05_Oscillatory Motion

1. Applications of Oscillatory Motion

A point mass $m$ attached to a massless string of length $L$.

Torque equation: $\tau = I\alpha \Rightarrow -mg\sin\theta \cdot L = mL^2 \frac{d^2\theta}{dt^2}$
Equation of motion: $\frac{d^2\theta}{dt^2} = -\frac{g}{L}\sin\theta$
Small-angle approximation ($\theta \ll 1 \Rightarrow \sin\theta \approx \theta$):

\[ \frac{d^2\theta}{dt^2} = -\frac{g}{L}\theta \quad \text{(Simple Harmonic Oscillation)} \]
Solution: $\theta = \theta_{max}\cos(\omega t + \phi)$
Angular frequency and Period:

\[ \omega = \sqrt{\frac{g}{L}}, \quad T = \frac{2\pi}{\omega} = 2\pi\sqrt{\frac{L}{g}} \]

An extended rigid body oscillating about a pivot point at a distance $d$ from its center of mass.

图示：GP Oscillatory Motion

schematic diagram from the hand-written notes of Mr. ztWang
Torque equation: $\tau = I\alpha \Rightarrow -mgd\sin\theta = I\frac{d^2\theta}{dt^2}$
Equation of motion: $\frac{d^2\theta}{dt^2} = -\frac{mgd}{I}\sin\theta$
Solution (for small angles): $\theta = \theta_{max}\cos(\omega t + \phi)$
Angular frequency: $\omega = \sqrt{\frac{mgd}{I}}$

图示：GP Oscillatory Motion2

A disc suspended by a wire that twists.

Projecting uniform circular motion of a point $P'$ onto the x-axis yields:

\[ x = A\cos(\omega t + \phi) \rightarrow \text{Simple Harmonic Motion} \]
Projecting onto the y-axis yields:

\[ y = A\sin(\omega t + \phi) \]
Notes:
1. Projection of uniform circular motion $\iff$ Simple harmonic motion.
2. Uniform circular motion $\iff$ Combination of two orthogonal simple harmonic motions.

Motion reduced by an external force (e.g., fluid resistance).

Fluid resistance (Low speed): $F_d = -bv$, where $b$ is the damping constant.
Equation of motion:

\[ F_{net} = -kx - bv = ma \Rightarrow m\frac{d^2x}{dt^2} + b\frac{dx}{dt} + kx = 0 \]

图示：GP Oscillatory Motion3

Underdamped ($\frac{b}{2m} < \sqrt{\frac{k}{m}}$):

- Solution: $x = Ae^{-\frac{b}{2m}t}\cos(\omega' t + \phi)$

- Damped frequency: $\omega' \equiv \sqrt{\frac{k}{m} - \left(\frac{b}{2m}\right)^2}$

图示：GP Oscillatory Motion4

Overdamped ($\frac{b}{2m} > \sqrt{\frac{k}{m}}$):
- Solution: $x = A_1\exp\left[\left(-\frac{b}{2m} + \sqrt{\left(\frac{b}{2m}\right)^2 - \omega_0^2}\right)t\right] + A_2\exp\left[\left(-\frac{b}{2m} - \sqrt{\left(\frac{b}{2m}\right)^2 - \omega_0^2}\right)t\right]$
Critically Damped ($\frac{b}{2m} = \sqrt{\frac{k}{m}}$):
- $\omega' = 0$
- Solution: $x = (A_1 + A_2 t)e^{-\frac{b}{2m}t}$
- Note: Critical damping is the fastest way to restore the system to its equilibrium position, making it ideal for devices such as ammeters.

Oscillator driven by an external sinusoidal force.

Equation of motion:

\[ F = -kx - bv + F_{ext}\cos(\omega t) = ma \Rightarrow \ddot{x} + \frac{b}{m}\dot{x} + \frac{k}{m}x = \frac{F_{ext}}{m}\cos(\omega t) \]
General Solution:

\[ x = \underbrace{A'e^{-\frac{b}{2m}t}\cos(\omega' t + \phi')}_{\text{Transient solution}} + \underbrace{A\cos(\omega t - \phi)}_{\text{Steady solution}} \]
Steady State Parameters:
- Amplitude: $A = \frac{F_{ext}/m}{\sqrt{(\omega_0^2 - \omega^2)^2 + (b\omega/m)^2}}$
- Phase angle: $\tan\phi = \frac{b\omega/m}{\omega_0^2 - \omega^2}$
Notes:
1. For $t \gg 1$, only the steady solution survives.
2. $\omega$ is the frequency of the external force.
3. $\phi$ is the relative phase with respect to $F_{ext}$.

