Simple harmonic oscillations — OCR A-Level Physics
Test yourself on Simple harmonic oscillations with OCR A-Level practice questions.
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Simple harmonic oscillations explained
These six quantities describe any oscillation.
Read the full explanation
Displacement x is the signed distance from the equilibrium position, so it is negative on one side and positive on the other. Amplitude A is the maximum magnitude of displacement, always positive. Period T is the time for one complete cycle, measured in seconds. Frequency f is the number of cycles per second, f = 1/T, in hertz. Angular frequency ω converts cycles to radians per second: ω = 2π/T = 2πf, so a 2.0 Hz oscillation has ω = 2π × 2.0 = 4π rad s⁻¹ ≈ 12.6 rad s⁻¹. Phase difference φ compares two oscillations of the same frequency, expressed in radians or degrees; a quarter-cycle lag is π/2 rad or 90°.
(b) angular frequency ω; ω = 2π/T or ω = 2πf
Angular frequency ω is the rate of change of phase of an oscillation, measured in radians per second. One complete cycle corresponds to 2π radians, so ω = 2π/T, where T is the period in seconds. Because f = 1/T, it follows that ω = 2πf, where f is the frequency in hertz. For example, an oscillation with period T = 0.50 s has ω = 2π/0.50 = 4π rad s⁻¹ ≈ 12.6 rad s⁻¹; the same oscillation has f = 1/0.50 = 2.0 Hz, and ω = 2π × 2.0 = 4π rad s⁻¹, confirming consistency. Angular frequency links the time description of an oscillation to the sinusoidal description x = A cos(ωt) or x = A sin(ωt).
(c)
This row is a specification sub-heading marker rather than a full assessed statement, so treat it as a signpost to the material that follows in section 5.3.1. Read it alongside the surrounding statements on simple harmonic motion, the defining equation a = −ω²x, and the graphical and energy descriptions of oscillations. Your task is to locate the sub-clauses that sit under this heading in the specification and check you can explain each one in your own words. Make a short checklist: define SHM, quote and use a = −ω²x, sketch and interpret displacement–time, velocity–time and acceleration–time graphs, and describe energy interchange between kinetic and potential stores.
(i) simple harmonic motion; defining equation a x 2 ~ =-
Simple harmonic motion (SHM) is oscillation in which the acceleration is directly proportional to the displacement from equilibrium and always directed towards that equilibrium position. The defining equation is a = −ω²x, where a is acceleration in m s⁻², ω is angular frequency in rad s⁻¹ and x is displacement in m. The minus sign shows that acceleration and displacement are always in opposite directions, so the restoring acceleration points back towards equilibrium. For example, if ω = 4.0 rad s⁻¹ and x = 0.10 m, then a = −(4.0)² × 0.10 = −1.6 m s⁻², directed towards equilibrium. The proportionality a ∝ −x distinguishes SHM from other oscillations.
(ii) techniques and procedures used to determine the period/frequency of simple harmonic oscillations
To determine the period T of a simple harmonic oscillator, time many complete oscillations with a stopwatch and divide the total time by the number of oscillations; frequency follows from f = 1/T. For a mass on a spring, displace the mass slightly and release it, then time 20 oscillations from a fixed fiducial mark, starting the stopwatch as the mass passes that mark. Repeat and average to reduce random timing error. For a pendulum, keep the amplitude small so the motion stays simple harmonic, and time from the equilibrium position where the bob moves fastest. Record the total time to the nearest 0.01 s, divide by the number of oscillations, and state T with its uncertainty.
(d) solutions to the equation a = -ω²x e.g. x = A cos(ωt) or x = A sin(ωt)
The defining equation of simple harmonic motion is a = -ω²x, where a is acceleration, x is displacement from equilibrium and ω is angular frequency. Its solutions are sinusoidal functions of time: x = A cos(ωt) or x = A sin(ωt), where A is the amplitude. Differentiating x = A cos(ωt) gives v = -Aω sin(ωt) and a = -Aω² cos(ωt), which equals -ω²x. The cosine form starts at maximum displacement when t = 0; the sine form starts at equilibrium moving in the positive direction. Both describe the same motion with a different choice of starting time, and the period is T = 2π/ω.
(e) velocity v = ±ω√(A² - x²) hence vmax = ωA
For a simple harmonic oscillator the speed at displacement x is given by v = ±ω√(A² - x²), where A is the amplitude and ω is the angular frequency. The ± sign shows that the oscillator passes through a given displacement in either direction. Speed is greatest at the equilibrium position, where x = 0, giving vmax = ωA. Speed is zero at the extremes of the motion, where x = ±A, because the oscillator momentarily stops before reversing. The equation follows from energy conservation or from combining the sinusoidal displacement and velocity expressions, and it allows the speed to be found at any displacement without knowing the time.
