional to negative displacement from a clearly defined equilibrium. Speed is greatest at equilibrium, acceleration magnitude is greatest at the endpoints, and acceleration always points back toward equilibrium. Amplitude measures equilibrium to one extreme, while the full range is twice that value.
Angular frequency, ordinary frequency, and period describe the same timing with different units. Position generally gives two phase candidates, so the sign of initial velocity selects the physical one. Radians keep the phase rate consistent with angular frequency. Graphs and equations should agree at positive maximum, equilibrium crossings, and negative maximum, providing three rapid checks.
Changing mass or stiffness changes the ideal spring frequency, whereas changing amplitude does not under the linear model. Amplitude still changes maximum speed, acceleration, and energy, so timing independence does not mean the motion is otherwise unchanged. Units also reveal missing derivative factors: maximum speed has one inverse-time factor and maximum acceleration has two.
What this chapter covers
- 01
Restoring-force definition
- 02
Amplitude and peak-to-peak range
- 03
Angular frequency and period
- 04
Phase and phase constant
- 05
Velocity as a derivative
- 06
Acceleration opposite displacement
Worked example · free
Select phase from two initial conditions
- 1Set cosine of phase equal to minus 0.5.
- 1Identify plus and minus 2 pi over 3 as candidates.
- 1Use positive velocity to require negative sine.
- 1Choose minus 2 pi over 3 radians.
Key terms
- Equilibrium Position
- An equilibrium position is where the net restoring force vanishes.
- Restoring Force
- A restoring force points toward equilibrium after displacement.
- Period
- Period is the time required for one complete oscillation.
- Frequency
- Frequency is the number of complete cycles per second.
- Phase
- Phase is the angle specifying an oscillator's position within its cycle.
Simple Harmonic Motion FAQ
Is every periodic motion simple harmonic?
No. Simple harmonic motion specifically requires acceleration or restoring force proportional and opposite to displacement. A motion can repeat without satisfying that linear restoring condition. Test the force law around equilibrium rather than judging only from the repeated appearance of a graph.
Why is amplitude not the whole vertical range?
Amplitude measures maximum displacement from equilibrium to one extreme. The distance from negative maximum to positive maximum crosses equilibrium and is twice the amplitude, called the peak-to-peak range.
What should happen at maximum positive displacement?
Velocity is zero because the oscillator reverses direction, while acceleration has maximum magnitude toward equilibrium. With positive displacement, that acceleration is negative in the usual axis.
Exam move
Start by testing the defining restoring law rather than assuming that every repeated motion is simple harmonic. Choose equilibrium as zero, displace the system to either side, and check whether acceleration is proportional and opposite to displacement. Sketch one cosine cycle and label positive maximum, equilibrium crossings, and negative maximum with velocity and acceleration signs.
Add a velocity graph from the slope of displacement and an acceleration graph from the opposite of displacement. At each landmark, state which quantities vanish and which reach maximum magnitude. Use these relationships to catch sign errors without differentiating again. Solve phase questions by applying both initial conditions. Position usually produces two angle candidates; the sign of velocity selects the correct one.
Substitute the chosen phase back into both expressions at time zero and express an equivalent phase within the requested interval. Keep calculators in radians whenever angular frequency appears, and distinguish angular frequency from ordinary frequency and period by writing units beside every value. Read graph axes before extracting amplitude or period.
Measure amplitude from equilibrium to one extreme, not across the full peak-to-peak range, and measure period between equivalent phase points rather than arbitrary crossings. Build short comparison sets where amplitude, spring stiffness, or mass changes one at a time. Predict which changes affect angular frequency in the ideal model and which do not.
Use dimensional checks on maximum speed and acceleration, verifying one and two powers of inverse time respectively. Finally, translate between an equation, a graph, and a verbal starting state in both directions. The goal is to recognise one physical motion across several representations while preserving equilibrium, direction, and phase.
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