Cambridge A Level Physics: SHM and Damped/Resonant Oscillations Notes

Describing Oscillations

• An oscillation is the repetitive variation with time t of the displacement x of an object about the equilibrium position (x = 0).
• Pendulum oscillation between points A and B; on a displacement–time graph the motion is represented by a wave with amplitude equal to the displacement x.
• Equilibrium position (x = 0) is the position when there is no resultant force acting on the object.
• Displacement (x) is the distance of a point from the equilibrium position; it is a vector quantity and can be positive or negative; units: metres (m).
• Amplitude (A or x) is the maximum value of the displacement on either side of the equilibrium position; units: metres (m).
• Wavelength (λ) is the length of one complete oscillation measured from the same point on two consecutive waves; units: metres (m).
• On a displacement–time graph, λ is not directly shown for a single oscillator (SHM does not have a traveling distance in space like a traveling wave).
• Phase in waves can be measured in fractions of a wavelength, degrees, or radians.
• One complete oscillation corresponds to 1 wavelength, 360° or 2π radians.
• Phase difference between two points/waves:
  • In phase: crests/troughs align; phase difference = 0 (or 2π, etc.).
  • Anti-phase: crest aligns with trough; phase difference = π radians (180°).
• Phase difference of 1/4 λ is an example of a relative lead/lag between waves.

Phase and Phase Difference (Waves vs SHM)

• Phase difference describes how far in front or behind one wave is relative to another.
• A phase difference of 1/4 λ corresponds to a phase shift of 90° or π/2 radians.
• In SHM, if two points on the same oscillator are compared, their instantaneous displacements have a fixed phase relation (e.g., x(t) and v(t) are 90° out of phase at certain instants).

Worked Example: Angular frequency from time period

• Example given: T = 2.6 − 0.5 = 2.1 s (time period from a graph).
• Angular frequency ω is found from: oxed{ \omega = rac{2\, ext{π}}{T} }
• If T = 2.1 s, then ω ≈ rac{2\, ext{π}}{2.1} \approx 3.0 ext{ rad s}^{-1}

Simple Harmonic Motion (SHM)

Conditions for SHM

• Oscillations are periodic (repeating).
• There is a central equilibrium point (the fixed point).
• Displacement, velocity and acceleration change continuously.
• There is a restoring force always directed toward the fixed point.
• Magnitude of the restoring force is proportional to displacement: a \propto -x where a is acceleration and x is displacement.
• More precisely: a = -\omega^2 x where \omega is the angular frequency.
• This differential form leads to sinusoidal motion in time.
• For SHM, the acceleration is always directed opposite to displacement and proportional in magnitude to the displacement.

Key SHM Equations

• Acceleration: a = -\omega^2 x
  • Maximum acceleration occurs at maximum displacement (x = ±x_0).
• Displacement (from equilibrium, starting at x = 0 at t = 0):
  • If starting from equilibrium, x = x0 \sin(\omega t) where x0 is the amplitude.
• Velocity (time derivative of displacement):
  • v(t) = \frac{dx}{dt} = x_0 \omega \cos(\omega t)
  • Maximum velocity occurs when displacement is zero (at the equilibrium position), i.e., at x = 0.
  • Maximum speed is v{\max} = \omega x0 at the equilibrium position.
• Alternative form relating speed to displacement:
  • v = \pm \omega \sqrt{x_0^2 - x^2} (derived from energy or the derivative form).
• Displacement–time graph forms:
  • If starting at the equilibrium, the displacement-time graph is a sine curve: x(t) = x_0 \sin(\omega t).
  • The velocity-time graph is a cosine curve and 90° out of phase with displacement.
  • The acceleration-time graph is a negative sine curve and 90° out of phase with velocity.

Worked Examples (SHM displacement and angular frequency)

