Vibrations and Resonance ICSE Class 10: Natural, Damped and Forced Vibrations

What Vibrations and Resonance is about

Weightage: CISCE publishes no chapter or unit weightage for ICSE Physics. Vibrations and Resonance is in the 3. Sound (ii) part of the 2027 syllabus and can be asked in both Section A and Section B of the 80-mark paper.Source: CISCE ICSE Physics (Science Paper 1) syllabus, examination year 2027

Key concepts

Natural (free) vibrations

When a body is disturbed and left to itself, it vibrates at its own natural frequency, with constant frequency and (ideally, without any resistance) constant amplitude. These are natural or free vibrations.

The natural frequency depends on the body's shape, size, material and, for a string, its length, tension and thickness:

• a shorter, tighter or thinner string has a higher natural frequency;
• a shorter air column has a higher natural frequency;
• a shorter pendulum has a higher frequency.

Examples: a plucked string, a struck tuning fork, a simple pendulum (in vacuum, ideally).

Worked example

A simple pendulum makes 20 oscillations in 40 s. Find its time period and natural frequency.

Try it first, then show the solutionSolution

1. $T=\frac{40}{20}=2$ s.
2. $f=\frac{1}{T}=0.5$ Hz.

Answer: T = 2 s; f = 0.5 Hz

Damped vibrations

In practice, air resistance and friction take energy away, so the amplitude decreases with time until the vibration stops. These are damped vibrations. The frequency stays (practically) the natural frequency; only the amplitude falls.

The damping is greater in a denser medium (a pendulum in water stops much sooner than in air) and for bodies with larger surface area.

Examples: a swinging pendulum slowly coming to rest; the sound of a struck tuning fork dying away; a car's shock absorbers deliberately damping its bounce.

Worked example

Why does the sound of a struck tuning fork die away even if nobody touches it?

Try it first, then show the solutionSolution

1. The prongs do work against air resistance and internal friction.
2. Their energy is lost, so the amplitude (and loudness) decreases: damped vibrations.

Answer: Energy is lost to air resistance, so the amplitude decreases.

Forced vibrations

When a periodic external force acts on a body, the body vibrates with the frequency of the applied force, not its own natural frequency. These are forced vibrations. Their amplitude is usually small (unless the frequencies match), and they stop when the force is removed.

Examples: a tabletop vibrating when the stem of a vibrating tuning fork is pressed on it (the sound becomes louder because the large table surface sets more air vibrating); the sound board of a guitar or sitar; the diaphragm of a loudspeaker.

Worked example

The stem of a vibrating tuning fork of 256 Hz is pressed on a table. At what frequency does the tabletop vibrate, and why is the sound louder?

Try it first, then show the solutionSolution

1. The table is forced by the fork, so it vibrates at 256 Hz.
2. The large table surface sets a much larger mass of air into vibration, so the sound is louder (forced vibrations).

Answer: 256 Hz; larger vibrating area

Resonance: the special case

Resonance occurs when the frequency of the applied periodic force equals the natural frequency of the body. The body then vibrates with a very large amplitude (a loud sound for an air column or string).

Key words for a full answer: *forced vibration*, *frequency of the applied force*, *matches/equals the natural frequency*, *large amplitude*.

Resonance is a special case of forced vibrations: all resonance is forced vibration, but most forced vibrations are not resonance.

Worked example

A vibrating tuning fork is held over a tube whose air column can be lengthened by a sliding piston. At one length a loud sound is heard. Name and explain the effect.

Try it first, then show the solutionSolution

1. The fork forces the air column to vibrate.
2. At this length, the natural frequency of the air column equals the frequency of the fork.
3. So the air column vibrates with maximum amplitude: resonance, a loud sound.

Answer: Resonance: natural frequency of the air column = frequency of the fork

Demonstrations of resonance

1. Pendulums on a common string: several pendulums of different lengths hang from a stretched string. One (the driver) is set swinging. The pendulum with the same length as the driver swings with a large amplitude (resonance); the others swing only a little (forced vibrations).
2. Air column: a vibrating tuning fork over a tube partly filled with water. As the water level (and so the air column length) changes, at one particular length the sound becomes loud. A fork of higher frequency needs a shorter air column.
3. String on a sound box: a paper rider on a stretched wire jumps off when a tuning fork placed on the box has the same frequency as the wire's natural frequency.

Worked example

In the common-string experiment, pendulum A is set swinging. B has the same length as A; C and D are shorter and longer. Which swings with the largest amplitude, and why?

Try it first, then show the solutionSolution

1. B has the same length, hence the same natural frequency as A.
2. So B is in resonance with A and swings with the largest amplitude; C and D perform forced vibrations of small amplitude.

Answer: B, by resonance

Uses and dangers of resonance

Uses:

• Musical instruments: air columns (flute) and sound boxes (guitar) resonate to make notes loud.
• Tuning a radio: the circuit is adjusted until its natural frequency matches the frequency of the wanted station (electrical resonance).
• Finding an unknown frequency by matching it with a known one.

