Chapter Review

Oscillations

SHM Systems · Energy and Resonance

Defining Simple Harmonic Motion

SHM is oscillatory motion where the restoring force is directly proportional to displacement from equilibrium and always directed towards it. The defining signature is $a = -\omega^2 x$.

Key Points

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If acceleration is proportional to negative displacement ($a \propto -x$), the motion is SHM
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Amplitude does not affect period or frequency in ideal SHM
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At equilibrium ($x = 0$): acceleration is zero, velocity is maximum
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At extremes ($x = \pm A$): acceleration is maximum ($|a_{max}| = \omega^2 A$), velocity is zero
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Angular frequency $\omega = 2\pi/T = 2\pi f$ — differs from frequency $f$ by a factor of $2\pi$

Formula

$$a = -\omega^2 x$$

SHM Equations and Circular Motion

SHM is the projection of uniform circular motion onto a diameter. The displacement, velocity, and acceleration follow sinusoidal functions governed by amplitude, angular frequency, and phase.

Key Points

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Displacement: $x = A\sin(\omega t + \phi)$, where $\phi$ is the initial phase
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Velocity: $v = A\omega\cos(\omega t + \phi) = \omega\sqrt{A^2 - x^2}$
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Acceleration: $a = -A\omega^2\sin(\omega t + \phi) = -\omega^2 x$
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$v_{max} = A\omega$ at equilibrium; $a_{max} = A\omega^2$ at extremes
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If $\omega$ doubles, $v_{max}$ doubles but $a_{max}$ quadruples
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Phase $\phi = 0$ means starting from mean position; $\phi = \pi/2$ means starting from positive extreme

Formula

$$x = A\sin(\omega t + \phi)$$

Mass-Spring System

A mass on a spring executes SHM with period determined by the ratio of inertia to stiffness. The same formula applies for both horizontal and vertical springs.

Key Points

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$T \propto \sqrt{m}$ and $T \propto 1/\sqrt{k}$ — square-root relationships
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Quadrupling mass doubles $T$; quadrupling $k$ halves $T$
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Vertical spring: gravity only shifts equilibrium point, period is unchanged
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Finding $k$ from static extension: $k = mg/x$
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Maximum speed: $v_{max} = A\omega = A\sqrt{k/m}$

Formula

$$T = 2\pi\sqrt{\frac{m}{k}}$$

Simple Pendulum

A simple pendulum approximates SHM for small angles, with period depending only on length and gravitational field strength — independent of mass.

Key Points

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Valid only for small angles where $\sin\theta \approx \theta$ (radians)
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Mass cancels out: increasing bob mass increases both inertia and restoring force equally
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$T \propto \sqrt{L}$ and $T \propto 1/\sqrt{g}$ — quadrupling $L$ doubles $T$
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Length $L$ is measured to the center of the bob, not its top
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No gravity ($g = 0$) means no oscillation ($T \to \infty$)

Formula

$$T = 2\pi\sqrt{\frac{L}{g}}$$

Energy Conservation in SHM

SHM systems continuously exchange kinetic and potential energy while total mechanical energy remains constant, proportional to the square of the amplitude.

Key Points

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Total energy: $E = \frac{1}{2}kA^2 = \frac{1}{2}mv_{max}^2$ — constant throughout
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$E \propto A^2$: doubling amplitude quadruples total energy
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At equilibrium: all energy is kinetic ($KE = E$, $PE = 0$)
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At extremes: all energy is potential ($PE = E$, $KE = 0$)
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Work done stretching a spring uses average force: $W = \frac{1}{2}kx_o^2$ (not $kx_o^2$)

Formula

$$E_{total} = \frac{1}{2}kA^2$$

Energy Partition at Any Position

At any displacement $x$, kinetic and potential energy can be found individually, and their sum always equals total energy.

Key Points

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$PE = \frac{1}{2}kx^2$ and $KE = \frac{1}{2}k(A^2 - x^2)$
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At $x = A/2$: PE is $E/4$, KE is $3E/4$ — energy goes as $x^2$, not $x$
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KE equals PE at $x = A/\sqrt{2} \approx 0.707A$
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PE/E ratio shortcut: just compute $(x/A)^2$
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On energy-displacement graph: PE is upward parabola, KE is inverted parabola, total is a flat line

Formula

$$KE = \frac{1}{2}k(A^2 - x^2)$$

Free and Forced Oscillations

Free oscillations occur at the system's natural frequency without external driving. Forced oscillations occur when an external periodic force drives the system, which eventually oscillates at the driving frequency.

Key Points

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Natural frequency depends only on system parameters: $f_0 = \frac{1}{2\pi}\sqrt{k/m}$ (spring) or $\frac{1}{2\pi}\sqrt{g/L}$ (pendulum)
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Free oscillations in real systems gradually die out due to friction
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In forced oscillations, the steady-state frequency equals the driving frequency, not the natural frequency
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Examples of forced oscillations: vehicle engine vibrations, pushing a swing periodically

Resonance

Resonance occurs when the driving frequency matches the natural frequency of the system, causing maximum energy transfer and extraordinarily large amplitude.

Key Points

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Condition: $f_{driving} = f_0$ — perfectly synchronized energy input
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At resonance, velocity is in phase with the driving force, maximizing power transfer
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Below $f_0$: system barely responds; above $f_0$: system cannot keep up
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Applications: radio tuning (selective frequency absorption), microwave ovens (2450 MHz matches water molecules)
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Dangers: soldiers break step on bridges to avoid matching the bridge's natural frequency

Formula

$$f_{driving} = f_0$$

Damped Oscillations and Sharpness of Resonance

Damping dissipates energy from oscillating systems, causing amplitude to decay over time. The amount of damping determines how sharp or broad the resonance peak is.

Key Points

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Light damping: many oscillations with slowly decaying exponential amplitude envelope
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Critical damping: fastest return to equilibrium without overshooting (used in shock absorbers)
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Overdamping: slow, non-oscillatory return to equilibrium
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Low damping → sharp resonance: tall, narrow peak (selective, like radio tuning)
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High damping → flat resonance: short, broad peak over a wide frequency range

Formulas

SHM Defining Equation

Acceleration proportional to displacement, opposite in direction.

Formula

$a = -\omega^2 x$

Mass-Spring Period

Period from inertia (m) and stiffness (k).

Formula

$T = 2\pi\sqrt{\frac{m}{k}}$

Pendulum Period

Period from length and gravity, mass-independent.

Formula

$T = 2\pi\sqrt{\frac{L}{g}}$

SHM Displacement

General equation with initial phase.

Formula

$x = A\sin(\omega t + \phi)$

SHM Velocity at Position x

Speed depends on how far from the turning point.

Formula

$v = \omega\sqrt{A^2 - x^2}$

Total Mechanical Energy

Constant energy proportional to amplitude squared.

Formula

$E = \frac{1}{2}kA^2$

KE at Displacement x

Kinetic energy as function of position.

Formula

$KE = \frac{1}{2}k(A^2 - x^2)$