• Example 1: A mass on a spring with amplitude 4.3 cm; period 0.8 s; released from the center (x=0). Find displacement at t = 0.3 s.
  • Known: x_0 = 4.3 cm = 0.043 m, T = 0.8 s.
  • Angular frequency: \omega = \frac{2\pi}{T} = \frac{2\pi}{0.8} = 7.85\ \text{rad s}^{-1}.
  • Displacement: x = x_0 \sin(\omega t) = 0.043 \sin(7.85 \times 0.3) \approx 0.0304\ \text{m} = 3.0\ \,\text{cm}. (positive means same side of equilibrium as the starting point)
• Example 2: Pendulum-like SHM with amplitude x_0 = 15 cm and frequency f = 6.7 Hz; find speed at a displacement x = 12 cm from equilibrium.
  • Amplitude: x_0 = 15 cm = 0.15 m; displacement for speed: x = 12 cm = 0.12 m; frequency f = 6.7 Hz.
  • Angular frequency: \omega = 2\pi f = 2\pi(6.7) ≈ 42.10\ \text{rad s}^{-1}.
  • Displacement equation: the speed when displacement is x is found via v = \sqrt{\omega^2 (x_0^2 - x^2)}\,, or equivalently using the derivative form with the appropriate phase.
  • Substitution leads to a numerical value (example result shown in notes): v ≈ 3.8\ \text{m s}^{-1}.

Energy in SHM

• Total energy of an SHM system: E = \frac{1}{2} m \omega^2 x_0^2
  • m = mass of oscillator, x_0 = amplitude, ω = angular frequency.
• Energies involved depending on system type:
  • For a horizontal mass–spring system:
  • Elastic potential energy: E_p = \frac{1}{2} k x^2 where k is the spring constant and x is displacement from equilibrium.
  • Kinetic energy: E_k = \frac{1}{2} m v^2.
  • For a pendulum (small angles): gravitational potential energy, E_p = m g h; kinetic energy as mass moves through equilibrium.
• Energy interchange over a cycle:
  • At maximum displacement (x = ±x0): Ep = E{p,max} = \frac{1}{2} k x0^2 or, for gravity-based pendulum, E_p is maximum at amplitude.
  • At equilibrium (x = 0): Ek = E{k,max} = \frac{1}{2} m v{\max}^2, with v{\max} = ω x_0.
  • The total energy E remains constant (in the absence of damping): E = Ek + Ep = \frac{1}{2} m \omega^2 x_0^2 = \text{constant}.
• Energy–displacement graphs for half a period:
  • Kinetic energy is minimum (often zero) at the turning points (x = ±x_0).
  • Potential energy is maximum at turning points (x = ±x_0).
  • Kinetic energy is maximum at the equilibrium position (x = 0).
  • The total energy is represented by a horizontal straight line on an energy vs. time or energy vs. displacement plot.

Example: Energy calculation for SHM

• Ball between two fixed points A and B connected by springs; frequency f = 4.8 Hz; amplitude x_0 = 1.5 cm; mass m = 23 g = 0.023 kg.
• Total energy: E = \frac{1}{2} m \omega^2 x_0^2 with \omega = 2\pi f = 2\pi(4.8) = 30.16\ \,\text{rad s}^{-1}. Then
  • E = \frac{1}{2} (0.023) (30.16)^2 (0.015)^2 \approx 2.4\ \text{mJ}.

SHM Graphs

• Displacement–time graph for SHM starting from equilibrium is a sine curve: x(t) = x_0 \sin(\omega t).
• Velocity–time graph is the gradient of the displacement–time graph: hence a cosine curve, 90° out of phase with displacement.
• Acceleration–time graph is the gradient of the velocity–time graph, which gives a negative sine curve, 90° out of phase with velocity.
• General relationships:
  • Velocity is the rate of change of displacement: v = \frac{dx}{dt}.
  • Acceleration is the rate of change of velocity: a = \frac{dv}{dt}.
• Phase relationships:
  • Displacement leads velocity by 90° when starting from equilibrium; velocity leads acceleration by 90°; etc.

Energy in SHM (Graphs and Interpretations)

• Displacement-energy relationships:
  • Displacement x oscillates between -x0 and +x0 (amplitude A).
  • Potential energy is maximum at x = ±x_0 (pendulum: at amplitude; springs: at extension/compression max).
  • Kinetic energy is maximum at x = 0 (equilibrium).
• Total energy E remains constant in an ideal SHM system (no damping).

Damped & Forced Oscillations, Resonance

Damping

• In real systems, oscillators lose energy due to resistive forces (e.g., friction, air resistance), leading to damping.
• Damping reduces amplitude over time; frequency of damped oscillations remains approximately the same as the undamped frequency.
• Types of damping (based on how quickly amplitude decays):
  • Light damping: amplitude decays exponentially with time; oscillations continue but with decreasing amplitude.
  • Critical damping: returns to equilibrium in the shortest possible time without oscillating.
  • Heavy damping: returns to equilibrium without oscillating, but more slowly than critical damping.
• Practical note: distinguish resistive (damping) force from restoring force; damping opposes motion, restoring force returns toward equilibrium.
• Example: A weighing scale needle should damp quickly but not overshoot; critically damped or heavily damped arrangements can be recommended to avoid oscillations.

