Category: Part 4: Oscillations & Waves

Problem Statement Solve the oscillation/wave problem: A conical pendulum (string length $l$, angle $\theta_0$ to vertical) rotates steadily. Show small perturbations about the steady state result in oscillations and find their frequency. For steady rotation with $\dot\phi = \Omega = \sqrt{g/(l\cos\theta_0)}$ and cone half-angle $\theta_0$. Perturbing t Given Information See problem statement for all given…

Problem Statement Solve the oscillation/wave problem: A double pendulum (both masses $m$, both lengths $l$) has two normal modes. Find their frequencies. For small oscillations with angles $\theta_1$, $\theta_2$, the equations of motion are: $$2\ddot\theta_1 + \ddot\theta_2 + \frac{2g}{l}\theta_1 = 0$$ $$\ddot\theta_1 + \ddot\theta_2 + \frac{g}{l}\thet Given Information See problem statement for all given quantities.…

Problem Statement Solve the oscillation/wave problem: A mass $m$ at the end of a uniform cantilever (length $l$, flexural rigidity $EI$) performs vertical oscillations. Find the frequency (ignore cantilever mass). For a cantilever clamped at one end and loaded at the free end, the deflection $\delta$ under force $F$ is: $$\delta = \frac{Fl^3}{3EI}$$ So Given…

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 cylindrical log (length $L$, density $\rho_w/2$) floats vertically in water (density $\rho_w$). Find the period of vertical oscillations. At equilibrium, half the log is submerged (since $\rho_{\rm log} = \rho_w/2$). Submerged length $h_0 = L/2$. For displacement $x$ downward from equilibrium, the Given Information See problem statement for…

Problem Statement Solve the oscillation/wave problem: Describe the transient behavior of a forced oscillator starting from rest. When does steady-state dominate? The complete solution is the sum of the particular (steady-state) solution and the homogeneous (transient) solution: $$x(t) = \underbrace{A\cos(\omega t – \psi)}_{\text{steady-state}} + \under Given Information See problem statement for all given quantities. Physical…

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: Find the period of a simple pendulum for amplitude $\theta_0$ (not necessarily small). Give the first-order correction. The exact period is: $$T = 4\sqrt{\frac{l}{g}}\int_0^{\theta_0}\frac{d\theta}{\sqrt{2(\cos\theta – \cos\theta_0)}}$$ Using $\cos\theta = 1-2\sin^2(\theta/2)$ and the substitution $ Given Information See problem statement for all given quantities. Physical Concepts & Formulas This problem…

Problem Statement Solve the oscillation/wave problem: A simple pendulum (length $l$) has its support oscillating horizontally as $x_0 = a\cos(\omega_0 t)$. Find the equation of motion and resonance condition. In the lab frame, the equation for the pendulum angle $\theta$ (small angle): $$l\ddot\theta = -g\theta – \ddot{x}_0 = -g\theta + a\omega_0^2\cos Given Information See problem…

Problem Statement Solve the oscillation/wave problem: A rectangular plate (mass $m$) is suspended by two vertical threads (length $l$, separation $a$) and can twist about the vertical axis. Find the period of torsional oscillations. When the plate twists by angle $\phi$ (small), each thread tilts, raising the plate by $\Delta h = a^2\phi^2/(8l)$ (for s…