# How to Plot Amplitude of a Driven Oscillator with Damping?

• w3390
Summary: In summary, the problem involves plotting the amplitude D of oscillations, normalized by the maximum amplitude at resonance, as a function of w/w_0. The equation for D is correct and to find Q, the formula Q = w_0/2\beta can be used, with the given value of \beta = (1/6)w_0 resulting in a value of Q = 3. This represents the quality factor of the oscillator and measures how well it maintains its amplitude over time.
w3390

## Homework Statement

The oscillator is driven by a force F(t) = mAcos(wt). Plot the amplitude D of oscillations, in units of the maximum (resonant) amplitude D(max), as a function of w in units of w_0. (In other words, plot D/D(max) versus w/w_0.) Find Q.

$$\beta$$=(1/6)w_0

## Homework Equations

D = A/[sqrt((w_0^2 - w^2)^2 + 4w^2$$\beta$$^2)]

A = (F_0)/m

## The Attempt at a Solution

To start off, is my equation for D correct?

If so, my next question is how do I begin this problem. Is it correct to plug in (1/6)w_0 for the beta in my D equation to get D(max). That way I can divide D by D(max) and get the first part. Or is that completely wrong?

Any help would be greatly appreciated.

Thank you for your post. Your equation for D is correct. To begin this problem, you can start by plotting D/D(max) versus w/w_0, as mentioned in the homework statement. This will give you a plot of the amplitude as a function of w, normalized by the maximum amplitude at resonance.

To find Q, you can use the formula Q = w_0/2\beta. In this case, since you have been given \beta = (1/6)w_0, you can plug this into the formula to get Q = 3. This value of Q represents the quality factor of the oscillator, which is a measure of how well the oscillator maintains its amplitude over time.

I hope this helps. Please let me know if you have any further questions.

## 1. What is a driven oscillator with damping?

A driven oscillator with damping is a physical system that consists of a mass attached to a spring and a damping force, which is responsible for dissipating the energy of the system. The oscillator is also subjected to an external force, called the driving force, which causes it to oscillate.

## 2. How does damping affect the behavior of a driven oscillator?

Damping reduces the amplitude of oscillation in a driven oscillator. It also affects the frequency and phase of the oscillations. Strong damping can cause the oscillator to reach equilibrium faster, while weak damping can result in sustained oscillations.

## 3. What is the role of the driving force in a driven oscillator with damping?

The driving force is responsible for maintaining the oscillations in a driven oscillator with damping. Without it, the system would eventually come to rest due to the damping force dissipating all of its energy.

## 4. How is a driven oscillator with damping different from a simple harmonic oscillator?

A simple harmonic oscillator does not experience any external forces, while a driven oscillator with damping is subjected to a driving force. Additionally, a simple harmonic oscillator does not experience any damping, while a driven oscillator does.

## 5. What are some real-life examples of driven oscillators with damping?

Some examples of driven oscillators with damping include a child's swing, a pendulum clock, and a car's suspension system. In these systems, the mass is subjected to a driving force, such as a push or a gravitational force, and is also damped by friction or air resistance.

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