Dangers:

• Soldiers break step while crossing a bridge so that their marching frequency cannot match the bridge's natural frequency.
• A singer's sustained note can shatter a glass if it matches the glass's natural frequency.
• Buildings and bridges are designed (with dampers) so that wind or earthquakes do not set up resonance.

Worked example

Why are soldiers asked to break step while crossing a suspension bridge?

Try it first, then show the solutionSolution

1. Marching in step applies a periodic force to the bridge.
2. If its frequency equals the bridge's natural frequency, resonance gives a very large amplitude that could damage the bridge.
3. Breaking step removes the regular periodic force.

Answer: To avoid resonance between marching and the bridge

Physics in real life: Vibrations and Resonance

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.

Topics in this chapter

• Resonance: Meaning, Demonstrations and Uses
• Forced Vibrations: Meaning and Examples
• Natural and Damped Vibrations

Vibrations and Resonance formula sheet

Frequency and period

$f=\frac{1}{T}$

Natural frequency of a vibrating body.

Wave equation

$v=f\lambda$

Wavelength of sound from a tuning fork.

Condition for resonance

$f_{\text{applied}}=f_{\text{natural}}$

Gives maximum amplitude.

Air column (qualitative)

$\text{shorter air column}\Rightarrow \text{higher natural frequency}$

Not the water column.

Vibrations and Resonance: diagrams to practise

Practise each one on paper: draw it from the checklist, then tick off every label.

How the board asks Vibrations and Resonance

The question patterns to expect, and the method that works for each.

Tricks to remember Vibrations and Resonance

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

• N-D-F-R

  Natural (own frequency, free), Damped (amplitude dies), Forced (someone else's frequency), Resonance (forced + equal frequencies = huge amplitude).

  Use it when: Classifying vibrations.

• Match and magnify

  Resonance: frequencies match, amplitude is magnified.

  Use it when: Writing the full explanation.

• Air, not water

  In a bottle or burette, the AIR column resonates; less air, higher pitch.

  Use it when: Air-column questions.

• Break step, break resonance

  Soldiers break step to avoid resonance.

  Use it when: Applications.

Vibrations and Resonance: board-level solved questions

Ordered from board level to HOTS. Cover the solution, write your own, then compare step by step.

Case-study practice (Section B style)

Read the situation, then answer every part. In the board paper this is a 10-mark question split into 3 + 3 + 4 marks.

Where the marks are

How a typical answer is split into step marks. Show every step: a right final answer alone rarely earns full marks.

Where students lose marks in Vibrations and Resonance

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

• ✗Mistake: Forced vibrations vs resonance

  CISCE examiners' comments, 2024: for a tabletop vibrating with a tuning fork's stem pressed on it, candidates chose resonance or damped vibrations instead of forced vibrations; in an air-column experiment others wrote forced vibrations instead of resonance.

  ✓Do this: Any body driven by a periodic force performs forced vibrations; it is resonance only if the frequencies match and the amplitude is maximum.

  Usually costs: 1 mark.

• ✗Mistake: Missing key words in the explanation

  CISCE examiners' comments, 2024 and 2025: candidates omitted 'natural', 'matching' and 'amplitude', or said the fork's frequency matched that of the paper rider or the wooden box.

  ✓Do this: Write: the frequency of the applied force equals the natural frequency of the [air column / wire], so it vibrates with maximum amplitude.

  Usually costs: 1 mark.

• ✗Mistake: Water column instead of air column

  CISCE examiners' comments, 2024 and 2025: candidates referred to the frequency of the burette or the water column, and in an assertion–reason item took frequency to depend on the water column rather than the air column.

  ✓Do this: It is the air column above the water that vibrates; shorter air column, higher frequency.

  Usually costs: 1 mark.

• ✗Mistake: Right answer, wrong reason

  CISCE examiners' comments, 2024: for a higher-frequency fork, candidates wrote 'decrease' with no reason or a wrong one, or stated an inverse relation but then answered 'increase'.

  ✓Do this: Give the reason in one line: higher frequency needs a shorter air column because natural frequency increases as the column shortens.

  Usually costs: 1 mark.

• ✗Mistake: Not reading the diagram

  CISCE examiners' comments, 2025: candidates failed to notice that the wires in a sound-box diagram had different thicknesses (so different natural frequencies).

  ✓Do this: Look for differences in length, thickness and tension: each changes the natural frequency.

  Usually costs: 1 to 2 marks.

Practice questions on Vibrations and Resonance

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

1. Q1 · 1 mark · MCQBoard level

   In damped vibrations:
   (a) the frequency keeps increasing
   (b) the amplitude decreases with time
   (c) the amplitude increases with time
   (d) the body vibrates at the frequency of an external force

   Hint

   Energy is lost.

