Natural vibrations, Damped vibrations, Forced vibrations and Resonance | ICSE Class 10 Physics Notes

This note covers the meaning of vibrations, amplitude and frequency, natural vibrations, damping, forced vibrations, resonance, simple demonstrations, practical applications, displacement graphs and comparisons between the different kinds of vibration.

What quantities describe a vibration?

A vibration, or oscillation, is a to-and-fro motion about a mean position. The mean position is the central position about which the motion occurs. At a stable equilibrium position, a body left at rest stays there, and a small displacement brings a force tending to return it.

A restoring force is a force that tends to bring a displaced body back towards its equilibrium position. A displacement tells us how far, and in which direction, the body is from that position. It changes as the body moves.

What do amplitude, period and frequency mean?

The amplitude is the magnitude of the maximum displacement from the mean position. Magnitude means size without a direction. An extreme position is a position at the end of the body's excursion, where its displacement has the greatest magnitude.

One complete vibration is a complete cycle of the motion: the body returns to its starting position with the same direction of motion. Simply reaching the mean position again need not complete a cycle, because the body can pass through it in opposite directions.

The time period is the time taken for one complete vibration. The frequency is the number of complete vibrations per unit time. The SI, or International System of Units, unit of period is the second, written s. The SI unit of frequency is hertz, written Hz.

Definition: 1 Hz = 1 complete vibration per second. Frequency describes how often the motion repeats; amplitude describes the size of its maximum displacement.

Let f represent frequency and T represent time period. Their relation is f = 1/T. When T is expressed in seconds, f is expressed in hertz. A longer time for each cycle therefore means fewer cycles per second.

Periodic motion repeats after equal intervals of time. A vibrating string and a swinging pendulum provide examples of to-and-fro motion. Uniform circular motion is periodic but is not itself a to-and-fro vibration about a mean position.

Keep the two questions separate when describing a vibration: “How far does it move?” asks about amplitude, while “How often does it repeat?” asks about frequency. This distinction is essential when explaining damping and resonance.

What are natural vibrations and natural frequency?

Definition: Natural or free vibrations are the vibrations of a body displaced from its equilibrium position and then left to vibrate without a continuing external periodic force.

An external periodic force is an externally applied push or pull that repeats at regular intervals. The initial displacement or brief push starts a free vibration. After that initial disturbance, no repeated external driving action is needed in the ideal description.

The natural frequency is the frequency at which a system vibrates freely. It depends on the properties of the vibrating system. A system here means the body, or connected parts, whose vibration is being considered.

How does the body keep moving after release?

When a body is displaced, a restoring force draws it towards equilibrium. Its motion carries it beyond that position. It then moves back, producing a to-and-fro motion. The repeated motion is not evidence of a fresh external push on every cycle.

For an ideal vibration without energy loss, the amplitude remains constant. Mechanical energy is the sum of energy associated with motion and energy associated with position, stretching or compression. These are called kinetic energy and potential energy, respectively.

Energy passes between these forms during the vibration. In the ideal case, the total mechanical energy remains constant. In a real system, friction and other causes remove energy from the vibration, so the freely vibrating body does not continue indefinitely with unchanged amplitude.

How does a pendulum illustrate natural vibration?

A simple pendulum is an ideal small body, called a bob, suspended by a massless, unstretchable string from a fixed support. Displace the bob slightly and release it. The bob then swings about its mean position without repeated pushes.

For small swings at the same place, a longer simple pendulum has a longer period and therefore a lower natural frequency. This dependence concerns the properties of the system. It does not mean that repeated pushes are needed to set the free frequency.

In the diagram, θ, the Greek letter theta, denotes the angle between the displaced string and the vertical line through the support.

A string hangs from a fixed support. The bob is shown below the support and displaced to the right. A dashed vertical line marks the reference direction, θ marks the angular displacement, and a curved arrow indicates the swinging motion.

See Fig. 13.2(b) in your NCERT textbook

Why do damped vibrations become smaller?

Definition: Damped vibrations are vibrations whose amplitude decreases with time because mechanical energy is lost from the vibrating system.

Damping is the reduction of vibration through energy loss. Friction is a resistance to relative motion between surfaces. Resistance from the surrounding air can also oppose motion. These effects reduce the energy available for the body's successive swings.

