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Chapter 28: Problem 78

Your professor tells you about the days before digital computers when engineers used electric circuits to model mechanical systems. Suppose a \(5.0-\mathrm{kg}\) mass is connected to a spring with \(k=1.44 \mathrm{kN} / \mathrm{m} .\) This is then modeled by an \(L C\) circuit with \(L=2.5 \mathrm{H} .\) What should \(C\) be in order for the \(L C\) circuit to have the same resonant frequency as the mass-spring system?

Short Answer

Expert verified
The value of C in microfarads (µF) for the LC circuit to have the same resonant frequency as the mass-spring system is 2.0576 µF.

Step by step solution

01

Understand the Physical Correspondences and Formulas

The mass-spring system mechanical resonance frequency formula is \( f_m = \frac{1}{2\pi} \sqrt{\frac{k}{m}} \) and the electrical LC Circuit resonance frequency formula is \( f_e = \frac{1}{2\pi \sqrt{LC}} \). Here, \( f_m \) is the frequency of the mass-spring system, m is the mass, k is the spring constant, \( f_e \) is the frequency of the LC circuit, L is the inductance and C is the capacitance.
02

Substitute Known Values into the Formulas

Substitute the known values into the formulas: For the mass-spring system, \( f_m = \frac{1}{2\pi} \sqrt{\frac{1.44 \times 10^3}{5}} \). For the LC circuit, we do not know C, so the formula remains \( f_e = \frac{1}{2\pi \sqrt{2.5C}} \).
03

Equate Frequencies and Solve for Unknown

Set \( f_m \) equal to \( f_e \) and simplify: \( \frac{1}{2\pi} \sqrt{\frac{1.44 \times 10^3}{5}} = \frac{1}{2\pi \sqrt{2.5C}} \). Simplify and solve for C to find the value of capacitance that makes the LC circuit have the same resonant frequency as the mass-spring system. Solve \( C = \frac{1}{\left(2.5 \times \left(\frac{1.44 \times 10^3}{5}\right)\right)} \)

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Most popular questions from this chapter

A \(470-\Omega\) resistor, \(10-\mu \mathrm{F}\) capacitor, and \(750-\mathrm{mH}\) inductor are each connected across \(6.3-\mathrm{V}\) rms, \(60-\mathrm{Hz}\) AC power. Find the rms current in each.

An AC voltage of fixed amplitude is applied across a series \(R L C\) circuit. The components are such that the current at half the resonant frequency is half the current at resonance. Show that the current at twice the resonant frequency is also half the current at resonance.

An LC circuit with a 20 - \(\mu\) F capacitor oscillates with period \(5.0 \mathrm{ms} .\) The peak current is \(25 \mathrm{mA} .\) Find (a) the inductance and (b) the peak voltage.

A 15 - \(\mu\) F capacitor carries \(1.4 \mathrm{A}\) rms. What's its minimum safe voltage rating if the frequency is (a) \(60 \mathrm{Hz}\) and (b) \(1.0 \mathrm{kHz} ?\)

A car-battery charger runs off the \(120-\mathrm{V}\) rms AC power line and supplies \(10-\mathrm{A}\) DC at \(14 \mathrm{V}\). (a) If the charger is \(80 \%\) efficient in converting the line power to the DC power it supplies to the battery, how much current does it draw from the AC line? (b) If electricity costs \(9.5 \notin / \mathrm{kWh}\), how much does it cost to run the charger for 10 hours if the power factor is \(1 ?\)
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