Hooke's Law

For elastic deformation, stress is proportional to strain. For an ideal spring: \[ \vec{F}=-k\vec{x} \] All materials display a linear dependence of restoring force on displacement for sufficiently small displacements. For some objects, \(F(x)\) is linear over large ranges of strain which is defined as: \[ \varepsilon=\frac{\Delta l}{l} \] Springs and elastic or rubber bands can typically be linear for \(\epsilon\leq 2\), and most other materials are linear for \(\epsilon\leq 0.01\).

Simple Harmonic Motion (SHM)

Mass on a Spring

• Hooke's Law \(\to F=-kx\)
• N's Second Law \(\to F=m\ddot{x}\)
• \(\implies -kx=m\ddot{x}\)

We can try a solution of the form \(x(t)=A\cos(\omega_0t+\phi_0)\) \[ m\ddot{x}=-mA\omega_0^2\cos(\omega_0t+\phi_0) \] \[ \implies -kx=-m\omega^2_0x\qquad\&\qquad\omega_0=\sqrt{\frac{k}{m}} \] The energy stored in this spring is the following: \[ E=\frac{1}{2}mv^2+\frac{1}{2}kx^2=\frac{1}{2}kA^2 \]

Angular SHM

Consider a rigid body with moment of inertia \(I\) about some axis connected to a spring or suspended from a fibre.

• This creates a torque \(\tau=-\kappa\theta\) for rotation from equilibrium angle.
• \(\kappa\) is a constant called the torsion constant.

The equation of motion is given by: \[ -\kappa\theta=I\frac{d^2\theta}{dt^2} \] The form of this equation is the same as for a mass on a spring. So we are dealing with a form of angular SHM. The angular frequency \(\omega_0\) and frequency \(f\) are given by: \[ \omega_0=\sqrt{\frac{\kappa}{I}}\qquad:\qquad f=\frac{\omega_0}{2\pi} \] Motion described by: \(\theta(t)=\Theta\cos(\omega_0t+\phi_0)\) Note: \(\omega_0\neq d\theta/dt\), as it is oscillating sinusoidally.

Damping Oscillations

A solution to the equation of motion assuming damping force is linear with respect to velocity is: \[ x(t)=Ae^{\frac{-b}{2m}t}\cos(\omega't+\phi) \] ...where \(b\) is the damping term and is small and \(w'\) (The new angular velocity is): \[ \omega'=\sqrt{\omega^2-\frac{b^2}{4m^2}}=\sqrt{\frac{k}{m}-\frac{b^2}{4m^2}} \] A system is critically-damped if \(b=2\sqrt{km}\), under-damped if less, or over-damped if more.

Forced Oscillation

The equation of motion in a forced oscillation is: \[ x(t)=A\sin(\omega_dt+\delta) \] where \(\omega_d\) is the driving frequency The amplitude of an oscillator driven by a sinusoidally varying force is: \[ A=\frac{F_0}{\sqrt{(k-m\omega_d^2)^2+b^2\omega_d^2}} \]

Pendulums

Simple Pendulum

A simple pendulum consists of a small finite sized mass supported on a weightless string or rod. One external force, gravity, acts vertically downwards and the tension in the string supports the mass. The mass moves in a circular arc, radius L around the pivot point X.

If \(\theta\) is small, \(\implies mg\sin\theta\approx mg\theta\) Equation of motion: \[ L\frac{d^2\theta}{dt^2}=-g\theta \] ...with solution \[ \theta(t)=A\cos(\omega_0t+\phi_0) \] Natural frequency: \[ \omega_0=\sqrt{\frac{g}{L}} \]

Physical Pendulum

• Know the moment of inertia for a point mass at some distance (\(L\)) from a pivot: \(I=mL^2\)
• Remember the parallel axis theorem: \(I_{||}=I_{cm}+M_{total}d^2\)
  
  

Equation of motion: \[ \frac{d^2\theta}{dt^2}=-\frac{mgL_{cm}}{I}\theta \] ...with solution \[ \theta(t)=\Theta\cos(\omega_0t+\phi_0) \] Natural frequency: \[ \omega_0=\sqrt{\frac{mgL_{cm}}{I}} \]

Waves

Speed of Waves

String: Tension: \(T\), mass per unit length: \(\mu\) \[ v=\sqrt{\frac{T}{\mu}} \] Bulk Material: Bulk modulus: \(B\), density: \(\rho\) \[ v=\sqrt{\frac{B}{\rho}} \] Rod: Youngs modulus: \(Y\), density: \(\rho\) \[ v=\sqrt{\frac{Y}{\rho}} \]

Expression for Travelling Waves

A wave travelling to the right (change sign of \(kx\) to change direction) is expressed as: \[ y(x,t)=A\sin(\omega t-kx) \] Where \(k\) is the wave number: \[ k=\frac{2\pi}{\lambda} \] The velocity of the wave is: \[ v=f\lambda=\frac{\omega}{k} \]

