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 (–ω^{2}t^{2})

** 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 (-ω^{2}t^{2}) is an exponential function. Exponential functions do not repeat themselves. Therefore, it is a non-periodic motion.

(f) The given function 1 + ωt + ω^{2}t^{2} 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) sin^{2} 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) sin^{2} 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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