PhysicsICSE Class 10 · Vibrations and Resonance

Natural and Damped Vibrations: Natural Frequency and Falling Amplitude (ICSE Class 10)

Updated 15 October 2026 · Fact-checked

Natural (free) vibrations are the vibrations of a body disturbed and left to itself, at its own natural frequency with constant amplitude in the ideal case. In practice friction and air resistance take away energy, so the amplitude decreases steadily while the frequency stays the same: these are damped vibrations.

Understand Natural and Damped Vibrations

Natural (free) vibrations. Pull a pendulum aside and let go, pluck a stretched string, or strike a tuning fork, then leave it alone. It vibrates at a frequency that belongs to the body itself: its natural frequency. If no resistance acted, it would go on vibrating for ever with the same frequency and the same amplitude. These ideal vibrations are natural or free vibrations.

What decides the natural frequency. It depends on the body's shape, size, material and how it is held:

Stretched string: higher natural frequency if it is shorter, tighter (greater tension) or thinner (less mass per unit length).
Air column: a shorter air column has a higher natural frequency.
Simple pendulum: a shorter pendulum has a higher natural frequency (it swings faster). The mass of the bob does not change it.
Tuning fork: thicker, shorter prongs give a higher frequency; each fork is made for one natural frequency.

The natural frequency does not depend on how hard the body is struck; a harder blow changes only the amplitude.

Frequency and time period. If $n$ vibrations take time $t$ :

$f = \frac{n}{t}, T = \frac{t}{n} = \frac{1}{f} .$

Damped vibrations. In reality a vibrating body pushes against the air (or another medium) and rubs at its supports. These resistive (frictional) forces take energy away from the body, mostly turning it into heat and sound. As a result:

the amplitude decreases steadily with time until the body stops;
the frequency remains (very nearly) the natural frequency.

Vibrations whose amplitude keeps decreasing because of resistance are damped vibrations.

How quickly do they die out? The damping is greater when:

the medium is denser or more viscous: a pendulum swinging in water or oil stops far sooner than one in air;
the body has a larger surface area pushing against the medium: a pendulum with a cardboard sail attached stops sooner.

In a vacuum, with frictionless supports, there would be no damping: ideal natural vibrations.

Damping as a nuisance and as a help. A struck bell or tuning fork 'dies away' because of damping. Sometimes we want damping: shock absorbers in vehicles damp the bouncing of the springs so that a ride over a bump does not keep the car oscillating; the needle of a measuring instrument is damped so that it settles quickly; tall buildings and bridges have dampers to reduce swaying.

Comparing the three kinds.

Type	External periodic force?	Frequency	Amplitude
Natural (free)	No	Natural frequency	Constant (ideal)
Damped	No	Natural frequency (practically)	Decreases with time
Forced	Yes	Frequency of the applied force	Small, unless at resonance

A damped vibration is still a free vibration: no periodic force keeps it going. That is how it differs from a forced vibration.

Natural and Damped Vibrations: formulas and key facts

Natural frequency from a count: $f = \frac{n}{t}$; n complete vibrations in time t.
Time period: $T = \frac{t}{n} = \frac{1}{f}$; Unchanged as the vibration is damped.
Wavelength of the sound: $v = f λ$; The pitch of a damped fork does not change; only loudness falls.
What raises the natural frequency: String shorter, tighter or thinner
Air column shorter
Pendulum shorter
Tuning fork shorter, thicker prongs; The strength of the blow changes only the amplitude.

String	shorter, tighter or thinner
Air column	shorter
Pendulum	shorter
Tuning fork	shorter, thicker prongs

How to solve Natural and Damped Vibrations questions

Questions ask you to define, classify, explain the fall in amplitude, or calculate a frequency or period.

1Check for an external periodic force: none means natural (ideal) or damped (real) vibration.
2If the amplitude falls, say why: resistance of the medium and friction take away energy.
3State clearly that the frequency stays the natural frequency during damping.
4For a count, use $f = \frac{n}{t}$ and $T = \frac{1}{f}$ ; give T in s (or ms) and f in Hz.
5For a change in natural frequency, use the factors: length, tension and thickness for a string; length for an air column or pendulum.

Natural and Damped Vibrations in real life

Each concept, met in an everyday situation. Read the situation, try the problem, then check the solution. The one-line hook is what to remember.