Particle Velocity in a Wave

Particles of the medium (e.g. rope) move back and forth as the wave propagates. Particle velocity at a fixed position \(x\) is the derivative of \(y\) at that position with respect to time. \[ \left(\frac{\partial y}{\partial t}\right)=A\omega\cos(\omega t-kx) \] We can use this to derive the Wave Equation: \[ \frac{\partial^2y}{\partial t^2}=v^2\frac{\partial^2y}{\partial x^2} \]

Reflections

A fixed end will invert the wave, whereas a free end will mirror the wave back as it came to the end. Advanced stuff:

Principle of Linear Superposition

When two wave functions overlap the resulting displacement is just the algebraic sum of the individual displacements: \[ y(x,t)=y_1(x,t)+y_2(x,t) \]

Standing Waves

Trigonometry simplifies the result to two waves travelling opposite directions: \[ y(x,t)=2A\sin(\omega t+\phi_1)\cos(kx+\phi_2) \] Positions with zero displacement at all times = ‘Nodes’ Positions where SHM has maximum amplitude = ‘Antinodes’ From A Level, it is clear that: \[ \lambda_n=\frac{2L}{n} \] hence, \[ k=\frac{2\pi}{\lambda}=\frac{\pi n}{L} \] Different modes (overtones) are calculated with the following formula: \[ f_n=\frac{nv}{2L}=nf_1 \] When on a string: \[ f_n=\frac{n}{2L}\sqrt{\frac{T}{\mu}} \]

Pressure Waves

Path Difference

Path difference (\(d\)) is given by: \[ d=\left||\vec{a}|-|\vec{b}|\right| \] from the following diagram:

If \(d=n\lambda\), full constructive interference if \(d=n\lambda/2\), full destructive interference

Interference

2D Interference:

Beats

Beats are another common example of interference. Consider two overlapping waves with slightly different frequencies. The phase difference varies with position, and so does the amplitude, as before…

The Mathematics of Beats

Consider two travelling sound waves of equal amplitude \[ y_1=A\sin(\omega_1t-k_1x)\qquad\qquad y_2=A\sin(\omega_2t-k_2x) \] Simplify algebra by choosing \(x=0\) (doesn’t affect conclusion), Resultant motion is: \[\begin{align} &y=A\sin(\omega_1t)+A\sin(\omega_2t)\\ &\implies y=2A\sin\left(\frac{\omega_1+\omega_2}{2}t\right)\cos\left(\frac{\omega_1-\omega_2}{2}t\right)\\ &\omega_a=\frac{\omega_1+\omega_2}{2}\qquad\qquad\omega_d=\frac{\omega_1-\omega_2}{2}\\ &\implies y=2A\sin(\omega_at)\cos(\omega_dt)\;\;(x=0) \end{align}\] ...A wave with frequency \(\omega_a\) and modulating amplitude \(2A\cos( \omega_dt)\) Perceived sound has average frequency \(f_a\), with an amplitude slowly varying at frequency \(f_d\). Loudness varies at frequency \(2f_d\) = the beat frequency \(f_b\): \[ f_b=2f_d=f_1-f_2 \]

Doppler Effect

Source emits frequency \(f_s\) and moves with speed \(v_s\) relative to the air directly towards listener. The listener hears frequency \(f_L\) and moves with speed \(v_L\) relative to the air directly towards source. Speed of sound in still air is \(v\): \[ f_L=f_s\left(\frac{v\pm v_L}{v\mp v_s}\right) \] Note:

• If the listener approaches the source we use \(+\) in the numerator.
• If the source approaches the listener we use \(–\) in the denominator.
• For a receding listener or source we use the opposite sign.

Doppler Effect for Electromagnetic Waves

There is also a Doppler effect for electromagnetic waves travelling in a vacuum. Here there is no medium and so the effect depends only on the relative motion of the source with respect to the receiver. The frequency detected at the receiver, \(f_r\), is related to the frequency emitted at the source, \(f_s\), by: \[ f_r=\sqrt{\frac{c\pm v}{c\mp v}}f_s \] Where \(v\) is the relative speed and \(c\) is the speed of light in a vacuum. If the listener and source are approaching we use \(+\) in the numerator and \(–\) in the denominator. If the listener and source are receding we use \(–\) in the numerator and \(+\) in the denominator. The frequency detected at the receiver, \(f_r\), is related to the frequency emitted at the source, \(f_s\), by: \[ \frac{\lambda_r}{\lambda_s}=\frac{f_s}{f_r} \]

Supersonics

A cone of constructive interference forms: \[ \sin\theta=\frac{v}{v_{source}} \] The Mach number (\(M\)) is: \[ M=\frac{v_{source}}{v} \]

Power In Transverse Waves

\[ P(x,t)=\sqrt{\mu T}\omega^2A^2\cos^2(\omega t-kx) \]