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Chapter 14: Problem 64

A sinusoidally varying driving force is applied to a damped harmonic oscillator of force constant \(k\) and mass \(m\) . If the damping constant has a value \(b_{1},\) the amplitude is \(A_{1}\) when the driving angular frequency equals \(\sqrt{k / m}\) . In terms of \(A_{1},\) what is the amplitude for the same driving frequency and the same driving force amplitude \(F_{\text { max }},\) if the damping constant is \((a) 3 b_{1}\) and (b) \(b_{1} / 2 ?\)

Short Answer

Expert verified
a) \(\frac{A_1}{3}\); b) \(2A_1\).

Step by step solution

01

Understand the Problem

The problem involves a damped harmonic oscillator subjected to a sinusoidally varying driving force. Initially, the system has certain parameters. With these parameters, when the driving angular frequency is \(\omega=\sqrt{k/m}\), the amplitude is \(A_1\). The problem asks to find out how the amplitude changes when the damping constant \(b\) changes to \(3b_1\) and \(b_1/2\).
02

Recall the Amplitude Equation for Damped Driven Oscillators

The amplitude \(A\) of a forced damped harmonic oscillator is given by: \[ A = \frac{F_{\text{max}}/m}{\sqrt{(\omega_0^2 - \omega^2)^2 + (\frac{b\omega}{m})^2}} \]where \(\omega_0 = \sqrt{\frac{k}{m}}\) is the natural frequency and \(\omega\) is the driving frequency. Since \(\omega = \omega_0\), the equation simplifies to: \[ A = \frac{F_{\text{max}}/m}{\frac{b\omega_0}{m}} = \frac{F_{\text{max}}}{b\omega_0} \]
03

Express Initial Amplitude in Terms of Parameters

Given the initial conditions where \(b = b_1\), the amplitude \(A_1 = \frac{F_{\text{max}}}{b_1 \omega_0}\).
04

Calculate Amplitude for 3b_1 Damping

Replace \(b_1\) with \(3b_1\) in the simplified amplitude equation:\[ A = \frac{F_{\text{max}}}{3b_1 \omega_0} = \frac{A_1}{3} \]The amplitude is one-third of the initial amplitude \(A_1\).
05

Calculate Amplitude for b_1/2 Damping

Similarly, replace \(b_1\) with \(\frac{b_1}{2}\) in the amplitude equation:\[ A = \frac{F_{\text{max}}}{(b_1/2) \omega_0} = 2\frac{F_{\text{max}}}{b_1 \omega_0} = 2A_1 \]The amplitude is twice the initial amplitude \(A_1\).
06

Summarize Results

For a damping constant of \(3b_1\), the amplitude is \(A = \frac{A_1}{3}\). For a damping constant of \(b_1/2\), the amplitude is \(A = 2A_1\).

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Key Concepts

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

Driving Force
In the world of physics, when we talk about a damped harmonic oscillator, we often introduce an external agent called the 'driving force'. This driving force is usually sinusoidal and external, influencing the system to keep oscillating.

The role of the driving force is to supply energy to the oscillating system to compensate for the energy lost due to damping. Without it, the damping would slowly bring the oscillation to a halt. The driving force is characterized by its frequency and amplitude, which synchronizes with the system's natural frequency to produce maximum amplitude.
  • This synchronization is crucial as it determines the energy imparted to the system.
  • Matching frequencies mean more energy input and a larger amplitude.
Understanding the interplay between the driving force and the system's dynamics is key to mastering damped harmonic oscillations.
Amplitude Equation
The amplitude equation is fundamental in analyzing the behavior of a damped harmonic oscillator.

