Ch 32: AC Circuits

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook

Knight Calc 5th Edition

Ch 32: AC Circuits

Problem 69b

Chapter 32, Problem 69b

Prove that the energy dissipation is a maximum at ω = ω₀.

Verified step by step guidance

Start by recalling the formula for the power dissipated in a driven damped harmonic oscillator. The power dissipated is proportional to the square of the amplitude of oscillation and can be expressed as: $ P(\omega) = \frac{F_0^2}{2m} \cdot \frac{\gamma \omega^2}{(\omega_0^2 - \omega^2)^2 + \gamma^2 \omega^2} $, where $ F_0 $ is the driving force amplitude, $ \omega $ is the driving angular frequency, $ \omega_0 $ is the natural angular frequency, $ \gamma $ is the damping coefficient, and $ m $ is the mass.

To find the condition for maximum energy dissipation, take the derivative of $ P(\omega) $ with respect to $ \omega $ and set it equal to zero: $ \frac{dP(\omega)}{d\omega} = 0 $. This will help identify the critical points of the power function.

Simplify the derivative $ \frac{dP(\omega)}{d\omega} $ using the quotient rule and chain rule. Focus on the denominator $ (\omega_0^2 - \omega^2)^2 + \gamma^2 \omega^2 $ and the numerator $ \gamma \omega^2 $, as these terms determine the behavior of the function.

Solve the resulting equation for $ \omega $. After simplification, you will find that the maximum power dissipation occurs when $ \omega = \omega_0 $, which is the natural angular frequency of the system.

Verify that $ \omega = \omega_0 $ corresponds to a maximum by checking the second derivative $ \frac{d^2P(\omega)}{d\omega^2} $ or analyzing the behavior of the function around $ \omega_0 $. This confirms that energy dissipation is indeed maximized at $ \omega = \omega_0 $.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Energy Dissipation

Energy dissipation refers to the process by which energy is transformed from one form to another, often resulting in the loss of usable energy, typically as heat. In mechanical systems, this can occur due to friction, air resistance, or internal material resistance. Understanding how energy dissipates is crucial for analyzing system efficiency and performance.

Resonance Frequency (ω₀)

The resonance frequency, denoted as ω₀, is the frequency at which a system naturally oscillates when not subjected to external forces. At this frequency, the system can absorb maximum energy from external sources, leading to increased amplitude of oscillation. This concept is vital in understanding how systems respond to periodic driving forces and the conditions under which energy dissipation is maximized.

Damping

Damping is the effect that reduces the amplitude of oscillations in a system, often due to energy dissipation mechanisms. It plays a critical role in determining how quickly a system returns to equilibrium after being disturbed. The relationship between damping and resonance frequency is essential for analyzing energy dissipation, as maximum energy loss occurs when the driving frequency matches the system's natural frequency.

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Textbook Question

FIGURE CP32.68 shows voltage and current graphs for a series RLC circuit. What is the resistance R?

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A series RLC circuit with ε₀= 10.0 V consists of a 1.0 Ω resistor, a 1.0 μH inductor, and a 1.0 μF capacitor. What is V₁ at ω = ω₀ and at ω = ω₁?

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Textbook Question

A motor attached to a 120 V/60 Hz power line draws an 8.0 A current. Its average energy dissipation is 800 W. What is the motor's resistance?

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Consider the parallel RLC circuit shown in FIGURE CP32.72. What is I in the limits ω→0 and ω→∞?

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A motor attached to a 120 V/60 Hz power line draws an 8.0 A current. Its average energy dissipation is 800 W. How much series capacitance needs to be added to increase the power factor to 1.0?

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Show that the peak inductor voltage in a series RLC circuit is maximum at frequency $ω_{L} = {(\frac{1}{ω_{0}^{2}} - \frac{1}{2} R^{2} C^{2})}^{- 1 / 2}$ .

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