Slow drive ($\omega \ll \omega_0$):
- $A \approx \frac{F_{ext}/m}{\omega_0^2} = \frac{F_{ext}}{k}$
- $\phi \approx 0$ (Same phase as $F_{ext}$)
Fast drive ($\omega \gg \omega_0$):
- $A \approx 0$
- $\phi \approx \pi$ (Opposite phase as $F_{ext}$)
Resonance ($\omega = \omega_0$):
- $\tan\phi = \infty \Rightarrow \phi = \frac{\pi}{2}$
- Amplitude $A$ peaks. As $b \to 0$, $A(\omega = \omega_0) \to \infty$.

Potential energy for a pair of neutral atoms or molecules:

\[ U(x) = 4\epsilon \left[ \left(\frac{\sigma}{x}\right)^{12} - \left(\frac{\sigma}{x}\right)^6 \right] \]

Equilibrium position ($x_0$):

\[ \left.\frac{dU}{dx}\right|_{x_0} = 0 \Rightarrow x_0 = 2^{1/6}\sigma \approx 1.122\sigma \]
Effective spring constant ($k$):

\[ k = \left.\frac{d^2U}{dx^2}\right|_{x_0} = \frac{72\epsilon}{x_0^2} \]
Angular frequency:

\[ \omega = \sqrt{\frac{k}{\mu}} = \sqrt{\frac{72\epsilon}{\mu x_0^2}} \quad \text{($\mu$: effective mass)} \]

Two masses $m$ connected by a central spring $k$ and bound to walls by springs $k'$.

图示：GP Oscillatory Motion5

Equations of motion:

\[ \begin{cases} m\ddot{x}_1 = -k'x_1 - k(x_1 - x_2) \\ m\ddot{x}_2 = -k'x_2 + k(x_1 - x_2) \end{cases} \]
Normal modes: Frequency $\omega$ is the same for all oscillators, with a fixed relative phase.

Trying $x_1 = A_1\cos(\omega t + \phi_0)$ and $x_2 = A_2\cos(\omega t + \phi_0)$ yields the eigenvalue problem:

\[ \begin{pmatrix} k+k' & -k \\ -k & k+k' \end{pmatrix} \begin{pmatrix} A_1 \\ A_2 \end{pmatrix} = m\omega^2 \begin{pmatrix} A_1 \\ A_2 \end{pmatrix} \]

Solution 1: Translational mode
- Eigenvalue constraint checks out when $A_1 = A_2$.
- $\omega = \sqrt{\frac{k'}{m}}$
Solution 2: Vibrational mode
- Eigenvalue constraint checks out when $A_1 = -A_2$.
- $\omega = \sqrt{\frac{2k+k'}{m}}$

Adding and subtracting the differential equations:

$m\frac{d^2}{dt^2}(x_1+x_2) = -k'(x_1+x_2) \Rightarrow x_1+x_2 = C_1\cos(\sqrt{\frac{k'}{m}}t + \phi_1)$
$m\frac{d^2}{dt^2}(x_1-x_2) = -(2k+k')(x_1-x_2) \Rightarrow x_1-x_2 = C_2\cos(\sqrt{\frac{2k+k'}{m}}t + \phi_2)$
Overall motion: The linear combination of the normal modes. Setting $C_2 = 0$ isolates the translational mode, while $C_1 = 0$ isolates the vibrational mode.

A molecule with $N$ atoms has a total of $3N$ degrees of freedom (modes).

Linear molecule:
- Translation: 3 ($x, y, z$)
- Rotation: 2 ($\theta, \phi$)
- Vibration: $3N - 5$
Non-linear molecule:
- Translation: 3
- Rotation: 3 (Euler angles $\alpha, \beta, \gamma$)
- Vibration: $3N - 6$

图示：GP Oscillatory Motion6