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Natural Frequency

Key idea: Define natural and driving frequency, distinguish free from forced oscillations, and identify the steady forced-response frequency (A Level Physics).

  • GCE A-Level H2 Physics 2027
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Learning objectives

  • Distinguish free and forced oscillations, natural frequency and driving frequency.
  • Interpret resonance response curves, damping effects and practical applications.

1. Definitions (Must Know)

A. Natural frequency

The natural frequency, f₀, is the frequency at which a system oscillates when it is left to vibrate freely. It is set by properties of the system, such as mass and stiffness.

B. Free oscillation

An ideal free oscillation has no external periodic driving force and neither gains energy from nor loses energy to the environment. A real released oscillator is usually damped, so its free-motion amplitude decreases.

C. Forced oscillation (driving force)

A forced oscillation occurs when a system is acted on by a continuing external periodic force.

After any transient motion dies away, the system oscillates at the driving frequency, f_d, not automatically at its natural frequency.

D. Driving frequency

The driving frequency, f_d, is the frequency of the external periodic force. The natural frequency affects the response amplitude; the driver sets the steady-state frequency.

Free and forced oscillations comparedA free oscillator is displaced and released once, then moves at its natural frequency. A forced oscillator receives a continuing periodic input and, after transients decay, moves at the driving frequency.Free oscillationdisplace once, then releasefrequency = f₀no continuing periodic driverForced oscillationperiodicdriver, fdoscillatornatural f₀steady frequency = fdlargest response when fd is near f₀
Scroll diagram horizontally to read all labels.
Free motion reveals the system's natural frequency. In steady forced motion, the driver sets the frequency while the system properties and damping set the amplitude and phase.

2. Key Ideas (What Earns Marks)

  • Free oscillation reveals the system’s natural frequency f₀.
  • In steady forced motion, the oscillator follows f_d.
  • Changing mass, stiffness or geometry can change f₀; changing only the driver changes f_d.
  • The steady amplitude is largest when f_d is close to f₀; that response is developed in Resonance.
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3. Detailed Explanations

A. Free response and transient motion

Displacing a system and releasing it supplies the initial energy. With no continuing driver, it then oscillates freely. In the ideal syllabus model, energy remains within the oscillator; in a real system, damping removes mechanical energy and the amplitude decays.

When a periodic driver is first switched on, the motion may contain both a temporary transient response and the forced response. Damping removes the transient, leaving steady oscillation at f_d.

B. What determines the natural frequency

The natural frequency is a property of the oscillator. For example, increasing the mass of a spring–mass oscillator lowers its natural frequency, while increasing the spring stiffness raises it. The exact formula depends on the system and is not the definition.

C. Driver frequency versus response amplitude

The driver sets how often the steady motion repeats. It does not guarantee a large amplitude: amplitude also depends on how close f_d is to f₀ and on the damping.

4. Common Mistakes

  • Saying “free oscillation means no force acts” (the restoring force is essential).
  • Calling the driving frequency the natural frequency.
  • Assuming a forced oscillator always has a large amplitude.

5. Exam Tips

  • State whether the question describes a one-off release or a continuing periodic driver.
  • In steady forced motion, write: “oscillation frequency equals driving frequency”.
  • Treat f₀ as a system property and f_d as an externally chosen input.

6. Worked Examples

Modelled example 1

Identify free and forced motion

Core

Problem

A tuning fork is struck once and rings. A loudspeaker cone is supplied with a continuing sinusoidal signal. Classify each motion and identify the decisive distinction.
Study the worked solution

  1. Classify the tuning fork

    Method

    After the strike, it undergoes free motion.

    Reason

    The strike supplies initial energy but is not a continuing periodic driver.

    Working

    One-off excitation ⇒ free oscillation after release.

  2. Classify the loudspeaker

    Method

    The cone undergoes forced motion.

    Reason

    The continuing sinusoidal electrical signal supplies a periodic drive.

    Working

    Continuing periodic input ⇒ forced oscillation.

  3. State the distinction

    Method

    Free versus forced refers to continuing periodic driving, not to whether a restoring force acts.

