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Chapter 31: Problem 6

A \(0.50 \mathrm{~kg}\) body oscillates in SHM on a spring that, when extended \(2.0 \mathrm{~mm}\) from its equilibrium position, has an \(8.0 \mathrm{~N}\) restoring force. What are (a) the angular frequency of oscillation, (b) the period of oscillation, and (c) the capacitance of an \(L C\) circuit with the same period if \(L\) is \(5.0 \mathrm{H} ?\)

Short Answer

Expert verified
(a) \(89.44 \text{ rad/s}\), (b) \(0.0702 \text{ s}\), (c) \(1.58 \times 10^{-4} \text{ F}\).

Step by step solution

01

Determine Spring Constant

Use Hooke's Law which defines the restoring force for a spring as \( F = kx \). Given that the force \( F = 8.0 \text{ N} \) and the displacement \( x = 2.0 \times 10^{-3} \text{ m} \), we can find the spring constant \( k \):\[k = \frac{F}{x} = \frac{8.0}{2.0 \times 10^{-3}} = 4000 \text{ N/m}.\]
02

Calculate Angular Frequency

The angular frequency \( \omega \) is given by the formula \( \omega = \sqrt{\frac{k}{m}} \), where \( k = 4000 \text{ N/m} \) and \( m = 0.50 \text{ kg} \).\[\omega = \sqrt{\frac{4000}{0.50}} = \sqrt{8000} = 89.44 \text{ rad/s}.\]
03

Determine Period of Oscillation

The period \( T \) of oscillation is related to the angular frequency by \( T = \frac{2\pi}{\omega} \).\[T = \frac{2 \pi}{89.44} = 0.0702 \text{ s}.\]
04

Calculate Capacitance for LC Circuit

Use the relationship between the period and inductance-capacitance in an LC circuit, \( T = 2\pi\sqrt{LC} \). Solve for \( C \) using \( T = 0.0702 \text{ s} \) and \( L = 5.0 \text{ H} \).\[0.0702 = 2\pi\sqrt{5.0 \cdot C} \Rightarrow \sqrt{5.0 \cdot C} = \frac{0.0702}{2\pi}\]Square both sides and solve for \( C \):\[5.0 \cdot C = \left(\frac{0.0702}{2\pi}\right)^2 \Rightarrow C = \frac{1}{5.0} \cdot \left(\frac{0.0702}{2\pi}\right)^2 \approx 1.58 \times 10^{-4} \text{ F}.\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Hooke's Law
Hooke鈥檚 Law is a foundational principle in physics that explains the behavior of springs. It states that the force needed to extend or compress a spring is directly proportional to the distance it is stretched or compressed. The formula used is:
  • \[ F = kx \]
Here, \( F \) is the force applied to the spring, \( k \) is the spring constant, and \( x \) is the displacement of the spring from its equilibrium position.
If a spring has a high spring constant \( k \), it means it is stiffer and requires more force to compress or extend.
Understanding Hooke鈥檚 Law is crucial for analyzing systems involving oscillations and harmonics.
Angular Frequency
Angular frequency \( \omega \) is a key concept in oscillatory motion, helping describe how fast an object oscillates through its cycle. The formula for angular frequency in simple harmonic motion (SHM) is:
  • \[ \omega = \sqrt{\frac{k}{m}} \]
Where \( \omega \) is angular frequency, \( k \) is the spring constant, and \( m \) is mass.
It鈥檚 measured in radians per second (rad/s).
It tells us how many radians an object oscillates around the circle every second during SHM.
The higher the angular frequency, the faster the object oscillates, indicating a stiffer spring or lighter mass.
LC Circuit
An LC circuit is an electrical circuit consisting of an inductor (\( L \)) and a capacitor (\( C \)). These components store energy, with the inductor storing it as magnetic energy and the capacitor storing it as electric energy.
Such circuits are vital in tuning radios and generating oscillations. They switch energy back and forth between the inductor and capacitor, similar to a spring and mass system in mechanics, thus forming oscillatory systems.
The period of oscillation \( T \) in an LC circuit is:
  • \[ T = 2\pi\sqrt{LC} \]
This relation shows that the frequency of an LC circuit depends only on the inductance and capacitance values.
Capacitance
Capacitance is a measure of how much electric charge a capacitor can hold at a certain voltage. It is directly related to the capacitor's ability to store energy.
The basic unit of capacitance is the farad (F).
In the context of an LC circuit, larger capacitance implies more energy storage, leading to slower oscillations, as the energy takes longer to transfer between the capacitor and inductor.
Capacitance \( C \) can be determined by re-arranging the formula for the period of oscillation in an LC circuit:
  • \[ C = \frac{1}{L} \left(\frac{T}{2\pi}\right)^2 \]
This clearly ties in the oscillation properties and the component characteristics of the LC circuit.
Spring Constant
The spring constant \( k \) is a measure of a spring鈥檚 stiffness. It is a key parameter in Hooke's Law, defining how much force is needed to extend or compress a spring by a unit length.
A larger \( k \) means a stiffer spring that doesn't easily deform, whereas a smaller \( k \) indicates a more flexible spring.
The spring constant is pivotal when determining the system鈥檚 angular frequency and period of oscillation in simple harmonic motion.
  • When you know the force applied and displacement, the spring constant can be calculated using:\[ k = \frac{F}{x} \]
This understanding is crucial for tackling problems related to oscillatory systems.

