Oscillations

• Acceleration proportional to displacement and opposite to it: \(a = -\omega^2 x\). Velocity leads displacement by \(\pi/2\); acceleration is in antiphase. Energy \(\propto x_0^2\).
• Proportional and opposite: \(a = -\omega^2 x\). Sine, cosine, minus sine: v leads by a quarter period, a is in antiphase. Energy \(\propto x_0^2\), recalled not given. Resonance is the maximum amplitude of a forced oscillation.

• 17.1.1 I can 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
• 17.1.2 I can understand that simple harmonic motion occurs when acceleration is proportional to displacement from a fixed point and in the opposite direction
• 17.1.3 I can use a = −ω²x and recall and use, as a solution to this equation, x = x₀ sin ωt
• 17.1.4 I can use the equations v = v₀ cos ωt and v = ±ω√(x₀² − x²)
• 17.1.5 I can analyse and interpret graphical representations of the variations of displacement, velocity and acceleration for simple harmonic motion
• 17.2.1 I can describe the interchange between kinetic and potential energy during simple harmonic motion
• 17.2.2 I can recall and use E = ½mω²x₀² for the total energy of a system undergoing simple harmonic motion
• 17.3.1 I can understand that a resistive force acting on an oscillating system causes damping
• 17.3.2 I can understand and use the terms light, critical and heavy damping and sketch displacement–time graphs illustrating these types of damping
• 17.3.3 I can 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

Units, significant figures and working are part of the physics. Give a calculated answer to the same number of significant figures as the least precise data, or one more; keep full precision in the working and round only at the end; write the unit with every final answer. A fifth of the qualification is experimental: Papers 3 and 5 test AO3 only, and their questions may be set in contexts outside the syllabus content, so the practical work in this chapter is set out as method, recording, graphs and uncertainties rather than as theory.