Shock absorbers

Suppose a car without working shock absorbers goes over a speed breaker. It keeps bouncing up and down on its springs for several seconds afterwards.

What type of vibration is the bouncing, and what do good shock absorbers do?

Try it first, then show the solutionSolution

After the bump no periodic force acts, so the car body vibrates freely at the natural frequency of its springs.
Shock absorbers apply strong resistance, removing energy quickly so the amplitude falls to zero within a bounce or two.

Answer: Free vibrations of the springs; shock absorbers heavily damp them.

Memory hook: Damping kills the bounce, not the ride.

A swing slowing down

Imagine a child stops pumping on a park swing and lets it run on. Each swing is a little smaller than the last, yet each one still seems to take the same time.

Explain both observations.

Try it first, then show the solutionSolution

Air resistance and friction at the hooks take away energy, so the amplitude decreases: damped vibrations.
The time period depends on the length of the swing, not on the amplitude, so each swing still takes about the same time.

Answer: Damping lowers the amplitude; the natural frequency (and period) stays the same.

Memory hook: Smaller swings, same rhythm.

Tricks to remember Natural and Damped Vibrations

Short memory aids. Say each one aloud once, then use it on a question the same day.

Damping dims, never detunes
A damped note gets softer (amplitude falls) but never changes pitch (frequency stays).
Use it when: Frequency and amplitude in damped vibrations.
Short, Tight, Thin
A Short, Tight, Thin string sings high.
Use it when: Factors raising the natural frequency of a string.
Thick medium, quick stop
Water stops a pendulum faster than air; oil faster still.
Use it when: Effect of the medium on damping.
Free, Fading, Forced
Natural vibrations are Free, damped ones Fade, forced ones are pushed by a driver.
Use it when: Telling the three apart.

Natural and Damped Vibrations: solved examples

Ordered from board level to HOTS. Try each one before you open the solution.

Example 1: Damping in oil · 1 mark · MCQBoard level

A pendulum swinging in a jar of oil comes to rest much sooner than an identical one swinging in air, because:
(a) the oil lowers its natural frequency to zero
(b) the oil offers greater resistance, removing its energy faster
(c) the oil increases the mass of the bob
(d) the oil applies a periodic force to it

Try it first, then show the solutionSolution

A denser, more viscous medium resists the motion more, so energy is removed faster and the vibrations are more heavily damped.

Answer: (b) the oil offers greater resistance, removing its energy faster

Example 2: Natural vibrations and natural frequency · 3 marks · Short answerBoard level

(a) What are natural vibrations? (b) What is meant by the natural frequency of a body? (c) Name two factors on which the natural frequency of a stretched string depends.

Try it first, then show the solutionSolution

(a) Vibrations of a body that is disturbed and then left to itself, with no external periodic force acting. (1)
(b) The frequency at which the body vibrates when it is free (it depends on the body itself). (1)
(c) Any two: length, tension, thickness (mass per unit length), material. (1)

Answer: (a) Free vibrations with no external periodic force (b) The body's own frequency of free vibration (c) Length and tension (any two)

Example 3: Period of a pendulum · 3 marks · Short answerHarder board level

A simple pendulum makes 40 complete oscillations in 36 s. Find (a) its time period, (b) its natural frequency. (c) After a minute its swings are visibly smaller. Has its time period changed?

Try it first, then show the solutionSolution

(a) $T = \frac{36}{40} = 0.9$ s. (1)
(b) $f = \frac{1}{0.9} \approx 1.11$ Hz. (1)
(c) No: the vibrations are damped, so only the amplitude falls; the period stays 0.9 s. (1)

Answer: (a) 0.9 s (b) About 1.11 Hz (c) No

Example 4: A tuning fork's natural frequency · 3 marks · Short answerHarder board level

A data logger records 1280 complete vibrations of one prong of a tuning fork in 4 s. Find (a) the natural frequency of the fork, (b) its time period in ms. (c) If the fork is struck harder, which of these quantities changes?

Try it first, then show the solutionSolution

(a) $f = \frac{1280}{4} = 320$ Hz. (1)
(b) $T = \frac{1}{320} = 3.125 \times 1 0^{- 3}$ s $= 3.125$ ms. (1)
(c) Neither: a harder blow increases only the amplitude (loudness). (1)

Answer: (a) 320 Hz (b) 3.125 ms (c) Neither; only the amplitude increases

Example 5: Pendulum in air and in water · 3 marks · Short answerHarder board level

Two identical pendulums are set swinging with the same amplitude, one in air and one with its bob in water. (a) Which stops first, and why? (b) Name the type of vibration in each case. (c) What would happen in a vacuum with frictionless support?

