University Physics with Modern Physics Plus Mastering Physics with eText -- Access Card Package (14th Edition)
University Physics with Modern Physics Plus Mastering Physics with eText -- Access Card Package (14th Edition)
14th Edition
ISBN: 9780321982582
Author: Hugh D. Young, Roger A. Freedman
Publisher: PEARSON
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Chapter 31, Problem 31.66P

DATA You are given this table of data recorded for a circuit that has a resistor, an inductor with negligible resistance, and a capacitor, all in series with an ac voltage source:

f(Hz) 80 160
z (Ω) 15 13
ϕ (°) –71 67

Here f is the frequency of the voltage source, Z is the impedance of the circuit, and ϕ is the phase angle. (a) Use the data at both frequencies to calculate the resistance of the resistor. (Hint: Use the results of Exercise 31.21.) Calculate the average of these two values of the resistance, and use the result as the value of R in the rest of the analysis. (b) Use the data at 80 Hz and 160 Hz to calculate the inductance L and capacitance C of the circuit. (c) What is the resonance frequency for the circuit, and what are the impedance and phase angle at the resonance frequency?

31.66 IDENTITY and SET UP: For an L-R-C series circuit, tan ϕ = ω L 1 ω C R and the power factor is cosϕ = R/Z.

EXECUTE: (a) cosϕ = R/Z, so R = Z cosϕ.

At 80 Hz: R = (15 Ω) cos(−71°) = 4.88 Ω

At 160 Hz: R = (13 Ω) cos(67°) = 5.08 Ω

The average resistance is (4.88 Ω + 5.08 Ω)/2 = 5.0 Ω

(b) We use tan ϕ = ω L 1 ω C R with R = 5.0 Ω from part (a).

At 80 Hz: tan ( 71 ) = 2 π ( 80  Hz ) L 1 2 π ( 80  Hz ) C 5.0   Ω .

−14.52 = 160π Hz L − 1/[(160π Hz)C]. Eq. (1)

At 160 Hz: tan ( 67 ) = 2 π ( 160  Hz ) L 1 2 π ( 160  Hz ) C 5.0   Ω .

11.78 = 320π Hz L − 1/[(320π Hz)C]. Eq. (2)

Multiply Eq. (1) by −2 and add it to Eq. (2), giving 2(14.52) + 11.78 = (1/C)(1/80π − 1/320π).

C = 7.31 × 10−5 F, which rounds to C = 73 μF.

Substituting this result into either Eq. (1) or Eq. (2) gives L = 25.3 mH, which rounds to L = 25 mH.

(c) The resonance angular frequency is ω 0 = 1 L C , so the resonance frequency is

f 0 = ω 0 2 π = 1 2 π L C = 1 2 π ( 2.53 × 10 2  H ) ( 73.1 × 10 6  F ) = 117  Hz

At resonance, Z = R = 5.0 Ω and ϕ = 0.

EVALUATE: It is only at resonance that Z = R, not at the other frequencies.

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Chapter 31 Solutions

University Physics with Modern Physics Plus Mastering Physics with eText -- Access Card Package (14th Edition)

