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Here is a comprehensive, exam-focused Obsidian MD note for the Oscillations topic. It strictly follows the CIE A-Level Physics (9702) syllabus and is heavily infused with exact mark-scheme phrasing, edge-cases from past papers, and examiner tips.

You can copy and paste this directly into your Obsidian vault.

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⚛️ Oscillations (CIE A-Level Physics 9702)

17.1 Simple Harmonic Oscillations (SHM)

Definition: Simple Harmonic Motion (SHM)

Simple Harmonic Motion is a specific type of oscillation where:

1. Acceleration is directly proportional to displacement from a fixed point / equilibrium position ($a\propto x$).
2. Acceleration is always directed towards that fixed point / acts in the opposite direction to the displacement ($a\propto -x$).

Core Terminology

• Displacement ($x$): The distance of a point on the wave from its equilibrium position. It is a vector quantity (can be positive or negative).
• Amplitude ($x_0$): The maximum magnitude of displacement on either side of the equilibrium position.
• Period ($T$): The time interval for one complete repetition/oscillation.
• Frequency ($f$): The number of oscillations per unit time ($f=1/T$).
• Angular Frequency ($\omega$): The rate of change of angular displacement with respect to time.
  • $\omega =2\pi f=\frac{2\pi }{T}$
• Phase Difference ($\varphi$): A measure of how much a point or wave is in front or behind another. Measured in radians or degrees.

Restoring vs. Resistive Forces

• Restoring Force: The force responsible for causing SHM. It always acts towards the equilibrium position.
• Resistive Force: The force responsible for damping. It acts in the opposite direction to the velocity/motion of the oscillator.

Non-SHM Examples (Common Exam Trap)

A bouncing ball or a person jumping on a trampoline is NOT undergoing SHM. Mark Scheme Justification: When in the air, the restoring force acting on them is their weight, which is constant. Therefore, the force (and acceleration) is not proportional to their displacement from equilibrium.

Key Equations & Kinematics

The defining equation of SHM is: $a=-{\omega }^{2}x$

Proving SHM mathematically: If a question gives you a derived formula like $a=-\frac{2g}{L}x$ (liquid in a U-tube) or $a=-\frac{A\rho g}{m}x$ (floating block) and asks you to “Explain how this shows SHM”, use this exact phrasing:

“The equation is in the form $a=-\text{constant}\times x$. This shows that acceleration is directly proportional to displacement, and the negative sign indicates they are in opposite directions.” (The constant is always equal to ${\omega }^2$).

Displacement, Velocity, and Acceleration Equations:

• Displacement:
  • $x=x_0\sin (\omega t)$ (If the object starts oscillating from the equilibrium position).
  • $x=x_0\cos (\omega t)$ (If the object starts oscillating from the maximum amplitude).
• Velocity:
  • $v=v_0\cos (\omega t)$ (if displacement is a sine curve).
  • $v=\pm \omega \sqrt{x_0^2-x^2}$ (Used to find velocity at a specific displacement).
• Maximums:
  • Maximum Velocity / Speed ($v_0$): Occurs at $x=0$. Formula: $v_{max}=\omega x_0$
  • Maximum Acceleration ($a_0$): Occurs at $x=\pm x_0$. Formula: $a_{max}={\omega }^2{x}_0$

SHM and Circular Motion

Uniform circular motion can be linked to SHM. If a ball moves in a horizontal circle and a light shines on it, the shadow it casts on a screen performs Simple Harmonic Motion. The radius $R$ of the circle equals the amplitude $x_0$, so $x=R\sin (\omega t)$.

Graphical Representations

1. Time Graphs ($t$ on x-axis):
   • Displacement ($x-t$): A sine or cosine curve.
   • Velocity ($v-t$): The gradient of the $x-t$ graph. It is $90^{\circ }$ ($\frac{\pi }{2}$ rad) out of phase with displacement.
   • Acceleration ($a-t$): The gradient of the $v-t$ graph. It is $180^{\circ }$ ($\pi$ rad) out of phase with displacement (a direct vertical reflection of the $x-t$ graph).
2. Acceleration-Displacement Graph ($a-x$):
   • A straight line passing exactly through the origin.
   • It has a negative gradient equal to $-{\omega }^2$.
3. Velocity-Displacement Graph ($v-x$):
   • Forms a closed loop / ellipse around the origin.
   • Intercepts the x-axis at $\pm x_0$ (velocity is 0 at max amplitude).
   • Intercepts the y-axis at $\pm v_0$ (velocity is max at equilibrium).

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17.2 Energy in Simple Harmonic Motion

Energy Interchange

In an undamped SHM system, the Total Energy ($E_T$) remains perfectly constant. Energy simply transfers between forms:

• Simple Pendulum: Gravitational Potential Energy (max at amplitude) $\rightleftharpoons$ Kinetic Energy (max at equilibrium).
• Horizontal Spring: Elastic Potential Energy (max at amplitude) $\rightleftharpoons$ Kinetic Energy (max at equilibrium).
• Vertical Spring: A constant interplay of EPE, GPE, and KE.

