Period and Angular Frequency - Definition, Formula, Unit, FAQs

Edited By Team Careers360 | Updated on Jul 02, 2025 04:33 PM IST

Many forms of energy, such as light and sound, travel in waves, as we all know. The frequency, amplitude, and speed of a wave are all qualities that define it. Any specific wave in wave mechanics has parameters such as frequency, time period, wavelength, amplitude, and so on. This page explains frequency, time period, and angular frequency in great depth.

Period and Angular Frequency - Definition, Formula, Unit, FAQs

The number of complete cycles of waves passing a spot in unit time is defined as the angular frequency. The time is the amount of time it takes for a complete wave cycle to pass a spot. The angular frequency is defined as the angular displacement of any wave constituent per unit time. It depicts the displacement y of any element along a string travelling in the positive x-direction with respect to time for a harmonic wave. In this case, the string element goes up and down in a basic harmonic motion.

Time period

The time is the amount of time it takes for a complete wave cycle to pass a spot. Periodic angular frequency is defined as the angular displacement of any wave constituent per unit time.

The particles in sinusoidal wave motion travel about the mean equilibrium or mean location with time, as seen above. The particles rise until they reach their highest position, the crest, and then descend until they reach their lowest point, the trough. The cycle follows a predictable pattern. The formula for calculating the time is as follows:

T=2π/ω

Also read -

What is frequency definition

Frequency meaning: In physics, frequency refers to the number of waves that pass-through a given point in one unit of time, as well as the number of cycles or vibrations that body in periodic motion goes through one unit of the time.

Frequency meaning

The number of wave crests passing a specific place every second is referred to as periodic angular frequency.

Time period formula

The term "period" refers to the amount of time it takes for something to occur. Periodic angular frequency is a quantity that has a rate. Frequency is defined as the number of cycles per second. The term "period" refers to the number of seconds in the cycle.

Formula for time period =1/f, f =frequency

Related Topics,

Time period definition

The time it takes for one complete cycle of vibration to pass a certain point is called a time period (abbreviated as 'T') 'Seconds' is the unit of time measurement. The reciprocal relationship between periodic angular frequency and time period can be written mathematically as T=1/f

Periodic Angular frequency

Periodic Angular frequency

The angular displacement of any wave element per unit time is known as angular frequency.

NCERT Physics Notes:

What is time period?

The time it takes for one complete cycle of vibration to pass a specific point is called a time period (abbreviated as 'T'). The time of a wave reduces as the frequency of the wave increases. 'Seconds' is the unit of time measurement. A pendulum's period is the time it takes to swing from one side to the other and back.

Angular frequency formula

$$
\omega=2 \pi / T
$$

is used to calculate the angular frequency physics. In radians per second, the angular frequency physics is measured. The frequency $f=1 / T$ is the inverse of the period. The number of complete oscillations per unit time is determined by the motion's frequency,

$$
f=1 / T=\omega / 2 \pi
$$

Also read:

Omega in physics $(\omega)$

The frequency physics of angular movement per unit time is measured by angular frequency (also known as radial or circular frequency). The angular displacement of any wave element per unit time is known as angular frequency. In wave terminology, the angular frequency of a sinusoidal wave refers to the angular displacement of any element of the wave per unit of time. Omega is the symbol for it.

$$
\omega=2 \pi / T
$$

Time definition

The length of time that an activity, process, or condition persists or continues. B) a non-spatial continuum measured in terms of events that occur one after the other from the past to the present to the future.

Units of time

'Seconds' is the unit of time measurement. The reciprocal relationship between frequency and time can be written mathematically as:

$\mathrm{f}=1 / \mathrm{T}$

Oscillatory Motion:

When an object moves back and forth in a periodic pattern, it is said to be oscillating. An oscillating motion is like how a pendulum oscillates from one end to the other (to-and-fro) about its mean position and keeps repeating its motion.

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Frequently Asked Questions (FAQs)

1. Define the term "Angular frequency."

The rate of change of the waveform phase or the angular displacement of any element of the wave per unit time is referred to as the angular frequency

2. What Is Rotational Motion and How Does It Work?

The rotational motion of any system occurs around the axis of rotation, and the system has a moment of inertia that seeks to counteract the change in motion.

3. What's the connection between ω and F?

In general, ω is the angular speed, or the rate at which an angle changes (as in a circular motion). The number of periodic oscillations or revolutions during a particular time period is equal to 1/T, or frequency (f). Angle speed, also known as angular frequency, refers to how much angle is covered in a given amount of time.

