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A-Level Simple harmonic motion and the energy of oscillations
What the A-Level syllabus expects for Simple harmonic motion and the energy of oscillations, and how to practise it.
What the syllabus expects
- Give simple cases of free oscillation, in which particles swing back repeatedly to equilibrium while neither drawing from nor shedding energy to their surroundings
- Study how an oscillator moves through experiment and graph work
- Handle these quantities confidently: phase, phase difference, angular frequency, frequency, period and amplitude, and give the period in terms of frequency and also of angular frequency
- Understand a = -ω2x as what defines simple harmonic motion: acceleration scales with displacement from equilibrium and always aims back toward that equilibrium point
- Identify x = x0 sin ωt as one solution to a = -ω2x and employ it
- Spot and apply v = v0 cos ωt together with v = ±ω√(x0²−x²)
- Using graphs, lay out how displacement, velocity and acceleration link up all through simple harmonic motion
- Explain how energy swaps back and forth between kinetic and potential forms as an oscillator vibrates
- Give real cases of damped oscillation, noting how light, critical and heavy damping each behave and why critical damping matters, as in a car's suspension
Scope: light corresponds to under-damping and heavy to over-damping - Sketch how a driven oscillator's amplitude varies with driving frequency, peaking at resonance where the drive nears the system's natural frequency
Scope: natural frequency is the frequency of free oscillation - Understand, without equations, how damping reshapes the frequency response and blunts the resonance peak
- Offer real instances of driven oscillation and resonance, noting where resonance proves useful and where it must be kept away
How it's examined
Questions on this topic most often ask you to find, evaluate, explain, show. About 5% of the past-paper style questions in Rae's bank for this subject sit in this topic.
Worked examples
Example 1 (3 marks)
In the oscillating LC circuit (with C = 1.0 μF charged to 50 V, L = 10 mH, and period T), find every instant during the first full period (0 to t = T) when the energy held in the capacitor equals the energy held in the inductor.
Show the worked answer
In an LC oscillation the charge is Q = Q0 cos(omega*t), so capacitor energy U_C = Q²/2C = (Q0²/2C)cos²(omega*t) and inductor energy U_L = (1/2)L*I² = (Q0²/2C)sin²(omega*t). They are equal when cos²(omega*t) = sin²(omega*t), i.e. tan(omega*t) = +/-1, so omega*t = pi/4, 3pi/4, 5pi/4, 7pi/4. With omega = 2pi/T this gives t = T/8, 3T/8, 5T/8, 7T/8. (Numerically T = 2*pi*sqrt(LC) = 6.28x10⁻⁴ s, so t ~ 7.9x10⁻⁵, 2.4x10⁻⁴, 3.9x10⁻⁴, 5.5x10⁻⁴ s.)
Example 2 (2 marks)
(a) Give the definition of simple harmonic motion. [2]
Show the worked answer
Simple harmonic motion is oscillatory motion in which the acceleration is directly proportional to the displacement from a fixed (equilibrium) point, and is always directed towards that point (i.e. opposite in direction to the displacement). Symbolically, a = -w² x.
Example 3 (2 marks)
For the oscillating LC circuit, explain why current keeps flowing in the circuit even after the capacitor has fully discharged.
Show the worked answer
When the capacitor is fully discharged, the current in the circuit is at its maximum value and all the energy is now stored in the magnetic field of the inductor. As this current tries to fall, the inductor produces a back-emf (Lenz's law) opposing the change, so it keeps the current flowing in the same direction. This continuing current transfers the inductor's magnetic energy back into the capacitor, charging it with the opposite polarity.
More A-Level H2 Physics topics
Physical quantities, units, measurement uncertainty and vector basics · Types of force, turning effects and conditions for equilibrium · Kinematics, uniformly accelerated motion, momentum and Newton's laws · Energy stores and transfers, work, kinetic and potential energy, fields, power and efficiency · Falling freely, the gravitational potential energy of a uniform field, and how air resistance changes the motion · Impulse and the conservation of momentum and energy · all of A-Level H2 Physics