• “x = 0.050 sin (10 × 0.10) = 0.050 sin 1.0 = 0.00087 m.” Correct \(\omega t\) is in radians. \(\omega\) is in rad s−1, so \(\omega t\) is an angle in radians: set the calculator to radian mode before any s.h.m. calculation. In radians, \(0.050 \sin 1.0 = 0.042\) m. A result about fifty times too small, as here, is the signature of degree mode.
• “The amplitude is the distance from one extreme to the other.” Correct The amplitude \(x_0\) is the maximum displacement from the equilibrium position: half the peak-to-peak distance.
• “S.h.m. is motion in which the acceleration is towards the centre.” Correct Two conditions, both needed: the acceleration is proportional to the displacement from a fixed point, and in the opposite direction. A bouncing ball is always pulled back, but its acceleration is not proportional to its displacement, so it is not s.h.m.
• “The acceleration is greatest at the middle, where the speed is greatest.” Correct \(a = -\omega^2 x\): the acceleration is zero at equilibrium and greatest, \(\omega^2 x_0\), at the extremes, where the speed is zero.
• “x = x0 sin ωt is on the formulas sheet.” Correct Only \(a = -\omega^2 x\), \(v = v_0 \cos \omega t\) and \(v = \pm\omega\sqrt{x_0^2 - x^2}\) are printed for s.h.m. The line \(x = x_0 \sin \omega t\) on the sheet is under alternating current/voltage. For s.h.m. you recall it, together with \(\omega = 2\pi/T\) and \(E = \tfrac{1}{2}m\omega^2 x_0^2\).
• “v = v0 cos ωt, whatever the starting point.” Correct The printed pair assumes timing starts at equilibrium, moving in the positive direction. Released from an extreme at t = 0, the pair is \(x = x_0 \cos \omega t\) and \(v = -v_0 \sin \omega t\).
• “Resonance is when something vibrates at its natural frequency.” Correct Resonance is the maximum amplitude of a forced oscillation, and it occurs when the driving frequency equals the natural frequency. Every free oscillation is at the natural frequency; that is not resonance.
• “The amplitude is the distance from one extreme to the other.” Repair It is the maximum displacement from the equilibrium position: half the peak-to-peak distance.
• “S.h.m. is when the acceleration is opposite to the displacement.” Repair It must also be proportional to the displacement, measured from a fixed point. Opposite but not proportional gives periodic motion that is not simple harmonic.
• “The acceleration is greatest at the equilibrium position because the speed is greatest there.” Repair \(a = -\omega^2 x\), so a = 0 at x = 0 and is greatest at the extremes, where v = 0.
• “x = 0.050 sin (10 × 0.10) = 8.7 × 10−4 m.” Repair That is degree mode. \(\omega t\) is in radians: \(0.050 \sin(1.0\ \text{rad}) = 0.042\) m.
• “Released from maximum displacement at t = 0, so v = v0 cos ωt.” Repair The printed \(v = v_0 \cos \omega t\) goes with \(x = x_0 \sin \omega t\). Starting at an extreme, \(x = x_0 \cos \omega t\) and \(v = -v_0 \sin \omega t\): zero at release, then negative as it returns.
• “v = ω√(x0² − x²) gives the velocity, so it is always positive.” Repair It gives the speed, with ±: every displacement is passed twice per cycle, once each way. State the direction from the situation.
• “Velocity and displacement are in antiphase.” Repair Velocity leads displacement by \(\pi/2\) (a quarter period). It is the acceleration that is in antiphase with the displacement.
• “E = ½mω²x0² is on the formulas sheet.” Repair It is recall, as are \(x = x_0 \sin \omega t\) and \(\omega = 2\pi/T\). Only \(a = -\omega^2 x\) and the two velocity equations are printed for s.h.m.
• “The kinetic energy–time graph has the same frequency as the oscillation.” Repair \(E_\mathrm{K}\) and \(E_\mathrm{P}\) each peak twice per oscillation and are never negative: they vary at twice the frequency.
• “Doubling the amplitude doubles the energy.” Repair \(E = \tfrac{1}{2}m\omega^2 x_0^2\), so doubling the amplitude quadruples the energy.
• “Light damping makes the period much longer, and critical damping means the system eventually stops oscillating.” Repair Light damping makes the amplitude decay while the period stays essentially unchanged. A critically damped system never oscillates at all: it returns to equilibrium in the shortest time without overshooting.
• “Resonance is when the system vibrates at its natural frequency.” Repair Resonance is the maximum amplitude of a forced oscillation, which occurs when the driving frequency equals the natural frequency.

• Read the command word before you decide how much to write. This syllabus has fifteen of them: calculate, comment, compare, define, describe, determine, explain, give, identify, justify, predict, show (that), sketch, state and suggest. Define wants a precise meaning — for a physical quantity, usually an equation in words with every quantity named. State and give want a fact and nothing more. Describe wants the points or the features. Explain wants the reasons and the relationships — a describe-level answer to an explain question is incomplete however well written it is. Show (that) gives you the result and asks for the structured evidence that leads to it, so every step must appear — and a final value worked to one more significant figure than the one printed makes it plain that you calculated it rather than copied it. Sketch wants a freehand graph with its key features — intercepts, asymptotes, the shape — correct, but no plotted scale.
• Say which one leads. A phase difference of \(\pi/2\) is incomplete when the question asks for the relationship: write “B lags A by \(\pi/2\) rad” or “A leads B by a quarter of a period”. The oscillation that reaches the same point in its cycle earlier leads.
• Choose the equation by what you are given. Given a time: \(v = v_0 \cos \omega t\) (or \(-v_0 \sin \omega t\) if t = 0 was at an extreme). Given a displacement: \(v = \pm\omega\sqrt{x_0^2 - x^2}\). Given both a speed and a displacement and asked for \(x_0\) or \(\omega\): rearrange the second, \(x_0^2 = x^2 + v^2/\omega^2\). The s.h.m. calculator drill practises all three.
• Units check. \(\mathrm{kg} \times (\mathrm{rad\,s^{-1}})^2 \times \mathrm{m^2} = \mathrm{kg\,m^2\,s^{-2}} = \mathrm{J}\); the radian has no dimensions. If your energy answer has an ω in rad s−1 and comes out in J, the powers are right.
• “Loses energy” is not an explanation. Say what the force does (opposes the velocity, so does work against the motion), where the energy goes (to the surroundings as internal energy), and why that reduces the amplitude (energy \(\propto x_0^2\)).
• Three conditions to keep in mind. Every equation holds for undamped s.h.m. (light damping changes the period only negligibly). Every \(\omega t\) is in radians. And \(x = x_0 \sin \omega t\) with \(v = v_0 \cos \omega t\) assume the clock starts at equilibrium moving positively; state your starting point whenever it matters.
• Interleave with the chapters that use this one. When you reach Topic 21 (alternating currents), re-answer: what do the peak value, the period and \(\omega\) mean in \(x = x_0 \sin \omega t\), and why must \(\omega t\) be in radians? When you revise Topic 8 (stationary waves on strings and in air columns), ask which natural frequencies a system has and what happens when it is driven at one of them. Recalling a topic inside a new context is worth more than another pass over this chapter on its own; at A Level, Paper 4 assumes the whole of the AS content, so nothing here is ever finished with.