Energy Equations

Because $E_T$ is constant, we can calculate it using the maximum kinetic energy (which occurs when $v=v_0$): $E_T=\frac{1}{2}m{v}_0^2$ Since $v_0=\omega x_0$, we get the standard total energy formula: $E_T=\frac{1}{2}m{\omega }^2{x}_0^2$

From this, we can derive the energy at any specific displacement $x$:

• Kinetic Energy: $E_K=\frac{1}{2}m{\omega }^2(x_0^2-x^2)$
• Potential Energy: $E_P=\frac{1}{2}m{\omega }^2{x}^2$

Energy Graphs

• Energy vs. Displacement ($E-x$):
  • $E_P$ Curve: A ‘U’ shaped parabola. Min at $x=0$, Max at $\pm x_0$.
  • $E_K$ Curve: An ‘n’ shaped inverted parabola. Max at $x=0$, Min (zero) at $\pm x_0$.
  • $E_T$ Line: A horizontal straight line capping the peaks of $E_K$ and $E_P$.
• Energy vs. Time ($E-t$):
  • Kinetic and potential energy cycle from max to min twice during one full displacement oscillation.
  • Rule: Frequency of energy oscillation = $2\times$ Frequency of displacement oscillation.

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17.3 Damped and Forced Oscillations, Resonance

Damping

Damping

The reduction in energy and amplitude of oscillations due to resistive forces acting on the oscillating system.

• Causes of Damping: Air resistance, friction (e.g., between wheels and a track), viscous drag (in liquids).
• Electromagnetic Damping (Eddy Currents): Must use exact mark scheme phrasing:
  1. The magnet’s motion causes cutting of magnetic flux.
  2. This induces an e.m.f. in the metal/coil.
  3. The e.m.f. causes induced (eddy) currents, leading to thermal energy dissipation in the resistor/metal.
  4. Energy is drawn from the oscillations, and the induced magnetic field opposes the motion of the magnet.

Three Types of Damping:

1. Light Damping: The amplitude gradually decreases/decays exponentially over time. The time period $T$ remains constant. (e.g., pendulum in air).
2. Critical Damping: The system returns to the equilibrium position in the shortest possible time without oscillating or overshooting. (e.g., car suspension).
3. Heavy Damping: The system returns to equilibrium very slowly, without oscillating. (e.g., door closing dampers).

Free vs. Forced Oscillations

• Free Oscillation: A body oscillates without any loss of energy (no resistive forces) and with no external forces applied. It oscillates at its natural frequency ($f_0$).
• Forced Oscillation: A body is made to vibrate by a continuous external periodic force (a driving force) inputting energy into the system at a driving frequency ($f$).

Resonance

Resonance

Occurs when the driving frequency of an external force equals the natural frequency of the oscillating system ($f=f_0$). This results in the system oscillating with a maximum amplitude and absorbing maximum energy.

The Resonance Curve (Amplitude vs. Driving Frequency):

• Features a sharp peak where driving frequency = natural frequency.
• The amplitude does not reach infinity at the peak because there are always some frictional/damping forces present in reality.

Effect of Damping on the Resonance Curve: If you increase damping on a resonating system:

1. The peak amplitude becomes lower.
2. The curve becomes broader/flatter (less sharp).
3. The peak shifts slightly to the left (resonance occurs at a slightly lower driving frequency).

Examples of Resonance:

• Useful: Piezoelectric quartz crystals vibrating for accurate timing/ultrasound generation; MRI scanners; microwaves matching the natural frequency of water molecules to heat food.
• Destructive (to be avoided): Vibrating metal body panels in cars; bridges oscillating wildly in the wind. Mitigation: Add stiffening struts or damping materials to change the natural frequency or absorb energy.

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🧠 Examiner Tips & Common Pitfalls

Calculations & Formulas

• RADIANS MODE: Whenever you calculate displacement using $x=x_0\sin (\omega t)$ or $x_0\cos (\omega t)$, your calculator MUST be in radians mode. The product $\omega t$ is an angle in radians, not degrees.
• Speed vs. Velocity: Exams frequently ask for the speed of the oscillator. Speed is a scalar. Calculate velocity using $v=\pm \omega \sqrt{x_0^2-x^2}$ and simply omit the $\pm$ sign for your final answer.
• Extracting $\omega$ from maximums: If a question gives you $v_{max}$ and $a_{max}$ (or you read them off a graph), you can easily find angular frequency: $\omega =\frac{a_{max}}{v_{max}}$.

Graphing Rules

• Drawing Arrows: If asked to label Wavelength ($\lambda$), Period ($T$), or Amplitude ($x_0$) on a graph, draw the arrows exactly from peak-to-peak or axis-to-peak. Sloppy, short, or floating arrows that don’t touch the lines will be penalized.
• Sketching Curves: Make sure your sine/cosine curves are drawn symmetrically. For $E-x$ graphs, ensure the curves are distinctly U-shaped (parabolic) and not V-shaped (linear).

Definitions & Explanations

• Restoring vs. Resistive: Do not mix these up. The restoring force pulls the mass back to $x=0$ and creates the SHM. The resistive force opposes the velocity and creates the damping.
• Proving SHM: Never just say “it oscillates”. You must state that the formula is in the form $a=-\text{constant}\times x$, which means $a\propto x$, and the minus sign proves they are in opposite directions.