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Most popular questions from this chapter

An ac generator produces emf \(\mathscr{8}=\mathscr{E}_{m} \sin \left(\omega_{d} t-\pi / 4\right)\), where \(\mathscr{E}_{m}=30.0 \mathrm{~V}\) and \(\omega_{d}=350 \mathrm{rad} / \mathrm{s}\). The current in the circuit attached to the generator is \(i(t)=I \sin \left(\omega_{d} t+\pi / 4\right)\), where \(I=620 \mathrm{~mA} .(\mathrm{a})\) At what time after \(t=0\) does the generator emf first reach a maximum? (b) At what time after \(t=0\) does the current first reach a maximum? (c) The circuit contains a single element other than the generator. Is it a capacitor, an inductor, or a resistor? Justify your answer. (d) What is the value of the capacitance, inductance, or resistance, as the case may be?

In a certain oscillating \(L C\) circuit, the total energy is converted from electrical energy in the capacitor to magnetic energy in the inductor in \(1.50 \mu \mathrm{s}\). What are (a) the period of oscillation and (b) the frequency of oscillation? (c) How long after the magnetic energy is a maximum will it be a maximum again?

An alternating emf source with a variable frequency \(f_{d}\) is connected in series with an \(80.0 \Omega\) resistor and a \(40.0 \mathrm{mH}\) inductor. The emf amplitude is \(6.00 \mathrm{~V}\). (a) Draw a phasor diagram for phasor \(V_{R}\) (the potential across the resistor) and phasor \(V_{L}\) (the potential across the inductor). (b) At what driving frequency \(f_{d}\) do the two phasors have the same length? At that driving frequency, what are (c) the phase angle in degrees, (d) the angular speed at which the phasors rotate, and (e) the current amplitude?

The frequency of oscillation of a certain \(L C\) circuit is \(200 \mathrm{kHz}\). At time \(t=0\), plate \(A\) of the capacitor has maximum positive charge. At what earliest time \(t>0\) will (a) plate \(A\) again have maximum positive charge, (b) the other plate of the capacitor have maximum positive charge, and (c) the inductor have maximum magnetic field?

An ac generator with \(\mathscr{E}_{m}=220 \mathrm{~V}\) and operating at \(400 \mathrm{~Hz}\) causes oscillations in a series \(R L C\) circuit having \(R=220\) \(\Omega, L=150 \mathrm{mH}\), and \(C=24.0 \mu \mathrm{F}\). Find (a) the capacitive reactance \(X_{C},(\mathrm{~b})\) the impedance \(Z\), and \((\mathrm{c})\) the current amplitude \(I .\) A second capacitor of the same capacitance is then connected in series with the other components. Determine whether the values of (d) \(X_{C}\), (e) \(Z\), and (f) \(I\) increase, decrease, or remain the same.
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