(f) the period of a simple harmonic oscillator is independent of its amplitude (isochronous oscillator)
An isochronous oscillator has a period that does not depend on its amplitude. For a mass on a spring the period is T = 2π√(m/k), and for a simple pendulum T = 2π√(l/g); neither expression contains the amplitude. Increasing the amplitude increases the distance travelled and the maximum speed in such a way that the time for one complete oscillation stays the same. This holds for an ideal simple harmonic oscillator, including a pendulum swinging through a small angle. At large amplitudes a real pendulum departs from simple harmonic motion and its period begins to increase, so the small-angle condition is part of the model.
(g) graphical methods to relate the changes in displacement, velocity and acceleration during simple harmonic motion.
Simple harmonic motion (SHM) is defined by acceleration proportional to displacement and directed towards the equilibrium position, a = −ω²x. Graphical methods let you relate displacement, velocity and acceleration at any instant. Plot x = A cos(ωt): the gradient at a point gives velocity, so v = −Aω sin(ωt), and the gradient of the velocity graph gives acceleration, a = −Aω² cos(ωt) = −ω²x. Velocity is maximum at equilibrium where x = 0, and zero at maximum displacement where a is maximum. Acceleration is always opposite in sign to displacement. Sketching these three curves on shared axes, or reading gradients and intercepts, reveals the phase relationships: v leads x by a quarter cycle and a is in antiphase with x.
Your focus
- Define displacement, amplitude, period, frequency, angular frequency and phase difference.
- Convert between period, frequency and angular frequency using ω = 2π/T = 2πf.
- Interpret a displacement–time graph to read off amplitude, period and phase difference.
Show all 27 objectives
- State and apply ω = 2π/T and ω = 2πf.
- Convert between period, frequency and angular frequency confidently.
- Use angular frequency in sinusoidal expressions for displacement.
- Identify the sub-clauses that sit under this specification heading.
- Explain each linked statement about simple harmonic oscillations in your own words.
- Use the heading as a checklist to plan revision of section 5.3.1.
- Define simple harmonic motion in terms of acceleration and displacement.
- Apply the defining equation a = −ω²x to calculate acceleration, displacement or angular frequency.
- Interpret the minus sign and the proportionality between acceleration and displacement.
- Describe a valid procedure for timing many oscillations and calculating the period of a simple harmonic oscillator.
- Calculate frequency from a measured period using f = 1/T with correct units.
- Evaluate sources of error in a timing experiment and suggest improvements to the procedure.
- Identify the amplitude and angular frequency from a given displacement-time equation.
- Show that x = A cos(ωt) or x = A sin(ωt) satisfies a = -ω²x by differentiation.
- Select the appropriate sine or cosine form from a stated initial condition.
- Calculate the speed of a simple harmonic oscillator at a given displacement using v = ±ω√(A² - x²).
- Determine the maximum speed from vmax = ωA and identify where it occurs.
- Explain why the speed is zero at the extremes of the motion.
- State that the period of a simple harmonic oscillator is independent of amplitude.
- Use the period equations for a mass-spring oscillator and a simple pendulum to justify amplitude independence.
- Explain the limits of the isochronous model for a real pendulum at large amplitudes.
- Sketch and interpret displacement–time, velocity–time and acceleration–time graphs for a simple harmonic oscillator.
- Use gradients of graphs to obtain velocity from displacement and acceleration from velocity.
- Explain how a = −ω²x is consistent with the shapes and phase relationships of the three graphs.
Simple harmonic oscillations exam tips
Marking Points
- Displacement is a signed vector-like quantity measured from the equilibrium position, not the total distance travelled.
- Amplitude is the maximum magnitude of displacement and is always positive.
- Period T is the time for one complete oscillation, measured in seconds.
- Frequency f = 1/T is the number of complete oscillations per second, measured in hertz.
- Angular frequency ω = 2π/T = 2πf, measured in rad s⁻¹.
- Phase difference compares the timing of two oscillations of the same frequency, in rad or degrees.
- Angular frequency ω is measured in radians per second (rad s⁻¹).
- One complete oscillation corresponds to a phase change of 2π radians.
- ω = 2π/T, where T is the period in seconds.
- ω = 2πf, where f is the frequency in hertz.
- Since f = 1/T, the two forms of the equation are equivalent.
- Angular frequency appears in sinusoidal descriptions such as x = A cos(ωt).
- SHM is defined by acceleration directly proportional to displacement from equilibrium and directed towards equilibrium.
- The defining equation is a = −ω²x.