• Cambridge International AS & A Level Physics 9702 has five components. Topic 17 is A Level content, so it is examined in Papers 4 and 5. A Level content: examined in Paper 4 (A Level structured, which also requires the AS content) and, as practical context, Paper 5. AS Level candidates take Papers 1, 2 and 3; A Level candidates take all five, either staged over two years (Papers 1–3 in year one, Papers 4 and 5 in year two) or together in one series. Examinations are available in the June and November series, and in March in India.
• Across both the AS Level and the A Level the assessment objectives are weighted AO1 40% (knowledge and understanding), AO2 40% (handling, applying and evaluating information) and AO3 20% (experimental skills and investigations). AS candidates are graded a–e; A Level candidates A*–E. The Data and formulas sheet is printed as page 2 of Papers 1 and 2 and as pages 2 and 3 of Paper 4: it gives the constants and a short list of formulas. Every other equation in this chapter is one the syllabus says you must recall, and this chapter says which is which.
• There is no multiple-choice paper on A Level content. A Paper 4 question on this topic can ask you to define or state the condition for s.h.m. (both parts: proportional, and opposite in direction), sketch x, v or a against t on one time axis or the three damping curves from one starting displacement, and explain damping in terms of energy or resonance in terms of the driving and natural frequencies.
• Calculations of ω, T, f, phase difference, x, v and a at a given time or displacement, maximum speed and acceleration, and energy. On the Data and formulas sheet: \(a = -\omega^2 x\), \(v = v_0 \cos \omega t\) and \(v = \pm\omega\sqrt{x_0^2 - x^2}\). To recall: \(\omega = 2\pi f = 2\pi/T\), \(x = x_0 \sin \omega t\) and \(E = \tfrac{1}{2}m\omega^2 x_0^2\). An a–x graph gives ω from its gradient, −ω².
• Paper 3 skill: the period of an oscillating system found by timing an appropriate number of consecutive oscillations from a fiducial marker at the equilibrium position, the mass or length varied, and T² plotted against it. Paper 5: a supplied relationship such as \(T = 2\pi\sqrt{l/g}\) rearranged to a straight line, with error bars, a worst acceptable line and the uncertainty in the constant found.
• Read the command word before you decide how much to write. This syllabus has fifteen of them: calculate, comment, compare, define, describe, determine, explain, give, identify, justify, predict, show (that), sketch, state and suggest. Define wants a precise meaning — for a physical quantity, usually an equation in words with every quantity named. State and give want a fact and nothing more. Describe wants the points or the features. Explain wants the reasons and the relationships — a describe-level answer to an explain question is incomplete however well written it is. Show (that) gives you the result and asks for the structured evidence that leads to it, so every step must appear — and a final value worked to one more significant figure than the one printed makes it plain that you calculated it rather than copied it. Sketch wants a freehand graph with its key features — intercepts, asymptotes, the shape — correct, but no plotted scale.