    Reason

    Both systems require restoring effects to oscillate.

    Working

    Decisive question: is an external periodic driver maintained?

Guided practice 2

Forced oscillation frequency

About 3 min

Problem

A child pushes a swing steadily at 0.80 Hz; the swing’s natural frequency is 0.60 Hz. What is the steady oscillation frequency?

Try this before viewing the solution

Unit: Hz

Hints

Hint 1: separate frequency from amplitude
Natural frequency controls where response is largest; driving frequency controls the steady repetition rate.
View solution step by step

  1. Identify the driver

    Method

    The driving frequency is 0.80 Hz.

    Reason

    The continuing pushes repeat at this rate.

    Working

    f_d = 0.80 Hz.

  2. State the steady response

    Method

    The swing oscillates at 0.80 Hz in steady state.

    Reason

    Forced motion follows the driver after transient free motion decays.

    Working

    f_steady = f_d = 0.80 Hz.

Common misconception 3

Change natural or driving frequency

Find and correct the mistake

Learner claim

A spring–mass oscillator is motor-driven. Its mass is increased while the motor setting stays unchanged. A learner says both natural and driving frequencies must decrease. Diagnose the claim.

Try this before viewing the solution

Direct frequency change

View solution step by step

  1. Track the system change

    Method

    The natural frequency changes and, for a spring–mass oscillator, decreases.

    Reason

    Increasing inertia changes the free response of the oscillator.

    Working

    System mass changes f₀.

  2. Track the external input

    Method

    The driving frequency remains fixed.

    Reason

    The motor setting, which determines f_d, is unchanged.

    Working

    Unchanged motor ⇒ unchanged f_d.

Examiner practice 4

Find the period from the natural frequency

2 marks

Examination question

A system has natural frequency 2.5 Hz. Define this quantity in context and calculate its free-oscillation period. [2 marks]

Try this before viewing the solution

View solution step by step

  1. Define natural frequency

    1 mark

    Method

    It is the frequency at which the system oscillates when left to vibrate freely.

    Reason

    The definition ties the frequency to free rather than driven motion.

    Working

    f₀ = 2.5 Hz for the free response.

  2. Calculate period

    1 mark

    Method

    T₀ = 0.40 s.

    Reason

    Period and frequency are reciprocals.

    Working

    T₀ = 1/f₀ = 1/2.5 = 0.40 s

Challenge 5

Frequency does not determine amplitude alone

Minimal support

Independent transfer

Two oscillators are driven at the same frequency but have different natural frequencies and damping. Decide whether their steady frequencies and amplitudes must match, and justify each conclusion.

Try this before viewing the solution

Hints

Hint 1: answer frequency and amplitude separately
Ask what the driver fixes, then what controls response size.
View solution step by step

  1. Compare steady frequencies

    Method

    The steady frequencies are equal to the common driving frequency.

    Reason

    After transients decay, each forced oscillator follows its driver.

    Working

    f₁ = f₂ = f_d in steady state.

  2. Compare amplitudes

    Method

    The amplitudes need not be equal.

    Reason

    Response size depends on proximity to each natural frequency and on damping.

    Working

    Different f₀ and damping can give different amplitudes at the same f_d.

7. Mind Stretchers

Mind stretcher 1: Transient and steady responsesExtension

Immediately after a periodic driver is switched on, why might the motion not yet repeat only at the driving frequency?

Show Answer

The initial conditions can also excite a transient response associated with the oscillator’s free motion. Damping removes this transient, leaving the steady forced response at the driving frequency.

Mind stretcher 2: Natural frequency is not a maximum frequencyExtension

Explain why “natural frequency” does not mean the greatest frequency at which the system can be forced to oscillate.

Show Answer

Natural frequency describes free motion; it is not an upper limit. A periodic driver can force steady motion at frequencies above or below f₀, although the response amplitude changes with frequency.

Continue with the next resource in this course.

Course and syllabus information
Course
GCE A-Level H2 Physics
Edition
GCE A-Level H2 Physics 2027