4. What is the definition of a natural frequency unit?

Natural frequency is measured in hertz, or occurrences per second, thus if it's five hertz, it implies it happens five times each second.

5. What is the definition of angular displacement?

The angle in radians (degrees, revolutions) through which a point or line has been rotated in a specific sense about a specified axis is defined as angular displacement. It's the angle at which a body moves along a circular path. Motion ceases to be a particle when a rigid body rotates about its own axis.

6. What is the period of an oscillation?

The period of an oscillation is the time taken for one complete cycle of the oscillatory motion. It represents how long it takes for the oscillating object to return to its initial position and state of motion.

7. How is angular frequency related to period?

Angular frequency (ω) is inversely proportional to the period (T). The relationship is given by ω = 2π/T. This means that as the period increases, the angular frequency decreases, and vice versa.

8. What are the units of period and angular frequency?

The unit of period is seconds (s), as it measures time. The unit of angular frequency is radians per second (rad/s), as it measures the rate of angular displacement.

9. Why is angular frequency called "angular"?

Angular frequency is called "angular" because it represents the rate of change of angular displacement in circular motion or oscillations. It describes how quickly an object rotates or oscillates in terms of angles per unit time.

10. Can period and frequency have negative values?

No, period and frequency cannot have negative values. Both are always positive quantities because they represent the duration of a cycle (period) or the number of cycles per unit time (frequency), which cannot be negative in physical reality.

11. How does changing the amplitude of an oscillation affect its period?

For simple harmonic motion, changing the amplitude does not affect the period. The period remains constant regardless of the amplitude. This is a key characteristic of simple harmonic oscillators like ideal springs and pendulums.

12. What's the difference between frequency and angular frequency?

Frequency (f) is the number of cycles per second, measured in Hertz (Hz). Angular frequency (ω) is 2π times the frequency, measured in radians per second (rad/s). The relationship is ω = 2πf.

13. How does mass affect the period of a simple pendulum?

Interestingly, the mass of the bob does not affect the period of a simple pendulum. The period depends on the length of the pendulum and the acceleration due to gravity, but not on the mass of the bob.

14. What is the formula for the period of a simple pendulum?

The formula for the period of a simple pendulum is T = 2π√(L/g), where T is the period, L is the length of the pendulum, and g is the acceleration due to gravity.

15. How does the period of a spring-mass system change if you double the spring constant?

If you double the spring constant (k), the period of the spring-mass system will decrease by a factor of √2. The period is inversely proportional to the square root of the spring constant, as given by the formula T = 2π√(m/k), where m is the mass.

16. What is meant by natural frequency?

Natural frequency is the frequency at which a system tends to oscillate in the absence of any driving or damping force. It is determined by the system's physical properties, such as mass and stiffness.

17. How is angular frequency related to velocity in circular motion?

In uniform circular motion, the angular frequency (ω) is related to the linear velocity (v) and the radius (r) of the circle by the equation v = ωr. This means that the linear velocity is the product of angular frequency and radius.

18. Can two oscillators with different amplitudes have the same period?

Yes, two oscillators can have different amplitudes but the same period. In simple harmonic motion, the period is independent of amplitude. It depends on other factors like mass and spring constant (for a spring-mass system) or length (for a pendulum).

19. What happens to the period of a pendulum if you take it to the moon?

The period of a pendulum on the moon would be longer than on Earth. Since the moon's gravity is about 1/6 of Earth's, and the period is proportional to the inverse square root of gravity (T ∝ 1/√g), the period would increase by a factor of about √6, or about 2.45 times.

20. How does angular frequency relate to energy in an oscillating system?

The energy in an oscillating system is proportional to the square of the angular frequency. For example, in a spring-mass system, the total energy is E = ½kA², where k = mω². This shows that systems oscillating at higher angular frequencies have more energy for the same amplitude.

21. What is the phase of an oscillation?

The phase of an oscillation describes the position and direction of motion of the oscillating object at a particular instant of time. It's often expressed as an angle, with one complete oscillation corresponding to 2π radians or 360 degrees.

22. How does damping affect the period of an oscillation?

In lightly damped systems, damping slightly increases the period of oscillation compared to an undamped system. However, this effect is often negligible in practice. In heavily damped systems, the motion may not be periodic at all.

