Oscillations

All three subtopics here are printed under "A Level subject content" in the 9702 syllabus, and every objective carries the tier "A Level". None of it is AS. It is examined on Paper 4. Paper 1 is the AS multiple-choice paper, and the whole 9702 bank on this site comes from Paper 1, so no practice is tagged to this topic.

Syllabus points

17.1 Simple harmonic oscillations

• Understand and use the terms displacement, amplitude, period, frequency, angular frequency and phase difference in the context of oscillations, and express the period in terms of both frequency and angular frequency.
• Understand that simple harmonic motion occurs when acceleration is proportional to displacement from a fixed point and in the opposite direction.
• Use a = -ω²x and recall and use, as a solution to this equation, x = x₀ sin ωt.
• Use the equations v = v₀ cos ωt and v = ±ω√(x₀² - x²).
• Analyse and interpret graphical representations of the variations of displacement, velocity and acceleration for simple harmonic motion.

17.2 Energy in simple harmonic motion

• Describe the interchange between kinetic and potential energy during simple harmonic motion.
• Recall and use E = ½mω²x₀² for the total energy of a system undergoing simple harmonic motion.

17.3 Damped and forced oscillations, resonance

• Understand that a resistive force acting on an oscillating system causes damping.
• Understand and use the terms light, critical and heavy damping and sketch displacement-time graphs illustrating these types of damping.
• Understand that resonance involves a maximum amplitude of oscillations and that this occurs when an oscillating system is forced to oscillate at its natural frequency.

The vocabulary

• Displacement x: the distance of the oscillator from its equilibrium position, in a stated direction. A vector, so it can be negative.
• Amplitude x₀: the maximum displacement from equilibrium. Always positive, and note that it is measured from the middle, not from one extreme to the other. A pendulum swinging through a total of 8 cm has an amplitude of 4 cm.
• Period T: the time for one complete oscillation, in seconds.
• Frequency f: the number of complete oscillations per unit time, in hertz. f = 1/T.
• Angular frequency ω: measured in rad s⁻¹, with

ω = 2π/T = 2πf

• Phase difference: the fraction of a cycle by which one oscillation leads or lags another, expressed as an angle. A whole cycle is 2π rad, so a quarter of a cycle is π/2 rad and half a cycle is π rad.

The name angular frequency, for something that is not going round in a circle, is not an accident. Simple harmonic motion is the projection of uniform circular motion onto a diameter, which is why the same ω appears here as in topic 12, and why sines and cosines describe it.

The defining condition

Simple harmonic motion occurs when the acceleration is proportional to the displacement from a fixed point and is always directed towards that point, which is to say in the opposite direction to the displacement.

a = -ω²x

The minus sign is the whole of the physics. It says the acceleration always points back towards equilibrium, so the further the oscillator is displaced, the harder it is pulled back. That is what makes the motion repeat.

Two consequences follow immediately, and both are examined:

• A graph of a against x is a straight line through the origin with a negative gradient, and that gradient is -ω². Being asked to find the period from such a graph is common: read the gradient, take the square root of its magnitude to get ω, then T = 2π/ω.
• The period does not depend on the amplitude. This is called isochronism, and it is why a pendulum clock keeps time as its swing dies away, and why the amplitude of a mass on a spring can be changed without changing its period.

The equations of motion

Taking the oscillator to start at the equilibrium position:

x = x₀ sin ωt

v = v₀ cos ωt, where v₀ = ωx₀

a = -ω²x₀ sin ωt

If instead it starts at maximum displacement, the sine and cosine swap: x = x₀ cos ωt. Read the question to see where t = 0 is, since choosing the wrong one costs the whole calculation.

The equation that avoids time altogether is often the most useful:

v = ±ω√(x₀² - x²)

The ± is there because the oscillator passes each position twice per cycle, once in each direction. Two special cases are worth reading off it:

• At x = 0, the equilibrium position, v = ±ωx₀, the maximum speed.
• At x = ±x₀, the extremes, v = 0. The oscillator is momentarily at rest.

Worked example. A mass oscillates with amplitude 5.0 cm and period 0.40 s. Find the maximum speed and the speed at a displacement of 3.0 cm.

ω = 2 × 3.142 / 0.40 = 15.71

so ω = 15.7 rad s⁻¹.

v₀ = 15.71 × 0.050 = 0.7855

so the maximum speed is 0.79 m s⁻¹.

At x = 0.030 m:

v = 15.71 × √(0.050² - 0.030²)

The bracket first:

0.0025 - 0.0009 = 0.0016

whose square root is 0.040, so

v = 15.71 × 0.040 = 0.6284

giving 0.63 m s⁻¹.

Notice the 3, 4, 5 triangle hiding in that: at 3/5 of the amplitude the speed is 4/5 of the maximum. Examiners like these numbers for exactly that reason.

Worked example, maximum acceleration.

a = 15.71² × 0.050 = 12.34

so the maximum acceleration is 12.3 m s⁻², at the extremes of the motion where the displacement is greatest.