Damping in practice (graphs and interpretation)

• Lightly damped: many oscillations with decreasing amplitude; energy is gradually dissipated.
• Critically damped: no oscillations; returns to equilibrium as quickly as possible.
• Heavy damping: no oscillations; long time to return to equilibrium.

Resonance

• Definition: resonance occurs when a system is forced to oscillate at its natural frequency, leading to maximum amplitude.
• Key terms:
  • Natural frequency (f_n): the frequency at which the system oscillates freely.
  • Driving frequency (f_d): the frequency of the external driving force.
• When fd ≈ fn (and especially when fd = fn): the system absorbs energy most efficiently from the driving force, achieving maximum amplitude.
• Practical example: pushing a swing at its natural frequency increases the amplitude of swings with each push.
• Important point: even when fd is close to fn, the amplitude increases but is maximized at resonance when fd = fn.
• Consequence: at resonance, energy transfer from driver to oscillator is most efficient.
• The notes emphasize that the driving force supplies energy at the driving frequency to maintain large oscillations when resonance occurs.

Connections to Foundations and Real-World Relevance

• SHM is a foundational model for many physical systems: pendulums, springs, guitar strings, tuning forks, building structures, and even microscopic systems under small oscillations.
• Understanding SHM helps in designing mechanical systems (suspensions, vibration isolation, clocks, musical instruments).
• The energy exchange between kinetic and potential forms in SHM demonstrates fundamental energy conservation and transfer in conservative systems, with damping accounting for real-world energy loss.
• The phase relationships among displacement, velocity, and acceleration underpin signal processing concepts (Fourier analysis, filtering) where sinusoidal components are fundamental.
• Resonance has broad implications: constructive resonance in musical instruments and structural design concerns (e.g., bridges and buildings must avoid resonant frequencies that could lead to large oscillations). Practical engineering often uses damping to mitigate resonance risks.

Key Formulas (Summary)

• Time period and frequency:
  • T = \frac{1}{f}
  • \omega = \frac{2\pi}{T}
  • \omega = 2\pi f
• SHM motion:
  • a = -\omega^2 x
  • If starting from equilibrium: x(t) = x_0 \sin(\omega t)
  • Velocity: v(t) = \frac{dx}{dt} = x_0 \omega \cos(\omega t)
  • Maximum speed: v{\max} = \omega x0
  • Alternative speed form: v = \pm \omega \sqrt{x_0^2 - x^2}
• Energy in SHM:
  • Total energy: E = \frac{1}{2} m \omega^2 x_0^2
  • Kinetic energy: E_k = \frac{1}{2} m v^2
  • Potential energy (elastic Spring): E_p = \frac{1}{2} k x^2
  • For pendulums: gravitational potential energy E_p = m g h
• Displacement–energy graph characteristics:
  • At x = 0: maximum kinetic energy; at x = ±x_0: maximum potential energy; total energy constant.
• Damping types (qualitative): light damping (exponentially decaying amplitude), critical damping (fastest return without oscillation), heavy damping (no oscillation, slow return).
• Resonance condition:
  • Driving frequency equals natural frequency: fd = fn
  • Maximum energy transfer occurs at resonance.

Quick Reference Tips

• Always ensure your calculator is in radians mode when using sine/cosine with angular frequency in rad s^{-1}.
• When converting between T, f, and ω, revisit: \omega = 2\pi f = \frac{2\pi}{T}.
• In SHM graphs, remember: x is zero at equilibrium, v is maximal there, a is maximal at maximum displacement (opposite direction to x).
• In energy graphs, total energy remains constant in ideal SHM; damping introduces energy loss over time.
• For practical damping decisions, consider how quickly a system should settle (critical/heavy damping) vs. allowing some overshoot (light damping).

Real-World Example Scenarios

• Swinging child: resonance occurs when you push at the swing’s natural frequency; small pushes can gradually increase amplitude if timed correctly.
• Car suspension: designed to be critically damped to avoid oscillations after hitting a bump, ensuring quick stabilization.
• Guitar string vibration: SHM governs the vibration of strings; energy exchanges between kinetic and potential forms determine timbre and sustain.
• Weighing scale needle: excessive oscillation reduces readability; damping (preferably critical) helps the needle settle quickly to a reading.