Category: Part 4: Oscillations & Waves

Problem Statement Solve the oscillation/wave problem: For a slightly anharmonic oscillator $U = \frac{1}{2}kx^2 + bx^3 + cx^4$, how does the period depend on amplitude to first order? For the cubic perturbation $bx^3$: by symmetry, it averages to zero in first order. The quartic term $cx^4$ shifts the effective frequency. Using perturbation theory (or Given…

Problem Statement Solve the oscillation/wave problem: Two blocks of masses $m_1$ and $m_2$ are connected by a spring ($k$). Block $m_1$ is fixed to a wall. Find (a) frequency when $m_1$ is fixed, (b) frequency when both are free. (a) $m_1$ fixed (wall): Only $m_2$ oscillates against the spring: $$\omega_a = \sqrt{\frac{k}{m_2}}$$ (b) Both free:…

Problem Statement Solve the oscillation/wave problem: Two masses $m_1$ and $m_2$ are connected by a spring of constant $k$. Find the frequency of oscillation of the relative coordinate. Let $x_1$ and $x_2$ be positions of the two masses, $x = x_2 – x_1 – l_0$ = extension. The CM moves at constant velocity (no external…

Problem Statement Solve the oscillation/wave problem: A mass $m$ between two fixed walls is connected to both walls by springs $k_1$ and $k_2$. Both springs are in their natural state at equilibrium. Find the oscillation frequency. When the mass moves by $x$ toward wall 2, spring 1 stretches (pulling back) and spring 2 compresses (pushing…

Problem Statement Solve the oscillation/wave problem: A mass $m$ on a frictionless inclined plane (angle $\alpha$) is connected to a spring (constant $k$) along the plane. Find the oscillation frequency and equilibrium position. Along the incline, taking positive direction up the slope. Equilibrium: spring compressed/stretched by $x_0 = mg\sin\alpha/k$ Given Information See problem statement for…

Problem Statement Solve the oscillation/wave problem: For a uniform rod pivoted at distance $d$ from its CM, find the value of $d$ that minimizes the period. For a uniform rod, $k^2 = l^2/12$ (radius of gyration squared about CM). The equivalent pendulum length: $$l_{\rm eq} = d + \frac{k^2}{d} = d + \frac{l^2}{12d}$$ Minimizing: $dl_{\rm…

Problem Statement Solve the oscillation/wave problem: A mass $m$ hangs from a spring of natural length $l_0$ and stiffness $k$. The spring stretches by $x_0 = mg/k$ at equilibrium. Show the oscillation frequency is still $\omega = \sqrt{k/m}$. Let $y$ = displacement from the equilibrium position. The spring extension is $x_0 + y$, so the…

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 uniform rod of mass $m$ and length $l$ can rotate about one end. Find the period of small oscillations. This is a physical pendulum. The CM is at $l/2$ from the pivot. Moment of inertia about the end: $$I = \frac{ml^2}{3}$$ Restoring torque: $\tau = -mg(l/2)\theta$. Period: $$T…

Problem Statement Solve the oscillation/wave problem: Find the mean power absorbed by a forced harmonic oscillator from the driving force. The instantaneous power is $P = F\dot{x}$. With $F = F_0\cos(\omega t)$ and $\dot{x} = A\omega\sin(\omega t – \psi + \pi) = A\omega\sin(\psi – \omega t + \pi)$ more directly: $$\dot{x} = A\omega\sin(\omega t –…