23. Can the period of an oscillation change over time?

Yes, the period of an oscillation can change over time in certain situations. For example, if the properties of the system change (like a spring weakening), or if there are external influences like changing temperature or pressure, the period may vary.

24. What is meant by the term "isochronous oscillations"?

Isochronous oscillations are oscillations that have the same period regardless of their amplitude. Simple harmonic motion, like that of an ideal pendulum for small amplitudes, is isochronous.

25. How is angular frequency used in wave equations?

In wave equations, angular frequency (ω) is used to describe how quickly the wave oscillates in time. It appears in the general form of a wave equation as y = A sin(kx - ωt), where k is the wave number and t is time.

26. What's the relationship between period and wavelength for a wave?

For a wave, the period (T) and wavelength (λ) are related by the wave speed (v): v = λ/T. This means that for a given wave speed, a longer wavelength corresponds to a longer period, and vice versa.

27. How does the concept of angular frequency apply to quantum mechanics?

In quantum mechanics, angular frequency is related to the energy of a particle or photon by the equation E = ℏω, where E is energy, ℏ is the reduced Planck constant, and ω is the angular frequency. This relationship is fundamental to wave-particle duality.

28. What is meant by resonant frequency, and how is it related to natural frequency?

Resonant frequency is the frequency at which a system responds with maximum amplitude when subjected to an external periodic force. It's often very close to, or the same as, the natural frequency of the system, which is the frequency at which the system oscillates freely without external forces.

29. How does gravity affect the angular frequency of a simple pendulum?

The angular frequency of a simple pendulum is given by ω = √(g/L), where g is the acceleration due to gravity and L is the length of the pendulum. This means that increasing gravity increases the angular frequency, while increasing the length decreases it.

30. Can you have oscillations with zero angular frequency?

Zero angular frequency would correspond to an infinitely long period, which means no oscillation at all. In practice, extremely low frequencies approach this limit, such as in geological processes that occur over millions of years.

31. How is angular frequency used in AC circuits?

In AC (alternating current) circuits, angular frequency describes how quickly the voltage or current oscillates. It's used in equations for impedance, reactance, and in describing the behavior of capacitors and inductors in these circuits.

32. What's the difference between angular frequency and angular velocity?

While both are measured in radians per second, angular frequency (ω) describes oscillations or waves, while angular velocity describes the rotation rate of an object. In circular motion, they have the same numerical value, but they represent different concepts.

33. How does temperature affect the period of a pendulum clock?

Temperature changes can affect the period of a pendulum clock by changing the length of the pendulum. As temperature increases, the pendulum rod typically expands, increasing its length and thus increasing the period. This is why precision clocks often use temperature compensation mechanisms.

34. What is the significance of the 2π in the formula for angular frequency?

The 2π in ω = 2πf represents one complete rotation in radians. It converts from cycles per second (frequency) to radians per second (angular frequency), essentially scaling the frequency to match the natural units of trigonometric functions used to describe oscillations.

35. How does air resistance affect the period of a real pendulum?

Air resistance introduces damping to a pendulum's motion, which slightly increases its period compared to an ideal pendulum. However, this effect is usually small for typical pendulums and is often negligible unless the pendulum is very light or moving very fast.

36. What is meant by the term "harmonic oscillator"?

A harmonic oscillator is a system that, when displaced from its equilibrium position, experiences a restoring force proportional to the displacement. Simple examples include an ideal spring-mass system or a simple pendulum for small amplitudes. These systems exhibit simple harmonic motion.

37. How does the period of a spring-mass system change if you double the mass?

If you double the mass in a spring-mass system, the period will increase by a factor of √2. The period is proportional to the square root of the mass, as given by the formula T = 2π√(m/k), where m is the mass and k is the spring constant.

38. What is the physical meaning of angular frequency in terms of motion?

Angular frequency represents how quickly an oscillating or rotating system completes one cycle. Physically, it's the rate of change of angular displacement. In simple harmonic motion, it determines how rapidly the system moves through its cycle of positions and velocities.

39. How are period and angular frequency related to the restoring force in simple harmonic motion?

In simple harmonic motion, the restoring force is proportional to displacement (F = -kx). The angular frequency is related to this force constant by ω = √(k/m), where m is mass. Since T = 2π/ω, we see that a stronger restoring force (larger k) leads to a higher angular frequency and shorter period.

