OCR Specification focus:
‘Angular frequency relations: ω = 2π/T and ω = 2πf for oscillatory systems.’
Understanding angular frequency is essential in simple harmonic motion (SHM), as it links time-based motion to circular motion concepts and allows calculation of frequency, period, and system behaviour.
Understanding Angular Frequency
Angular frequency describes how rapidly an object oscillates or moves through a cycle. It provides a direct link between linear oscillations (back-and-forth motion) and circular motion, both of which repeat in a predictable, periodic way.
Angular Frequency (ω): The rate of change of phase in an oscillating system, measured in radians per second (rad s⁻¹).
An oscillating system such as a mass on a spring, a pendulum, or an electrical circuit moves repeatedly about an equilibrium position. Each full oscillation corresponds to a complete cycle, and angular frequency quantifies how quickly these cycles occur in terms of radians rather than full cycles per second.
In one full oscillation, the system completes 2π radians of motion, mirroring the angular displacement of one full circle.
Graph of sine (and cosine) across a single cycle from 0 to 2π on the horizontal axis. The horizontal span 0 → 2π corresponds to one complete oscillation, i.e. period T in radians. This directly motivates ω = 2π/T and, using f = 1/T, the relation ω = 2πf. (Includes the cosine curve as extra detail beyond the syllabus focus on ω–T–f.) Source.
Hence, angular frequency forms the essential bridge between time-based motion (seconds per cycle) and angular motion (radians per second).
Relationship Between Period, Frequency, and Angular Frequency
The period (T) is the time for one complete oscillation, and the frequency (f) is the number of oscillations per second. These two quantities are inversely related. Angular frequency combines these fundamental time measures into a rotational equivalent.
EQUATION
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Angular Frequency in Terms of Period
ω = 2π / T
ω = Angular frequency (rad s⁻¹)
T = Period of oscillation (s)
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This equation shows that angular frequency is inversely proportional to the period. A shorter period means a faster oscillation and, therefore, a higher angular frequency.
In addition to period, angular frequency is often related directly to frequency:
EQUATION
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Angular Frequency in Terms of Frequency
ω = 2πf
ω = Angular frequency (rad s⁻¹)
f = Frequency (Hz or s⁻¹)
—-----------------------------------------------------------------
These two equations express the same physical concept in different forms, depending on whether you measure oscillations by time per cycle (T) or cycles per second (f).
Between these relations, students can easily move between different descriptions of oscillatory motion. The angular form, expressed in radians, is particularly useful in equations of simple harmonic motion, where angular measures naturally describe periodic variation.
Connecting Angular Frequency to Circular Motion
The use of 2π in both relations reflects the full angular measure of one complete revolution. This connection is not coincidental—SHM can be viewed as a projection of uniform circular motion onto one axis. Imagine a point moving around a circle at constant speed. The horizontal or vertical projection of that point undergoes SHM, oscillating between maximum positive and negative displacements.
In this analogy:
A point P undergoes uniform circular motion with angular velocity ω; its projection on a fixed axis performs simple harmonic motion. One full revolution corresponds to 2π radians, i.e. one complete oscillation with period T. This geometric link is the basis for ω = 2π/T = 2πf. (The figure also indicates velocity vectors; these are helpful context but not required by the syllabus.) Source.
The circular motion completes one revolution in time T.
The angular displacement of the point increases by 2π radians in each cycle.
Therefore, angular frequency, ω = 2π/T, expresses the rate of change of that angular displacement.
Through this comparison, angular frequency becomes not only a measure of oscillation rate but also a direct indicator of how fast the phase of motion changes in radians per second.
Physical Significance of Angular Frequency
Angular frequency is crucial because it appears in all core equations describing SHM. It determines:
How quickly displacement, velocity, and acceleration vary with time.
The maximum velocity (v_max = ωA) and maximum acceleration (a_max = ω²A) of the oscillator.
The shape and timing of oscillatory graphs, influencing how steep or rapid motion appears.
A higher angular frequency means faster oscillations, a shorter period, and more rapid changes in velocity and acceleration.
Comparing Linear and Angular Quantities
To visualise angular frequency’s role, it helps to distinguish between linear and angular descriptions:
Linear frequency (f) counts complete cycles per second (Hz).
Angular frequency (ω) measures how fast the phase angle increases, in radians per second.
Both describe the same underlying motion, but in different units. Angular frequency is often more convenient in mathematical descriptions because SHM equations use trigonometric functions dependent on angular arguments (ωt).
For instance, displacement in SHM is typically expressed as x = A cos(ωt + φ) or x = A sin(ωt), where ωt represents angular displacement in radians. The use of ω ensures consistent measurement within trigonometric expressions, as sine and cosine functions inherently depend on angular inputs.
Determining Angular Frequency Experimentally
Angular frequency can be determined by:
Measuring the time for one oscillation (T) using a stopwatch or data logger, then applying ω = 2π/T.
