Category: Part 4: Oscillations & Waves

Problem Statement Find the half-power bandwidth $\Delta\omega$ of a forced oscillator and express $Q$ in terms of $\omega_0$ and $\Delta\omega$. Given Information See problem statement for all given quantities. Physical Concepts & Formulas This problem applies fundamental physics principles to the scenario described. The solution requires identifying the relevant conservation laws and equations of motion,…

Problem Statement Solve the oscillation/wave problem: Show that at amplitude resonance, the phase lag of the oscillator behind the driving force is not exactly $90°$. When is it exactly $90°$? The phase lag of displacement behind force: $$\tan\psi = \frac{2\beta\omega}{\omega_0^2 – \omega^2}$$ At amplitude resonance $\omega_{\rm res} = \sqrt{\omega_0^2 Given Information See problem statement for…

Problem Statement Find the frequency at which the amplitude of forced oscillations is maximum (amplitude resonance). Given Information See problem statement for all given quantities. Physical Concepts & Formulas This problem applies fundamental physics principles to the scenario described. The solution requires identifying the relevant conservation laws and equations of motion, then solving systematically with…

Problem Statement Define the quality factor $Q$ for a damped harmonic oscillator. Express it in terms of (a) the natural frequency $\omega_0$ and damping coefficient $\beta$, and (b) energy stored vs. energy lost per cycle. Given Information See problem statement for all given quantities. Physical Concepts & Formulas This problem applies fundamental physics principles to…

Problem Statement Solve the oscillation/wave problem: A harmonic oscillator is driven by force $F = F_0\cos(\omega t)$. Find the steady-state amplitude and phase. The equation of motion: $$m\ddot{x} + r\dot{x} + kx = F_0\cos(\omega t)$$ The steady-state solution is $x = A\cos(\omega t – \psi)$ where: $$A = \frac{F_0/m}{\sqrt{(\omega_0^2-\omega^2)^2 + 4 Given Information See problem…

Problem Statement Define the quality factor $Q$ for a damped harmonic oscillator. Express it in terms of (a) the natural frequency $\omega_0$ and damping coefficient $\beta$, and (b) energy stored vs. energy lost per cycle. Given Information See problem statement for all given quantities. Physical Concepts & Formulas This problem applies fundamental physics principles to…

Problem Statement Define the quality factor $Q$ for a damped harmonic oscillator. Express it in terms of (a) the natural frequency $\omega_0$ and damping coefficient $\beta$, and (b) energy stored vs. energy lost per cycle. Given Information See problem statement for all given quantities. Physical Concepts & Formulas This problem applies fundamental physics principles to…

Problem Statement Define the quality factor $Q$ for a damped harmonic oscillator. Express it in terms of (a) the natural frequency $\omega_0$ and damping coefficient $\beta$, and (b) energy stored vs. energy lost per cycle. Given Information See problem statement for all given quantities. Physical Concepts & Formulas This problem applies fundamental physics principles to…

Problem Statement Define the quality factor $Q$ for a damped harmonic oscillator. Express it in terms of (a) the natural frequency $\omega_0$ and damping coefficient $\beta$, and (b) energy stored vs. energy lost per cycle. Given Information See problem statement for all given quantities. Physical Concepts & Formulas This problem applies fundamental physics principles to…

Problem Statement Solve the oscillation/wave problem: A tube (cross-section $S$, length $L$) is closed at one end and contains gas of density $\rho_0$ at pressure $p_0$. A piston of mass $m$ can slide freely. Find the natural frequency of small oscillations (adiabatic process, $\gamma$ = ratio of specific heats). For adiabatic compression/expansion wit Given Information…