Category: Part 4: Oscillations & Waves

Problem Statement Solve the oscillation/wave problem: Solve the oscillation/wave problem: Show that at resonance the rate of energy input by the source exactly equals the rate of energy dissipation in $R$. In steady-state oscillation, the total stored energy $W = \frac{1}{2}LI_0^2 = \frac{1}{2}Q_0^2/C$ is constant (not growing or decaying). Therefore t Given Information See problem…

Problem Statement Analyze the circuit: Analyze the circuit: Two LC circuits are coupled by mutual inductance $M$. Find the two normal-mode frequencies. For identical circuits ($L_1=L_2=L$, $C_1=C_2=C$) coupled by $M$: Equations of motion: $L\ddot{Q}_1 + M\ddot{Q}_2 + Q_1/C = 0$ and $M\ddot{Q}_1 + L\ddot{Q}_2 + Q_2/C = 0$. Symmetric mode: Given Information See problem statement…

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: Solve the oscillation/wave problem: In a forced RLC circuit at $\omega = 2\omega_0$, find the current phase relative to the driving EMF if $Q = 5.0$. Phase angle of current relative to EMF (current leads by $\psi$): $$\tan\psi = \frac{1/(\omega C) – \omega L}{R} = \frac{X_C – X_L}{R}$$ At…

Problem Statement Solve the oscillation/wave problem: Solve the oscillation/wave problem: What are self-oscillations? How do they differ from damped and forced oscillations? Damped oscillations: System oscillates with decaying amplitude; energy is continuously dissipated. Forced oscillations: External periodic force replenishes energy; amplitude depend Given Information See problem statement for all given quantities. Physical Concepts & Formulas…

Problem Statement Solve the oscillation/wave problem: Solve the oscillation/wave problem: A mechanical system has mass $m=0.10$ kg, spring $k=40$ N/m, damping $r = 0.20$ N·s/m. Find the equivalent $L$, $C$, $R$ and the $Q$-factor. Using the mechanical–electrical analogy: $m\leftrightarrow L$, $k\leftrightarrow 1/C$, $r\leftrightarrow R$: $$L = m = 0.10 Given Information See problem statement for…

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 Analyze the circuit: Analyze the circuit: Describe low-pass ($RC$) and high-pass ($CR$) filters. Find the cutoff (3 dB) frequency for each. Low-pass RC filter (output across $C$): $$H(\omega) = \frac{V_{\rm out}}{V_{\rm in}} = \frac{1/i\omega C}{R + 1/i\omega C} = \frac{1}{1+i\omega RC}$$ $$|H| = \frac{1}{\sqrt{1+(\omeg Given Information See problem statement for all given quantities.…

Problem Statement Analyze the circuit: Analyze the circuit: Two coupled circuits have inductances $L_1$, $L_2$ and mutual inductance $M$. The coupling coefficient is $k = M/\sqrt{L_1 L_2}$. When is coupling maximum? Faraday’s law for circuit 1 and 2: $$\mathcal{E}_1 = -L_1\dot{I}_1 – M\dot{I}_2$$ $$\mathcal{E}_2 = -L_2\dot{I}_2 – M\dot{I Given Information See problem statement for all…

Problem Statement Solve the capacitor/capacitance problem: Solve the capacitor/capacitance problem: Show that a pure inductor and a pure capacitor consume zero average power from an AC source. Pure inductor ($R=0$, $X_L = \omega L$): current lags voltage by $90°$, i.e., $\psi = -\pi/2$. $$\langle P_L\rangle = V_{\rm rms}I_{\rm rms}\cos(-\pi/2) = 0$$ Pure ca Given Information…