Counting the number of oscillations per second (f) using a sensor or oscilloscope, then applying ω = 2πf.
In both methods, care must be taken to average several oscillations for improved accuracy, since small timing errors can significantly affect ω. Modern sensors, photogates, and motion trackers provide highly precise measurements, especially useful for systems with short periods or high frequencies.
Practical examples include:
A mass–spring system, where ω = √(k/m), showing how stiffness (k) and mass (m) influence oscillation rate.
A simple pendulum, where ω = √(g/l) for small angles, linking gravitational acceleration (g) and length (l).
Though these expressions stem from specific physical systems, both are consistent with ω = 2πf, confirming angular frequency’s universal nature.
The Role of Angular Frequency in Phase Relationships
Because SHM can be described by angular motion, angular frequency also governs phase—the measure of how far through its cycle an oscillator is at any given time. The phase changes uniformly at a rate equal to ω, meaning that a system with higher ω completes its cycles faster and returns to equilibrium more often.
Phase relationships between oscillating quantities, such as displacement, velocity, and acceleration, all depend on ω. When comparing two oscillators, differences in ω indicate differing rates of oscillation, which can cause phase differences over time, a critical factor in phenomena such as interference and resonance.
FAQ
Radians per second describe how quickly the phase angle of the oscillation changes, rather than how many full cycles occur each second.
While frequency (Hz) counts complete oscillations, angular frequency (rad s⁻¹) measures progress within a single cycle. This makes it especially useful when describing motion mathematically, since trigonometric functions such as sine and cosine depend on angular values in radians.
Essentially, 1 Hz equals 2π rad s⁻¹, showing that angular frequency provides a more direct link to the oscillating object’s position at any instant.
Angular frequency controls how quickly all oscillating quantities vary with time.
In x = A cos(ωt), ω determines how fast displacement changes.
In v = -ωA sin(ωt), it sets the steepness of the velocity curve.
In a = -ω²A cos(ωt), ω² shows that acceleration increases rapidly with higher ω.
Thus, ω acts as a “speed multiplier” for oscillations, affecting the timing and sharpness of each motion phase.
If the graph shows one or more full oscillations:
Measure the period (T) — the time between successive peaks or identical points.
Apply ω = 2π/T to calculate angular frequency.
Alternatively, for non-complete cycles, the slope of the phase curve (the rate of change of angle in radians) can indicate ω. Graphing the motion helps visualise how quickly the system progresses through its cycle.
Although ω always represents 2π radians per cycle, its value depends on the system’s physical properties:
Mass–spring system: ω = √(k/m), where k is the spring constant and m the mass.
Simple pendulum (small angles): ω = √(g/l), where g is gravity and l is length.
These relationships show that stiffness or gravity influences how rapidly an oscillator moves through its cycle, while angular frequency itself remains the same type of measure across systems.
Using ω simplifies equations and aligns with angular arguments in trigonometric functions.
Trigonometric functions like sine and cosine require radians, not cycles, as their input.
Expressing motion as ωt ensures units are consistent with angular phase.
Differentiating or integrating with respect to time becomes simpler, since derivatives of sine and cosine naturally introduce ω and ω² factors.
This makes ω more suitable for advanced analysis, especially in mechanics, wave theory, and alternating current circuits.
Practice Questions
Question 1 (2 marks)
A pendulum swings with a period of 2.5 s.
(a) Calculate the angular frequency of the pendulum.
(b) State the unit of angular frequency.
Mark Scheme:
(a) Uses correct relation: ω = 2π / T → (1 mark)
Substitutes T = 2.5 s to give ω = 2.51 rad s⁻¹ (accept 2.5 rad s⁻¹ with correct working) → (1 mark)
(b) Unit: rad s⁻¹ → (1 mark if included, otherwise total 2 marks maximum)
Total: 2 marks
Question 2 (5 marks)
A mass–spring system oscillates with a frequency of 1.6 Hz.
(a) Determine the angular frequency of the oscillation.
(b) Explain, with reference to angular frequency, what happens to the oscillations if the spring constant is increased while the mass remains the same.
(c) Describe how angular frequency relates to the period of oscillation and how these quantities determine how rapidly the system moves through its cycle.
Mark Scheme
(a) Correct use of relation ω = 2πf → (1 mark)
Substitutes f = 1.6 Hz → ω = 10.1 rad s⁻¹ (allow 10 rad s⁻¹) → (1 mark)
(b) Recognises that ω = √(k/m) for a mass–spring system → (1 mark)
States that increasing k increases ω → (1 mark)
Explains that this results in faster oscillations or a shorter period → (1 mark)
(c) States that ω = 2π/T, showing ω and T are inversely related → (1 mark)
Explains that a higher ω means the system completes cycles more quickly (greater rate of change of phase) → (1 mark)