- The minus sign indicates that acceleration is always opposite in direction to displacement.
- Acceleration is measured in m s⁻², displacement in m and angular frequency in rad s⁻¹.
- The constant of proportionality between a and x is ω², so a graph of a against x is a straight line through the origin with negative gradient.
- SHM requires a restoring force or acceleration that always acts towards the equilibrium position.
- Times many complete oscillations with a stopwatch or timer and divides the total time by the number of oscillations to obtain T.
- Uses a fiducial mark and starts or stops timing at the same point in the cycle, such as the equilibrium position, to avoid parallax and reaction-time bias.
- Repeats the measurement and averages the values, or plots a suitable graph, to reduce the effect of random errors.
- Calculates frequency from f = 1/T, with T in seconds and f in hertz.
- For a pendulum, keeps the amplitude small so the motion remains simple harmonic and the period is independent of amplitude.
- Estimates the uncertainty in T, for example by finding the range of repeated values or the resolution of the stopwatch divided by the number of oscillations timed.
- Recognises that x = A cos(ωt) and x = A sin(ωt) are both solutions of a = -ω²x.
- Identifies A as the amplitude and ω as the angular frequency, with ω = 2πf = 2π/T.
- Differentiates a displacement solution to obtain velocity and acceleration, showing that a = -ω²x follows.
- Distinguishes the initial conditions: cosine starts at maximum displacement, sine starts at equilibrium.
- Uses the relationship between ω, period and frequency to convert between forms.
- States or applies v = ±ω√(A² - x²) with A as amplitude, x as displacement and ω as angular frequency.
- Recognises that vmax = ωA occurs at x = 0, the equilibrium position.
- Recognises that v = 0 at x = ±A, the extremes of the oscillation.
- Interprets the ± sign as showing the two possible directions of motion at a given displacement.
- Calculates ω from the period or frequency before substituting into the velocity equation.
- States that the period of a simple harmonic oscillator is independent of amplitude.
- Uses T = 2π√(m/k) for a mass-spring oscillator and notes that amplitude does not appear.
- Uses T = 2π√(l/g) for a simple pendulum and notes that amplitude does not appear.
- Explains that a larger amplitude gives a larger maximum speed, so the period remains unchanged.
- Recognises that the isochronous property applies to ideal simple harmonic motion and that a real pendulum needs a small amplitude.
- Acceleration is proportional to displacement and directed towards equilibrium: a = −ω²x, with the minus sign showing opposition to displacement.
- Velocity is the gradient of the displacement–time graph, so v = −Aω sin(ωt) for x = A cos(ωt).
- Acceleration is the gradient of the velocity–time graph, giving a = −Aω² cos(ωt), consistent with a = −ω²x.
- At maximum displacement, velocity is zero and acceleration is maximum; at equilibrium, velocity is maximum and acceleration is zero.
- The velocity–time graph leads the displacement–time graph by a quarter of a period, and acceleration is in antiphase with displacement.
- Graphical methods include reading gradients, intercepts and phase differences from sketched or plotted curves.
Examiner Tips
- 💡Write the defining relationship for each quantity before substituting numbers, so the examiner can see your reasoning.
- 💡Check units carefully: T in s, f in Hz, ω in rad s⁻¹, phase difference in rad or degrees.
- 💡When comparing two oscillations, confirm they have the same frequency before quoting a phase difference.
- 💡Convert period to seconds before substituting into ω = 2π/T.
- 💡Keep answers in terms of π where possible to avoid premature rounding, then evaluate at the end.
- 💡Check that your value of ω is larger than f by a factor of about 6.28 (2π).
- 💡Use the specification sub-headings as a revision checklist and tick off each linked statement once you can explain it.
- 💡Write your own one-sentence summary of each sub-clause and test yourself without notes.
- 💡Link each sub-clause to a worked example or graph so the idea is anchored to a method.
- 💡Quote a = −ω²x and explain the meaning of the minus sign when asked to define SHM.
- 💡Check the sign of your calculated acceleration against the direction of displacement.
- 💡Remember that a graph of a against x for SHM is a straight line through the origin with negative gradient.
- 💡State the number of oscillations timed and show the division explicitly, for example T = total time ÷ 20.
- 💡Describe the fiducial mark and the point in the cycle used for timing, because this is a standard procedural mark.
- 💡Give the period to a sensible number of significant figures and include an uncertainty, such as T = 1.42 ± 0.02 s.
- 💡Check the initial condition described in the question before choosing sine or cosine.
- 💡Convert between period and angular frequency using ω = 2π/T before substituting into a solution.
- 💡Verify a proposed solution by differentiating twice and comparing with a = -ω²x.
