Author: dexter
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Irodov Problem 4.52: Forced Oscillations: Power Absorbed at Resonance
Problem Statement Solve the oscillation/wave problem: Solve the oscillation/wave problem: Find the mean power P absorbed by a forced oscillator from the driving force F = F₀ cos(ωt) as a function of driving frequency ω. Show that P is maximum at ω = ω₀ and find P_max. Driving force: F₀ cos(ωt) System: mass m, damping…
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Irodov Problem 4.51: Forced Oscillations: Maximum Velocity Amplitude at Resonance
Problem Statement Solve the oscillation/wave problem: Solve the oscillation/wave problem: A forced oscillator is driven by a force F = F₀ cos(ωt). Show that the velocity amplitude v_a = aω reaches its maximum at ω = ω₀ (not at the displacement resonance frequency ω_res). Find v_a,max. Driving force: F₀ cos(ωt) System parameters: β, ω₀, mass…
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Irodov Problem 4.50: Quality Factor from Resonance Curve Half-Power Width
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 Natural (undamped) angular frequency $\omega_0 = \sqrt{k/m}$ Damping coefficient $\beta = b/(2m)$ where $b$ is the viscous drag…
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Irodov Problem 4.49: Forced Oscillations: Resonance Amplitude and Phase
Problem Statement Solve the oscillation/wave problem: Solve the oscillation/wave problem: A forced oscillator obeys ẍ + 2βẋ + ω₀²x = (F₀/m) cos(ωt). Find the steady-state amplitude a(ω) and phase lag φ(ω), and determine the resonance frequency ω_res. Driving amplitude: F₀/m Damping coefficient: β Natural frequency: ω₀ Driving frequency: ω (variable) St Given Information Mass $m$…
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Irodov Problem 4.48: Damped Oscillations: Amplitude at Energy Half-Life
Problem Statement Solve the nuclear physics problem: 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. Natural (undamped) angular frequency $\omega_0 = \sqrt{k/m}$ Damping coefficient $\beta = b/(2 Given Information Nuclide…
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Irodov Problem 4.47: Damped Oscillations: Amplitude Decrease After n Cycles
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 Natural (undamped) angular frequency $\omega_0 = \sqrt{k/m}$ Damping coefficient $\beta = b/(2m)$ where $b$ is the viscous drag…
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Irodov Problem 4.46: Damped Oscillations: Logarithmic Decrement and Quality Factor
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 Natural (undamped) angular frequency $\omega_0 = \sqrt{k/m}$ Damping coefficient $\beta = b/(2m)$ where $b$ is the viscous drag…
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HCV Ch28 P15 – Stefan-Boltzmann: Rate of Cooling by Radiation
Problem Statement A solid sphere of radius 5 cm, mass 1 kg, specific heat 500 J/kg·K, emissivity 0.9 is at 500 K in a surrounding at 300 K. Find its initial rate of cooling. ($\sigma = 5.67 \times 10^{-8}$ W/m²·K⁴) Given Information See problem statement for all given quantities. Physical Concepts & Formulas This problem…
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HCV Ch28 P14 – Convection: Heat Transfer Coefficient
Problem Statement Solve the thermodynamics problem: Solve the thermodynamics problem: A flat plate of area 0.5 m² at 60°C loses heat by convection to air at 20°C. The convective heat transfer coefficient is $h = 25$ W/m²·K. Find the rate of heat loss. $A = 0.5$ m² $T_{surface} = 60°C$, $T_{air} = 20°C$, $\Delta T…
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Problem 2.180 — Fermi-Dirac vs Boltzmann: When Quantum Statistics Matters
Problem Statement Solve the quantum/modern physics problem: Solve the quantum/modern physics problem: At what temperature does quantum degeneracy become important for electrons in copper? Quantum effects become important when $k_BT \lesssim E_F$. Define the Fermi temperature: $$T_F = \frac{E_F}{k_B} = \frac{7.04\times1.6\times10^{-19}}{1.38\times10^{-23}} = Given Information See problem statement for all given quantities. Physical Concepts & Formulas…