40. Can you have oscillations with infinite angular frequency?

Infinite angular frequency would correspond to a zero period, which is physically impossible as it would require infinite energy and violate the principles of relativity. In practice, extremely high frequencies approach this limit, such as in some quantum mechanical processes.

41. How does the concept of period apply to non-periodic motion?

For non-periodic motion, the concept of period doesn't strictly apply since the motion doesn't repeat regularly. However, we can sometimes identify characteristic timescales or quasi-periods in complex, non-periodic systems, which play a similar role in describing the system's temporal behavior.

42. What is the relationship between angular frequency and the speed of a wave?

For a wave, the angular frequency (ω), wave number (k), and wave speed (v) are related by the equation ω = kv. This means that for a given wavelength (which determines k), a higher angular frequency corresponds to a faster wave speed.

43. How does the period of a torsional pendulum depend on its moment of inertia?

The period of a torsional pendulum is directly proportional to the square root of its moment of inertia. The formula is T = 2π√(I/κ), where I is the moment of inertia and κ is the torsion constant of the spring. Increasing the moment of inertia increases the period.

44. What is meant by the term "frequency domain"?

The frequency domain is a way of analyzing and representing signals or systems in terms of frequency rather than time. It's particularly useful for understanding oscillatory systems, where the angular frequency plays a key role in describing the system's behavior.

45. How does the period of a planet's orbit relate to its distance from the sun?

According to Kepler's Third Law, the square of a planet's orbital period is proportional to the cube of its semi-major axis (average distance from the sun). This means that planets farther from the sun have longer periods, with the relationship T² ∝ r³.

46. What is the difference between period and cycle in oscillations?

While often used interchangeably, "period" specifically refers to the time taken for one complete oscillation, while "cycle" refers to the complete sequence of states or configurations that an oscillating system goes through before repeating. The period measures the duration of one cycle.

47. How does angular frequency relate to the Doppler effect?

The Doppler effect causes a change in the observed frequency (and thus angular frequency) of a wave when there is relative motion between the source and observer. The change in angular frequency is proportional to the relative velocity between source and observer.

48. What is meant by the term "natural angular frequency"?

The natural angular frequency is the angular frequency at which a system oscillates when it is disturbed from equilibrium and allowed to vibrate freely without external forces or damping. It's determined by the system's physical properties, like mass and stiffness.

49. How does the period of a pendulum change with altitude?

As altitude increases, the acceleration due to gravity decreases slightly, which causes the period of a pendulum to increase. The relationship is T = 2π√(L/g), so as g decreases, T increases, albeit by a very small amount for typical altitude changes on Earth.

50. What is the relationship between angular frequency and the time constant in exponential decay?

In exponential decay, the time constant τ (tau) is the time it takes for the quantity to decrease to 1/e of its initial value. It's related to angular frequency by τ = 1/ω. A higher angular frequency corresponds to a shorter time constant and faster decay.

51. How does the concept of period apply to chaotic systems?

Chaotic systems don't have a well-defined period in the traditional sense, as their behavior is not regularly repeating. However, they may exhibit quasi-periodic behavior or have characteristic timescales that can be analyzed using techniques from nonlinear dynamics.

52. What is the significance of the period in analyzing standing waves?

In standing waves, the period is crucial for determining the nodes and antinodes. Half a period corresponds to the distance between adjacent nodes or adjacent antinodes. Understanding the period helps in analyzing the wave's spatial structure and its resonant frequencies.

53. How does angular frequency relate to the de Broglie wavelength in quantum mechanics?

In quantum mechanics, the de Broglie wavelength λ of a particle is related to its momentum p by λ = h/p, where h is Planck's constant. Since E = hf = ℏω (where ℏ is the reduced Planck constant), and E = p²/2m for a non-relativistic particle, we can relate ω to λ through these equations.

54. What is meant by the Q factor, and how does it relate to period and frequency?

The Q factor (quality factor) is a dimensionless parameter that describes how under-damped an oscillator or resonator is. It's defined as Q = ω₀τ/2, where ω₀ is the resonant frequency and τ is the time constant of the decay of the oscillations. A higher Q indicates a lower rate of energy loss relative to the stored energy of the system.

55. How does the concept of period apply to coupled oscillators?

In coupled oscillators, there can be multiple characteristic periods corresponding to different modes of oscillation. The system may exhibit beats or other complex periodic behaviors that arise from the interaction of these modes, each with its own period and frequency.