- 💡Check that ω has units of rad s⁻¹ before substituting into the velocity equation.
- 💡Use vmax = ωA as a quick check on the maximum speed at equilibrium.
- 💡Keep the square root until the final step to avoid premature rounding.
- 💡Quote the period equation for the oscillator in the question to show amplitude is absent.
- 💡Use the phrase isochronous oscillator when explaining amplitude independence.
- 💡Mention the small-angle condition when discussing a pendulum, because it defines the simple harmonic model.
- 💡Sketch all three curves on the same time axis so phase relationships are visible and easy to compare.
- 💡State the defining relationship a = −ω²x before using graphs, then link each feature to it.
- 💡When reading a graph, identify the gradient or intercept the question asks for rather than describing the whole curve.
Common Mistakes
- Confusing displacement with distance travelled: displacement is signed and measured from equilibrium, whereas distance travelled accumulates and is never negative.
- Treating amplitude as a signed quantity: amplitude is the maximum magnitude of displacement and is always positive.
- Using ω = 2πT instead of ω = 2π/T: check that a larger period gives a smaller angular frequency.
- Expressing phase difference in seconds: phase difference is an angle in radians or degrees, not a time.
- Writing ω = 2πT instead of ω = 2π/T: a longer period must give a smaller angular frequency.
- Forgetting the factor 2π and using ω = f or ω = 1/T: angular frequency is in rad s⁻¹, not Hz.
- Mixing units by substituting T in milliseconds without converting to seconds first.
- Assuming ω is measured in hertz: hertz measures frequency f, while ω is measured in rad s⁻¹.
- Treating a sub-heading marker as a standalone examinable fact: it signposts the following clauses, so study the clauses it introduces.
- Skipping the sub-clauses because the heading looks empty: the assessed content is in the statements that follow.
- Learning the heading wording by rote instead of checking understanding of each linked statement.
- Assuming the heading limits the topic to one equation: it introduces several linked ideas about simple harmonic oscillations.
- Omitting the minus sign in a = −ω²x: without it the equation no longer shows that acceleration opposes displacement.
- Writing a = −ωx or a = −ω²x²: the correct relationship is a = −ω²x, with ω squared and x to the first power.
- Confusing acceleration with velocity: in SHM the acceleration is maximum at the extremes of motion, where velocity is zero.
- Treating ω as frequency in hertz: ω is angular frequency in rad s⁻¹, related to frequency by ω = 2πf.
- Timing only one oscillation, which makes the reaction-time error a large fraction of the measured period; instead time many oscillations and divide.
- Starting and stopping the stopwatch at different points in the cycle, which adds a systematic timing offset; instead use the same fiducial mark each time.
- Using a large amplitude for a pendulum and assuming the period is unchanged; instead keep the amplitude small so the simple harmonic approximation holds.
- Forgetting to convert a total time in milliseconds to seconds before dividing; instead express every time in seconds before calculating T.
- Treating ω as the frequency f rather than the angular frequency; instead use ω = 2πf.
- Believing that only one of the sine and cosine forms is a valid solution; instead recognise that both satisfy the equation and differ only in initial conditions.
- Dropping the minus sign in a = -ω²x, which reverses the direction of the restoring acceleration; instead keep the negative sign because acceleration is always directed towards equilibrium.
- Confusing amplitude A with displacement x; instead note that x varies with time while A is the maximum value of x.
- Using vmax = ωA² or vmax = A/ω; instead use vmax = ωA, which has units of m s⁻¹.
- Substituting x = A into the velocity equation and expecting a maximum; instead note that v = 0 at x = ±A.
- Treating the ± sign as an error; instead explain that it represents motion in either direction through the same displacement.
- Using frequency f in place of angular frequency ω; instead convert with ω = 2πf.
- Claiming that a larger amplitude always gives a longer period; instead state that period is independent of amplitude for simple harmonic motion.
- Assuming the isochronous property holds for any pendulum amplitude; instead note that the small-angle approximation is required.
- Thinking that amplitude affects the period through the maximum speed; instead explain that distance and speed increase together so the period is unchanged.
- Confusing period with frequency when discussing amplitude independence; instead note that f = 1/T, so frequency is also independent of amplitude.
- Thinking velocity and displacement are in phase: they are a quarter cycle out of phase, so velocity is zero when displacement is maximum.
- Believing acceleration is maximum at equilibrium: acceleration is zero there because displacement is zero, while velocity is maximum.
- Forgetting the minus sign in a = −ω²x, which would wrongly suggest acceleration acts with displacement rather than towards equilibrium.
- Confusing the gradient of a velocity–time graph with velocity itself: the gradient gives acceleration, not displacement.