Warning: foreach() argument must be of type array|object, bool given in /var/www/html/web/app/themes/studypress-core-theme/template-parts/header/mobile-offcanvas.php on line 20

Another interpretation of the Q of a resonance comes from the following: Consider the motion of a driven damped oscillator after any transients have died out, and suppose that it is being driven close to resonance, so you can set ω=ωo. (a) Show that the oscillator's total energy (kinetic plus potential) is E=12mω2A2. (b) Show that the energy ΔEdis dissipated during one cycle by the damping force  Fdmp  is 2πmβωA2. (Remember that the rate at which a force does work is Fv. ) (c) Hence show that Q is 2π times the ratio E/ΔEdis: 

Short Answer

Expert verified
(a) Derive energy as E=12mω2A2; (b) Show dissipated energy as 2πmβωA2; (c) Quality factor is Q=ω2β.

Step by step solution

01

Express Total Energy

The total energy of a driven damped harmonic oscillator is the sum of its kinetic and potential energy. In this case, let the total energy be expressed as E=12mω2A2. You know that A is the amplitude of the oscillator, m is the mass, and ω is the angular frequency. Since these quantities define the energy stored in the harmonic motion, the expression for E is valid.
02

Derive Energy Dissipated Per Cycle

To find the energy dissipated in one cycle by the damping force, consider the damping force formula Fdmp=bv, where v=ωAsin(ωt). The work done by the damping force in one cycle can be calculated as ΔEextdis=0TFdmpvdt=0Tbv2dt. Given the cycle period T=2πω, compute that ΔEdis=2πmβωA2, where β=b2m. This matches the given formula for dissipated energy.
03

Relate Quality Factor to Energy Ratios

The definition of quality factor Q is related to the energy stored to energy dissipated per cycle as Q=2πEΔEdis. Substituting from earlier results, Q=2π12mω2A22πmβωA2. Simplifying this expression gives Q=ω2β. This captures the relationship that Q represents the number of radians per cycle times the ratio of stored to dissipated energy.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with Vaia!

Key Concepts

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

Quality Factor
The concept of the quality factor, often denoted as Q, is a crucial aspect of resonance in oscillators. It describes how underdamped an oscillator is and essentially measures the sharpness of its resonance peak. In simpler terms, Q indicates how well an oscillator can store energy versus how much it loses over time through processes like damping.
To calculate Q, we use the formula:- Q=2πEΔEdis- Where E is the total energy of the system, ΔEdis is the energy dissipated per cycle.
High Q values mean that the oscillator maintains its energy for many cycles, creating a sharp and intense resonant peak. Conversely, a low Q value indicates that energy dissipates quickly. In practical applications, such as electronic circuits and musical instruments, a higher Q often implies better performance and narrower bandwidth. Understanding Q is fundamental in designing and analyzing systems that rely on precise energy control.
Damped Harmonic Oscillator
A damped harmonic oscillator is a system that experiences a restoring force proportional to its displacement and a damping force that opposes its motion. The damping force gradually decreases the amplitude of oscillation over time, resulting in energy loss.
When we talk about damping, we refer to the parameter β, which relates to how quickly the system dissipates energy. This parameter is given by β=b2m, where b is the damping coefficient and m is the mass. There are different types of damping:
  • Underdamped: The system oscillates with a gradually decreasing amplitude.
  • Critically damped: The system returns to equilibrium as quickly as possible without oscillating.
  • Overdamped: The system returns to equilibrium without oscillating, but more slowly.
A damped harmonic oscillator is crucial in many fields, from physics to engineering, as it models the behavior of systems like car suspensions, bridges, or even electronic components. Understanding damping ensures stability and efficiency in these systems.
Energy Dissipation in Oscillators
Energy dissipation in oscillators happens due to the damping force, which converts mechanical energy into thermal energy, reducing the amplitude of oscillation. This process is important because it defines the lifespan and efficiency of oscillatory systems.
The energy dissipated in one cycle by the damping force, ΔEdis, can be calculated using:
- ΔEdis=2πmβωA2

Here, β is the damping ratio, ω is the angular frequency, and A is the amplitude. This equation shows how energy dissipation is influenced by factors like mass and frequency.

In practical scenarios, minimizing energy loss is crucial for optimizing performance. For example, in watches, minimizing energy dissipation allows for a longer time between windings. In engineering, understanding this concept assists in designing systems that either maximize energy retention or strategically dissipate energy based on the application.
Driven Oscillator
A driven oscillator involves an external force that causes it to oscillate, distinguishing it from a free oscillator, which relies solely on initial energy. The key aspect of a driven oscillator is that it receives continual energy input, often allowing it to maintain steady oscillations even in the presence of damping.

When driven near its natural frequency, the system can achieve resonance, where even small driving forces produce large amplitude oscillations. This can be beneficial, as in radio circuits designed to pick up weaker signals, or detrimental, like in bridges susceptible to collapse due to resonance.
The mathematics governing a driven oscillator involve adding the driving force to the equation of motion for the harmonic oscillator. This alters how energy flows into the system and counteracts damping loss, allowing for a stable amplitude over time. Design considerations often include how the driving force's frequency and phase align with the system's natural properties to achieve desired outcomes.
Ultimately, driven oscillators illustrate the balance between energy input, natural frequency, and damping, providing versatility across scientific and engineering applications.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

The potential energy of a one-dimensional mass m at a distance r from the origin is U(r)=U0(rR+λ2Rr) for 0<r<, with Uo,R, and λ all positive constants. Find the equilibrium position ro. Let x be the distance from equilibrium and show that, for small x, the PE has the form U= const +12kx2. What is the angular frequency of small oscillations?

Write down the potential energy U(ϕ) of a simple pendulum (mass m, length l ) in terms of the angle ϕ between the pendulum and the vertical. (Choose the zero of U at the bottom.) Show that, for small angles, U has the Hooke's law form U(ϕ)=12kϕ2, in terms of the coordinate ϕ. What is k?

The maximum displacement of a mass oscillating about its equilibrium position is 0.2m, and its maximum speed is 1.2m/s. What is the period τ of its oscillations?

A massless spring has unstretched length lo and force constant k. One end is now attached to the ceiling and a mass m is hung from the other. The equilibrium length of the spring is now l1. (a) Write down the condition that determines l1. Suppose now the spring is stretched a further distance x beyond its new equilibrium length. Show that the net force (spring plus gravity) on the mass is F=kx. That is, the net force obeys Hooke's law, when x is the distance from the equilibrium position a very useful result, which lets us treat a mass on a vertical spring just as if it were horizontal. (b) Prove the same result by showing that the net potential energy (spring plus gravity) has the form U(x)= const +12kx2

Let f(t) be a periodic function with period τ. Explain clearly why the average of f over one period is not necessarily the same as the average over some other time interval. Explain why, on the other hand, the average over a long time T approaches the average over one period, as T

See all solutions

Recommended explanations on Physics Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free