To determine when the angular frequency of a damped oscillator, denoted as ω', equals half of the angular frequency of an undamped oscillator, represented as ω (i.e., ω' = ω/2), we need to understand the relationship between these frequencies and the parameters that influence damped oscillation.

Understanding Damped and Undamped Oscillators

An undamped oscillator, such as a simple harmonic oscillator, exhibits a consistent frequency of oscillation determined by its mass and spring constant. The angular frequency of this undamped system is given by:

• ω = √(k/m)

where k is the spring constant, and m is the mass of the oscillator.

The Effect of Damping

When damping is introduced, often due to friction or resistance, the system's oscillation characteristics change. The damped frequency (ω') is given by the formula:

• ω' = √(ω² - (b/2m)²)

Here, b represents the damping coefficient, which quantifies the amount of damping in the system.

Deriving the Condition for Damped Frequency

To find the condition under which ω' = ω/2, we can set up the equation:

• ω' = ω/2

Substituting the expression for ω' into this equation gives us:

• √(ω² - (b/2m)²) = ω/2

Next, we can square both sides to eliminate the square root, yielding:

• ω² - (b/2m)² = (ω²/4)

Rearranging this leads us to:

• ω² - (ω²/4) = (b/2m)²

Combining like terms results in:

• (3ω²/4) = (b/2m)²

Finding the Damping Coefficient

From this equation, we can solve for the damping coefficient b:

• b = 2m√(3ω²/4)

This means that for the damped frequency (ω') to equal half of the undamped frequency (ω), the damping coefficient must be equal to:

• b = mω√3/2

Practical Implications

In practical terms, this relationship indicates that if you have an oscillator and you know its mass and undamped frequency, you can determine the precise damping coefficient required to achieve a damped frequency that is exactly half of the undamped frequency. This information is crucial in various engineering applications, such as designing suspension systems or tuning musical instruments, where managing oscillation characteristics is vital.

In summary, the condition for a damped oscillator's frequency to equal half that of an undamped oscillator is directly tied to the damping coefficient, which must be carefully calculated based on the system's mass and the natural frequency. This interplay between mass, damping, and frequency is fundamental in the study of oscillatory systems.