Try it first, then show the solutionSolution

(a) The one in water: water is much denser than air and offers more resistance, so energy is removed faster. (1)
(b) Both are damped vibrations (more heavily damped in water). (1)
(c) No energy would be lost; it would go on with constant amplitude: ideal natural vibrations. (1)

Answer: (a) The pendulum in water (b) Damped in both (c) Constant amplitude, ideal natural vibration

Example 6: A fading note · 3 marks · Short answerHOTS / competency

A struck tuning fork of frequency 320 Hz starts with a prong amplitude of 8 mm, and the amplitude halves every 10 s. (a) Find the amplitude after 30 s. (b) What is the frequency of the note after 30 s? (c) How would a listener notice the change?

Try it first, then show the solutionSolution

(a) After 10 s: 4 mm; after 20 s: 2 mm; after 30 s: 1 mm. (1)
(b) Still 320 Hz: damping does not change the frequency. (1)
(c) The note becomes softer (less loud) but keeps the same pitch. (1)

Answer: (a) 1 mm (b) 320 Hz (c) Softer, same pitch

Example 7: Case study: the guitarist's strings · 4 marks · Case-basedHOTS / competency

A guitarist plucks two strings of the same length and material: string A is thin and tight, string B thick and slack. Later she presses string A against a fret, shortening its vibrating length.
(a) Which string has the higher natural frequency at first? Give two reasons.
(b) What happens to A's natural frequency when it is shortened?
(c) Why does the sound of each string die away after plucking?
(d) Does the pitch of a plucked string fall as its sound dies away? Explain.

Try it first, then show the solutionSolution

(a) A: it is thinner (less mass per unit length) and under greater tension; both raise the natural frequency. (1)
(b) It increases: a shorter string has a higher natural frequency. (1)
(c) Air resistance and losses at the supports and guitar body take energy away, so the vibrations are damped. (1)
(d) No: damping reduces the amplitude (loudness), not the frequency (pitch). (1)

Answer: (a) A, thinner and tighter (b) It increases (c) Damping (d) No, only the loudness falls

Where students lose marks in Natural and Damped Vibrations

The slips examiners see most often, and what to write instead.

Mistake: Saying the frequency decreases in damped vibrations
Damping removes energy, which shows up as a smaller amplitude; the frequency stays the natural frequency.
Wrong working
'In damped vibrations the frequency goes on decreasing.'
Right working
'In damped vibrations the amplitude decreases; the frequency remains the natural frequency.'
Do this: Damping dims, never detunes.
Usually costs: Examiners asked teachers to stress that damped vibrations only lose amplitude (2024 Q1(vi) comments).
Mistake: Calling damped vibrations forced
No periodic force maintains a damped vibration; it is a free vibration losing energy.
Wrong working
'A swing slowing down is a forced vibration because air acts on it.'
Right working
'It is a damped (free) vibration; air resistance is not a periodic driving force.'
Do this: Forced needs a periodic driver.
Usually costs: The 1-mark classification.
Mistake: Thinking the bob's mass or the amplitude sets a pendulum's frequency
For a simple pendulum the natural frequency depends on its length (and g), not on mass or amplitude.
Wrong working
'A heavier bob swings faster.'
Right working
'Only a change of length changes the pendulum's natural frequency.'
Do this: Remember: shorter pendulum, higher frequency.
Usually costs: MCQ marks.
Mistake: Confusing time period with total time
The period is the time for one vibration: total time ÷ number of vibrations.
Wrong working
Writing $T = 36$ s for 40 oscillations in 36 s.
Right working
$T = \frac{36}{40} = 0.9$ s.
Do this: Always divide by the number of vibrations.
Usually costs: The calculation mark.

Exam tips

Define each type in one line, naming the frequency and what happens to the amplitude.
For 'why does a pendulum stop', write 'air resistance and friction at the support remove energy; the vibrations are damped'.
Give a practical use of damping (shock absorbers) when asked for an application.
Keep the three kinds of vibration in a table in your notes; examiners often ask for differences.

Practice questions on Natural and Damped Vibrations

Write each answer on paper first. Open the hint only if you are stuck, and the answer only when you have finished.