   Answer

   (b)

2. Q2 · 1 mark · MCQBoard level

   Resonance occurs when:
   (a) the amplitude of a body is zero
   (b) the applied frequency equals the natural frequency of the body
   (c) a body vibrates in vacuum
   (d) two bodies have equal masses

   Hint

   Matching frequencies.

   Answer

   (b)

3. Q3 · 1 mark · MCQHarder board level

   If the length of an air column is decreased, its natural frequency:
   (a) decreases
   (b) increases
   (c) stays the same
   (d) becomes zero

   Hint

   Less air, higher note.

   Answer

   (b)

4. Q4 · 1 mark · Assertion-reasonBoard level

   Assertion (A): A louder sound is heard when the stem of a vibrating tuning fork is pressed against a tabletop.
   Reason (R): The tabletop is forced to vibrate, and its large area sets more air into vibration.
   (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

   Forced vibrations.

   Answer

   (c)

5. Q5 · 2 marks · Very short answerHarder board level

   Give two examples of resonance in daily life.

   Hint

   Radio, instruments, bridges…

   Answer

   Any two: tuning a radio; a flute's air column; a glass shattering with a sustained note; a bridge swaying with marching.

6. Q6 · 2 marks · Short answerHarder board level

   A guitar string vibrates at 440 Hz. Find (a) its time period, (b) the wavelength of the sound it produces in air ($v=330$ m/s).

   Hint

   T = 1/f; λ = v/f.

   Answer

   (a) $T=\frac{1}{440}\approx 0.00227$ s (b) $\lambda =\frac{330}{440}=0.75$ m

7. Q7 · 3 marks · Long answerBoard level

   Define natural, damped and forced vibrations, giving one example of each.

   Hint

   Own frequency; dying amplitude; external force.

   Answer

   Natural: at the body's own frequency when left free (a plucked string). Damped: amplitude decreases due to resistance (a pendulum slowing in air). Forced: at the frequency of an applied periodic force (a tabletop under a tuning fork).

8. Q8 · 3 marks · Long answerHarder board level

   (a) Why does a pendulum swinging in air eventually stop? (b) Would it stop sooner or later in water? (c) What would happen in a perfect vacuum with a frictionless support?

   Hint

   Damping.

   Answer

   (a) Air resistance and friction at the support remove energy: damped vibrations. (b) Sooner: water resists more. (c) It would keep swinging with constant amplitude (natural vibrations).

9. Q9 · 3 marks · Long answerHOTS / competency

   As a tall bottle is filled from a tap, the sound it makes rises in pitch. Explain, naming the vibrating part and the phenomenon involved.

   Hint

   Air column shortens.

   Answer

   The falling water excites the air column above it. As the water rises, the air column shortens, so its natural frequency increases and the pitch rises; the column resonates strongly at its natural frequency.

10. Q10 · 3 marks · Long answerHarder board level

    Give two differences between forced vibrations and resonance.

    Hint

    Frequency relation and amplitude.

    Answer

    Forced: applied frequency may be any value, different from the natural frequency; resonance: applied frequency equals the natural frequency. Forced: amplitude usually small; resonance: amplitude very large.

11. Q11 · 1 mark · MCQHOTS / competency

    Tuning a radio receiver to a particular station is an example of:
    (a) damped vibrations
    (b) natural vibrations
    (c) resonance
    (d) echo

    Hint

    Matching frequencies.

    Answer

    (c)

12. Q12 · 4 marks · Long answerHOTS / competency

    An engineer finds that a footbridge has a natural frequency of 2 Hz. (a) Why could a crowd walking in step at 2 steps per second be dangerous? (b) Name the phenomenon. (c) Suggest two ways to make the bridge safe.

    Hint

    Change the frequency or add damping.

    Answer

    (a) The periodic force of the steps would match the bridge's natural frequency. (b) Resonance, causing large-amplitude swaying. (c) Fit dampers so vibrations die away (damped vibrations); stiffen or change the structure to shift its natural frequency; ask people not to walk in step.

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. 1.Define natural frequency.Show answer

   The frequency at which a body vibrates when left free after being disturbed.

2. 2.What changes in damped vibrations?Show answer

   The amplitude decreases with time.

3. 3.Frequency of forced vibrations?Show answer

   The frequency of the applied periodic force.

4. 4.Condition for resonance?Show answer

   Applied frequency equals the natural frequency.

5. 5.Effect of resonance?Show answer

   Very large amplitude (loud sound).

6. 6.Shorter air column: higher or lower frequency?Show answer

   Higher.

7. 7.Why do soldiers break step on a bridge?Show answer

   To avoid resonance.

Quick revision checklist

The night before the exam: can you do each of these without looking?

• ☐Can you define all four types with key words?
• ☐Can you give two examples of each?
• ☐Do you mention the air column, not the water?
• ☐Can you predict the air-column length for a higher-frequency fork?
• ☐Can you give three differences between damped and forced vibrations?

Vibrations and Resonance: frequently asked questions