Ch. 31 - Prob. 31.5DQCh. 31 - Equation (31.9) says that ab = L di/dt (see Fig....Ch. 31 - Is it possible for the power factor of an L-R-C...Ch. 31 - In an L-R-C series circuit, can the instantaneous...Ch. 31 - In an L-R-C series circuit, what are the phase...Ch. 31 - When an L-R-C series circuit is connected across a...Ch. 31 - Prob. 31.11DQCh. 31 - A light bulb and a parallel-plate capacitor with...Ch. 31 - A coil of wire wrapped on a hollow tube and a...Ch. 31 - A circuit consists of a light bulb, a capacitor,...Ch. 31 - A circuit consists of a light bulb, a capacitor,...Ch. 31 - Prob. 31.16DQCh. 31 - An ideal transformer has N1, windings in the...Ch. 31 - An inductor, a capacitor, and a resistor are all...Ch. 31 - You want to double the resonance angular frequency...Ch. 31 - Prob. 31.1ECh. 31 - A sinusoidal current i = I cos t has an rms value...Ch. 31 - The voltage across the terminals of an ac power...Ch. 31 - A capacitor is connected across an ac source that...Ch. 31 - An inductor with L = 9.50 mH is connected across...Ch. 31 - A capacitance C and an inductance L are operated...Ch. 31 - Kitchen Capacitance. The wiring for a refrigerator...Ch. 31 - (a) Compute the reactance of a 0.450-H inductor at...Ch. 31 - (a) What is the reactance of a 3.00-H inductor at...Ch. 31 - A Radio Inductor. You want the current amplitude...Ch. 31 - A 0.180-H inductor is connected in series with a...Ch. 31 - A 250- resistor is connected in series with a...Ch. 31 - A 150- resistor is connected in series with a...Ch. 31 - You have a 200- resistor, a 0.400-H inductor, and...Ch. 31 - The resistor, inductor, capacitor, and voltage...Ch. 31 - Prob. 31.16ECh. 31 - In an L-R-C series circuit, the rms voltage across...Ch. 31 - A resistor with R = 300 and an inductor are...Ch. 31 - The power of a certain CD player operating at 120...Ch. 31 - In an L-R-C series circuit, the components have...Ch. 31 - (a) Show that for an L-R-C series circuit the...Ch. 31 - (a) Use the results of part (a) of Exercise 31.21...Ch. 31 - An L-R-C series circuit with L = 0.120 H, R = 240...Ch. 31 - An L-R-C series circuit is connected to a 120-Hz...Ch. 31 - A series ac circuit contains a 250- resistor, a...Ch. 31 - In an L-R-C series circuit the source is operated...Ch. 31 - Analyzing an L-R-C Circuit. You have a 200-...Ch. 31 - An L-R-C series circuit is constructed using a...Ch. 31 - In an L-R-C series circuit, R = 300, L = 0.400 H,...Ch. 31 - An L-R-C series circuit consists of a source with...Ch. 31 - In an L-R-C series circuit, R = 150 , L = 0.750 H,...Ch. 31 - In an L-R-C series circuit, R = 400 , L = 0.350 H,...Ch. 31 - In an L-R-C series circuit, L = 0.280 H and C =...Ch. 31 - Section 31.6 Transformers 31.34Off to Europe! You...Ch. 31 - A Step-Down Transformer. A transformer connected...Ch. 31 - A Step-Up Transformer. A transformer connected to...Ch. 31 - A coil has a resistance of 48.0 . At a frequency...Ch. 31 - Prob. 31.38PCh. 31 - An L-R-C series circuit has C = 4.80 F, L = 0.520...Ch. 31 - Five infinite-impedance voltmeters, calibrated to...Ch. 31 - CP A parallel-plate capacitor having square plates...Ch. 31 - Prob. 31.42PCh. 31 - A series circuit has an impedance of 60.0 and a...Ch. 31 - A large electromagnetic coil is connected to a...Ch. 31 - In an L-R-C series circuit, R = 300 , XC = 300 ,...Ch. 31 - At a frequency 1, the reactance of a certain...Ch. 31 - A High-Pass Filter. One application of L-R-C...Ch. 31 - A Low-Pass Filter. Figure P31.48 shows a low-pass...Ch. 31 - An L-R-C series circuit is connected to an ac...Ch. 31 - An L-R-C series circuit is connected to an ac...Ch. 31 - In an L-R-C series circuit the magnitude of the...Ch. 31 - In an L-R-C series circuit, the phase angle is...Ch. 31 - An L-R-C series circuit has R = 500 . L = 2.00 H,...Ch. 31 - The L-R-C Parallel Circuit. A resistor, an...Ch. 31 - The impedance of an L-R-C parallel circuit was...Ch. 31 - A 400- resistor and a 6.00-F capacitor are...Ch. 31 - An L-R-C series circuit consists of a 2.50-F...Ch. 31 - An L-R-C series circuit has R = 60.0 , L = 0.800...Ch. 31 - In an L-R-C series circuit, the source has a...Ch. 31 - In an L-R-C series ac circuit, the source has a...Ch. 31 - A resistance R, capacitance C, and inductance L...Ch. 31 - The Resonance Width. Consider an L-R-C series...Ch. 31 - An L-R-C series circuit draws 220 W from a 120-V...Ch. 31 - DATA A coworker of yours was making measurements...Ch. 31 - DATA You are analyzing an ac circuit that contains...Ch. 31 - DATA You are given this table of data recorded for...Ch. 31 - CALC In an L-R-C series circuit the current is...Ch. 31 - CALC (a) At what angular frequency is the voltage...Ch. 31 - Prob. 31.69PPCh. 31 - If the electrode oscillates between two points 20...Ch. 31 - The signal from the oscillating electrode is fed...Ch. 31 - If the frequency at which the electrode is...
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