Class 11 Physics Oscillations Angular Frequency

Angular Frequency (ω)

Angular frequency refers to the angular displacement per unit time. It can also be defined as the rate of change of the phase of a sinusoidal waveform (e.g., in oscillations and waves). Angular frequency is larger than frequency ν (in cycles per second, also called Hz), by a factor of 2π.

Consider the oscillatory motion which is varying with time t and displacement x of the particle from the origin:

x (t) = cos (ωt + Φ)

Let Φ = 0

x (t) = cos (ωt)

After t=T i.e. x (t) = x (t+T)

A cos ωt = A cos ω (t + T)

Now the cosine function is periodic with period 2π, i.e., it first repeats itself after 2π. Therefore,

ω (t + T) = ωt + 2π

i.e. ω = 2π/ T

Where ω = angular frequency of SHM.

  • I. unit is radians per second.
  • It is 2π times the frequency of oscillation.
  • Two simple harmonic motions may have the same A and φ, but different ω.

Problem: - Which of the following functions of time represent (a) simple harmonic, (b) periodic but not simple harmonic, and (c) non-periodic motion? Give period for each case of periodic motion (ω is any positive constant):

(a) sin ωt – cos ωt

(b) sin3ωt

(c) 3 cos (π/4 – 2ωt)

(d) cos ωt + cos 3ωt + cos 5ωt

(e) exp (–ω2t2)

Answer: -

(a) SHM

The given function is:

sinωt – cos ωt

This function represents SHM as it can be written in the form: a sin (ωt + Φ)

Its period is: 2π/ω

(b) Periodic but not SHM

The given function is:

sin 3ωt = 1/4 [3 sin ωt - sin3ωt]

The terms sin ωt and sin ωt individually represent simple harmonic motion (SHM). However, the superposition of two SHM is periodic and not simple harmonic.

Its period is: 2π/ω

(c) SHM

The given function is:

This function represents simple harmonic motion because it can be written in the form: a cos (ωt + Φ) its period is: 2π/2ω = π/ω

(d) Periodic, but not SHM

The given function is cosωt + cos3ωt + cos5ωt. Each individual cosine function represents SHM. However, the superposition of three simple harmonic motions is periodic, but not simple harmonic. 

(e) Non-periodic motion

The given function exp (-ω2t2) is an exponential function. Exponential functions do not repeat themselves. Therefore, it is a non-periodic motion.

 (f) The given function 1 + ωt + ω2t2 is non-periodic.

 

Problem: Which of the followingfunctions of time represent (a) simple harmonic motion and (b) periodic but notsimple harmonic? Give the period for each case?

(1) sin ωt – cos ωt

(2) sin2 ωt

 Answer:

(a) sin ωt – cos ωt

= sin ωt – sin (π/2 – ωt)

= 2 cos (π/4) sin (ωt – π/4)

 = √2 sin (ωt – π/4)

This function represents a simple harmonicmotion having a period T = 2π/ω and aphase angle (–π/4) or (7π/4).

(b) sin2 ωt

= ½ – ½ cos 2 ωt

The function is periodic having a periodT = π/ω. It also represents a harmonicmotion with the point of equilibriumoccurring at ½ instead of zero.

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