# A lightly damped harmonic oscillator

• kraigandrews
In summary, the logarithmic decrement δ is defined as the natural logarithm of the ratio of successive maximum displacements of a damped oscillator. There is a simple relationship between δ and Q, and to find the spring constant and damping constant of a damped oscillator, you need to use the equations \beta=b/(2m), \omegao=(k/m)1/2, and Q=\omegao/(2\beta). To solve for these constants, you will need to use the equation d2x/dt2+2\betadx/dt+\omegao2x=0 and find a general solution.
kraigandrews

## Homework Statement

The logarithmic decrement δ of a lightly damped oscillator is defined to be the natural logarithm of the ratio of successive maximum displacements (in the same direction) of a free damped oscillator. That is, δ = ln(An/An+1) where An is the maximum displacement of the n-th cycle. Derive the simple relationship between δ and Q.
Find the spring constant k and damping constant b of a damped oscillator with mass m, frequency of oscillation f and logarithmic decrement δ.
[Data: m = 4.0 kg; f = 0.9 Hz; δ = 0.029.]
First, the spring constant k...

Also, the damping constant b...

## Homework Equations

$\beta$=b/(2m)
$\omega$o=(k/m)1/2
Q=$\omega$o/(2$\beta$)

## The Attempt at a Solution

Ok so I do not know where to start, I can solve the equation:

d2x/dt2+2$\beta$dx/dt+$\omega$o2x=0

but other than that I have no idea where to go. My best guess is that it is unnecessary to solve it because no initial conditions are given thus one would be unable to find the constant or any initial amplitude. any help is greatly appreciated.
thanks

You do need to solve the equation. Find a general solution given the parameters in the equation, and go from there. From reading the problem, I think you can assume the oscillator is underdamped.

## 1. What is a lightly damped harmonic oscillator?

A lightly damped harmonic oscillator is a system in which the damping force is relatively small compared to the restoring force. It exhibits oscillatory behavior with a gradually decreasing amplitude over time.

## 2. What is the equation of motion for a lightly damped harmonic oscillator?

The equation of motion for a lightly damped harmonic oscillator is given by m¨ + γv + kx = 0, where m is the mass, γ is the damping coefficient, v is the velocity, k is the spring constant, and x is the displacement from equilibrium.

## 3. How does the damping coefficient affect the behavior of a lightly damped harmonic oscillator?

The damping coefficient determines the rate at which the oscillations of a lightly damped harmonic oscillator decrease in amplitude. A larger damping coefficient results in faster decay of the oscillations, while a smaller damping coefficient allows for longer-lasting oscillations.

## 4. What is the natural frequency of a lightly damped harmonic oscillator?

The natural frequency of a lightly damped harmonic oscillator is ω0 = √(k/m), where k is the spring constant and m is the mass. This frequency represents the rate at which the system would oscillate if there was no damping present.

## 5. How does the amplitude of a lightly damped harmonic oscillator change over time?

The amplitude of a lightly damped harmonic oscillator decreases over time due to the presence of damping. The rate at which the amplitude decreases depends on the value of the damping coefficient - larger damping coefficients lead to faster decay of the amplitude, while smaller damping coefficients result in slower decay.

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