When a system is under the influence of a driving force, its amplitude can be expressed using a specific formula. For a damped system, this formula shows how the driving force and damping affect the system's amplitude:\[ A = \frac{F_{\text{max}}/m}{\sqrt{(\omega_0^2 - \omega^2)^2 + (\frac{b\omega}{m})^2}} \]Where:- \( A \) is the amplitude of oscillation- \( F_{\text{max}} \) is the maximum force- \( b \) is the damping constant- \( m \) is the mass of the system- \( \omega \) is the driving frequency- \( \omega_0 \) is the natural frequencyWhen the driving angular frequency \( \omega \) equals the natural frequency \( \omega_0 \), the equation simplifies significantly, allowing easier calculations of how amplitude responds to changes in damping.
  • This equation indicates the inversely proportional relationship between amplitude and the damping constant.
  • For instance, if damping increases, the amplitude decreases.
Damping Constant
The damping constant, denoted by \( b \), plays a crucial role in affecting the amplitude of an oscillating system. It represents how much resistance the oscillating system faces due to damping.
  • Damping is the process through which oscillation's energy is lost, often due to friction or resistance.
  • The higher the value of \( b \), the quicker the oscillation will fade unless there is a compensatory driving force.
Understanding the impact of the damping constant is important when analyzing how an oscillation decays over time, as seen in this exercise.

To illustrate, consider two scenarios:- When the damping constant is tripled, i.e., \( 3b_1 \), the oscillation's amplitude reduces to a third of its original (\( A_1 \)).- Conversely, when the damping is halved, i.e., \( b_1/2 \), the system experiences an increase in amplitude, doubling from its initial value.
This makes it evident that adjusting \( b \) directly alters the behavior and longevity of the oscillation.
Natural Frequency
The natural frequency \( \omega_0 \) defines how a system would oscillate if there were no damping or external force.
  • It represents the inherent frequency of the system due to its mass and stiffness.
  • Mathematically, it's expressed as \( \omega_0 = \sqrt{\frac{k}{m}} \), where \( k \) is the spring constant, and \( m \) is the mass.
In the absence of damping, this frequency illustrates how the system naturally wants to behave.

When a driving force matches this natural frequency, a phenomenon known as resonance occurs, leading to a substantial increase in amplitude. This setup is where the system can achieve its maximum amplitude without the constraints of damping alone.

Thus, understanding the natural frequency is critical for predicting and controlling oscillatory behavior in various applications, such as mechanical systems and electronic circuits.

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Most popular questions from this chapter

A 2.00 -kg, frictionless block is attached to an ideal spring with force constant 300 \(\mathrm{N} / \mathrm{m} .\) At \(t=0\) the spring is neither stretched nor compressed and the block is moving in the negative direction at 12.0 \(\mathrm{m} / \mathrm{s} .\) Find (a) the amplitude and (b) the phase angle. (c) Write an equation for the position as a function of time.

A small block is attached to an ideal spring and is moving in SHM on a horizontal, frictionless surface. When the amplitude of the motion is \(0.090 \mathrm{m},\) it takes the block 2.70 s to travel from \(x=0.090 \mathrm{m}\) to \(x=-0.090 \mathrm{m} .\) If the amplitude is doubled, to \(0.180 \mathrm{m},\) how long does it take the block to travel (a) from \(x=0.180 \mathrm{m}\) to \(x=-0.180 \mathrm{m}\) and (b) from \(x=0.090 \mathrm{m}\) to \(x=-0.090 \mathrm{m} ?\)

A glider is oscillating in \(\mathrm{SHM}\) on an air track with an amplitude \(A_{1 .}\) You slow it so that its amplitude is halved. What happens to its (a) period, frequency, and angular frequency; (b) total mechanical energy; (c) maximum speed; (d) speed at \(x=\pm A_{1} / 4 ;\) (e) potential and kinetic energies at \(x=\pm A_{1} / 4 ?\)

CALC The balance wheel of a watch vibrates with an angular amplitude \(\theta,\) angular frequency \(\omega,\) and phase angle \(\phi=0 .\) (a) Find expressions for the angular velocity \(d \theta / d t\) and angular acceleration \(d^{2} \theta / d t^{2}\) as functions of time. (b) Find the balance wheel's angular velocity and angular acceleration when its angular displacement is \(\Theta,\) and when its angular displacement is \(\Theta / 2\) and \(\theta\) is decreasing. (Hint: Sketch a graph of \(\theta\) versus t.)

You want to find the moment of inertia of a complicated machine part about an axis through its center of mass. You suspend it from a wire along this axis. The wire has a torsion constant of 0.450 \(\mathrm{N} \cdot \mathrm{m} / \mathrm{rad}\) . You twist the part a small amount about this axis and let it go, timing 125 oscillations in 265 s. What is the moment of inertia you want to find?
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