Reading the graphs

For an oscillator released from maximum displacement, so that x = x₀ cos ωt:

Three relationships hold whichever starting point is chosen, and they are what the graph questions test:

• Velocity is the gradient of the displacement graph. Where displacement is at a maximum, the gradient is zero, so the velocity is zero.
• Acceleration is the gradient of the velocity graph, and it is also -ω² times the displacement, so the acceleration graph is the displacement graph turned upside down and scaled.
• Velocity leads displacement by a quarter of a cycle, π/2 rad, and acceleration is exactly out of phase with displacement, a phase difference of π rad.

Sketching all three on the same time axis and checking those three statements is the fastest way to catch an error in an exam.

Energy

In an oscillator with no damping, energy is continuously exchanged between kinetic and potential, and the total stays constant.

• At the extremes, x = ±x₀: the oscillator is at rest, so the kinetic energy is zero and the potential energy is at its maximum, equal to the total.
• At the equilibrium position, x = 0: the speed is greatest, so the kinetic energy is at its maximum and the potential energy is at its minimum.
• In between, the sum is unchanged.

The total energy:

E = ½mω²x₀²

which comes straight from ½mv₀² with v₀ = ωx₀.

The most examined feature of this equation is that E is proportional to the square of the amplitude. Doubling the amplitude gives four times the energy, not twice.

The kinetic energy at any displacement is

E_k = ½mω²(x₀² - x²)

and the potential energy is the remainder,

E_p = ½mω²x²

so a graph of either against displacement is a parabola, and a graph of either against time oscillates at twice the frequency of the motion, because energy reaches a maximum twice per cycle.

Worked example. A 0.25 kg mass oscillates with amplitude 0.080 m at 2.0 Hz.

ω = 2 × 3.142 × 2.0 = 12.57

E = 0.5 × 0.25 × 12.57² × 0.080² = 0.1264

so the total energy is 0.13 J.

If the amplitude is doubled to 0.16 m, the energy becomes

0.1264 × 4 = 0.5056

that is 0.51 J.

Damping

Damping is the reduction in the amplitude of an oscillation caused by a resistive force, such as air resistance, friction or the viscosity of a liquid, which does work against the motion and dissipates energy from the system, usually as thermal energy.

The amplitude decreases, and because energy is proportional to the square of the amplitude, the energy falls faster still.

Three degrees of damping are named:

• Light damping. The system still oscillates, but the amplitude decreases gradually, following an exponential decay envelope. The period is very slightly longer than the undamped period. A pendulum in air.
• Critical damping. The system returns to equilibrium in the shortest possible time without oscillating. It passes through equilibrium once at most, and does not overshoot. This is what car suspension and the needle of an analogue meter are designed for.
• Heavy damping, sometimes called overdamping. The system returns to equilibrium slowly and without oscillating, taking longer than in the critical case. A door closer, or a pendulum swinging in treacle.

When sketching these, get two things right: light damping shows oscillations of decreasing amplitude with the period essentially unchanged, and neither critical nor heavy damping crosses the axis and comes back. Critical damping reaches equilibrium first.

Forced oscillations and resonance

Every system has a natural frequency f₀ at which it oscillates when displaced and released. If a periodic external force is applied, the system undergoes forced oscillations at the driving frequency, not at its own.

Resonance occurs when the driving frequency equals the natural frequency of the system. At resonance:

• the amplitude is a maximum,
• energy is transferred from the driver to the system most efficiently, and
• the displacement lags the driving force by π/2 rad.

A graph of amplitude against driving frequency shows a peak at f₀. Damping changes the peak in two ways that are asked for together: greater damping gives a lower and broader peak, and the peak shifts very slightly to a lower frequency.

Resonance is useful in a microwave oven, a radio tuning circuit, a musical instrument and magnetic resonance imaging. It is a hazard in a bridge or a building driven by wind or by an earthquake, in machinery at particular running speeds, and in a wine glass driven by a loud note. In every case the engineering answer is the same: change the natural frequency, or add damping.

Common mistakes

• Measuring amplitude as the full swing from one extreme to the other rather than from equilibrium.
• Dropping the minus sign from a = -ω²x, which loses the entire point of the definition.
• Saying the period depends on the amplitude.
• Leaving the calculator in degree mode when evaluating sin ωt or cos ωt. The argument is in radians.
• Choosing x = x₀ sin ωt when the oscillator was released from maximum displacement, where the cosine form is needed.
• Forgetting the ± in v = ±ω√(x₀² - x²), or squaring the bracket incorrectly by writing (x₀ - x)².
• Saying doubling the amplitude doubles the energy. It quadruples it.
• Saying kinetic energy is maximum at the extremes of the motion. It is maximum at equilibrium.
• Drawing an energy-against-time graph at the same frequency as the displacement, rather than twice it.
• Saying damping changes the natural frequency substantially, or drawing a critically damped curve that overshoots and oscillates.
• Saying resonance occurs when the driving frequency is greater than the natural frequency, rather than equal to it.
• Saying damping increases the resonant amplitude. It reduces it and broadens the peak.