Question

A harmonic oscillator has angular frequnecy and amplitude.

what are the magnitudes of the disaplcement

Answer 1

In a harmonic oscillator, the magnitude of displacement at any time is determined by the absolute value of the displacement equation, incorporating amplitude, angular frequency, and phase angle.

In a harmonic oscillator, the displacement of the oscillating particle is characterized by its amplitude and angular frequency. The equation describing the displacement (x) as a function of time (t) for a harmonic oscillator is given by:

`\[ x(t) = A \cos(\omega t + \phi) \]`

where:

• A is the amplitude,

• 
  `\(\omega\)` is the angular frequency,

• 
  `\(\phi\)` is the phase angle.

The amplitude (A) represents the maximum magnitude of displacement from the equilibrium position. It is the distance between the equilibrium position and the farthest point reached by the oscillating particle. The angular frequency
`(\(\omega\))` is related to the frequency (f) by the equation
`\(\omega = 2\pi f\)`, where (f) is the frequency of oscillation.

The displacement (x) varies sinusoidally with time, oscillating between positive and negative values. The magnitude of displacement is simply the absolute value of x, irrespective of its direction.

To find the magnitude of displacement at any given time, take the absolute value of the displacement equation:

`\[ |x(t)| = |A \cos(\omega t + \phi)| \]`

This expression gives the magnitude of the displacement at any instant, considering the amplitude, angular frequency, and phase angle.