Q1 · 1 mark · MCQBoard level
The natural frequency of a simple pendulum can be increased by:
(a) using a heavier bob
(b) increasing its amplitude
(c) shortening its length
(d) swinging it in water
Hint
Which one changes the body itself?
Answer
(c) shortening its length
Q2 · 1 mark · Very short answerBoard level
Define damped vibrations.
Hint
What happens to the amplitude?
Answer
Free vibrations whose amplitude decreases steadily with time because of resistive (frictional) forces.
Q3 · 2 marks · Short answerHarder board level
A body vibrates freely with a time period of 0.25 s. Find its natural frequency and the number of vibrations it makes in one minute (ignore damping).
Hint
$f = \frac{1}{T}$ .
Answer
$f = \frac{1}{0.25} = 4$ Hz; in 60 s: $4 \times 60 = 240$ vibrations.
Q4 · 2 marks · Short answerHarder board level
Give two differences between natural vibrations and damped vibrations.
Hint
Amplitude and energy.
Answer
(1) Natural (ideal): amplitude stays constant; damped: amplitude decreases with time. (2) Natural: no energy is lost; damped: energy is lost to the surroundings through resistance and friction. (Frequency is the natural frequency in both.)
Q5 · 2 marks · Short answerBoard level
Give one example each of a useful and an unwanted damping.
Hint
Vehicles; musical sounds.
Answer
Useful: shock absorbers damping a car's bounce (or dampers in tall buildings). Unwanted: the note of a bell or tuning fork dying away (or a swing slowing down).
Q6 · 2 marks · Short answerHarder board level
The 320 Hz fork of a laboratory gives a note in air where sound travels at 340 m/s. Find the wavelength. Will the wavelength change as the note dies away?
Hint
$λ = \frac{v}{f}$ .
Answer
$λ = \frac{340}{320} = 1.0625$ m. No: frequency and speed are unchanged, so the wavelength stays the same; only the amplitude falls.
Q7 · 2 marks · Short answerHarder board level
Why does a pendulum fitted with a large, light cardboard disc stop swinging sooner than the same pendulum without it?
Hint
Surface area and air resistance.
Answer
The disc's large surface area pushes against much more air, so the air resistance is greater; energy is removed faster and the vibrations are damped more quickly.
Q8 · 3 marks · Short answerHOTS / competency
A pendulum's amplitude falls to three-quarters of its value on each complete swing. Starting at 12 cm, find its amplitude after two swings. Explain why the time for each swing does not change.
Hint
Multiply by 0.75 twice.
Answer
$12 \times 0.75 \times 0.75 = 6.75$ cm. Damping reduces only the amplitude; the period depends on the length of the pendulum (its natural frequency), which is unchanged.
Q9 · 1 mark · Assertion-reasonHarder board level
Assertion (A): Striking a tuning fork harder makes its note louder but not higher in pitch.
Reason (R): The natural frequency of a body depends on the body itself, not on the strength of the blow.
(a) (A) is true and (R) is false.
(b) (A) is false and (R) is true.
(c) Both (A) and (R) are true, and (R) is the correct explanation for (A).
(d) Both (A) and (R) are true, but (R) is not the correct explanation for (A).
Hint
Amplitude vs frequency.
Answer
(c) Both are true, and (R) explains (A).

Test your recall

Answer in your head or on paper first, then open the card. Pulling an answer out of memory makes it stick far better than re-reading.

1.Frequency of a body vibrating freely?Show answer
Its natural frequency.
2.What decreases in damped vibrations?Show answer
The amplitude.
3.Where is damping greater, air or water?Show answer
Water.
4.A shorter pendulum has a … natural frequency.Show answer
Higher.
5.Device in vehicles that damps vibrations?Show answer
Shock absorber.

Natural and Damped Vibrations: frequently asked questions

What are natural vibrations?

They are the vibrations of a body that has been disturbed and left to itself. It vibrates at its own natural frequency, with constant amplitude in the ideal case of no resistance.

What are damped vibrations?

Damped vibrations are free vibrations whose amplitude keeps decreasing because air resistance and friction take away energy. Their frequency stays practically equal to the natural frequency.

On what does the natural frequency of a string depend?

On its length, tension and thickness (mass per unit length). A shorter, tighter or thinner string has a higher natural frequency.