A pendulum released in air does not keep reaching its original extreme positions. Its successive excursions become smaller, and it eventually comes to rest at its equilibrium position. The reduction in excursion, rather than a change in the number of swings alone, identifies damped vibration.

Where does the energy go?

Energy does not disappear when a vibration dies away. It is transferred out of the organised mechanical motion, for example as internal energy in the system and surroundings. Internal energy is energy associated with the microscopic motion and interactions of particles.

The term dissipation describes this spreading of mechanical energy into other forms through effects such as friction. Less mechanical energy remains in the vibration. Consequently, the body reaches smaller maximum displacements in later cycles.

Do not describe damping merely as “the body moves more slowly”. Even an undamped vibration has changing speed during each cycle. The significant observation is that the amplitude decreases over successive cycles, rather than remaining constant.

How do free and damped vibrations overlap?

“Free” identifies the absence of a continuing external periodic drive. “Damped” identifies the loss of energy and the decreasing amplitude. A real pendulum can therefore vibrate freely after release while its vibrations are also damped by resistance.

The ideal treatment of free vibration ignores those losses. It gives a useful comparison with real observations, but it should not be used to claim that an actual pendulum or vibrating string retains the same amplitude indefinitely.

Note: A body passing through its mean position has zero displacement at that instant. This does not mean that its amplitude has become zero or that its vibration has stopped.

A damped displacement graph shows oscillations on both sides of the mean position, with successive peaks approaching the central line. Its shrinking height represents the decreasing amplitude. The curve is not a single downward line, because the body continues moving to and fro while the vibration decays.

How are forced vibrations produced?

Definition: Forced vibrations are vibrations maintained by an external periodic force. The external agency repeatedly supplies energy to the vibrating body.

The body or agency providing the repeated action is called the driver. Its rate of repetition is the driving frequency. The body that responds to this action is the driven body. Identify these separately before describing the motion.

After the initial changes have died away, the driven body vibrates at the driving frequency. This settled response is called the steady state. The natural frequency remains a property of the body; the driving frequency describes the external action maintaining its motion.

How does forcing differ from a single disturbance?

A single displacement followed by release starts a free vibration. Repeated periodic pushes can maintain a forced vibration. Both involve an initial disturbance, but a forced vibration continues to receive energy from an external source during its motion.

Forcing and damping may act together. The external drive supplies energy, while resistance removes energy. A driven vibration can therefore continue even though the system would gradually come to rest if the driver were removed.

In a steady vibration of constant amplitude, the energy supplied over a cycle balances the energy lost over that cycle. Constant amplitude in a driven body does not prove that friction or air resistance is absent.

What happens when a vibrating tuning fork touches a tabletop?

A tuning fork is a metal instrument with two prongs that vibrate when struck. Its stem is the part used to hold it. If the stem of a vibrating fork touches a tabletop, the fork exerts a repeating force on the table.

The tabletop undergoes forced vibrations. Its larger vibrating surface sets more surrounding air into vibration, and the sound becomes louder. The fork acts as the driver and the tabletop as the driven body.

This observation illustrates forcing. It does not, by itself, establish resonance, because no matching of the tabletop's natural frequency with the fork's frequency has been demonstrated. A repeated force can produce a response even when the resonance condition is not satisfied.

When explaining a forced vibration, state the source of the periodic force and the body that responds. Then distinguish the frequency of the response from its amplitude. The existence of a response and the production of a particularly large response are different observations.

When does forced vibration become resonance?

Definition: Resonance is the large-amplitude response produced when the frequency of an external periodic force matches a natural frequency of the vibrating system. It is a special case of forced vibration.

The body has a natural frequency even before the driver is applied. The driver supplies a repeated disturbance. When their frequencies match, the repeated action transfers energy effectively to the vibrating body, producing a much greater amplitude than a comparable drive far from that frequency.

What condition should an explanation state?

For the elementary resonance condition, write driving frequency = natural frequency. This is a comparison between two frequencies, not between two amplitudes. The body and driver need not have the same maximum displacement.

Real strings and air columns show resonance when the external frequency is close to one of their natural frequencies. “Close to” matters in a practical description: a large response is not confined to a mathematical equality with no surrounding range.

Some systems have more than one natural frequency. A vibrating string is an example. Thus, do not assume that every object has just one possible natural vibration. At this level, identify the natural frequency relevant to the motion being driven.

Why is the amplitude important?

The frequency condition explains why resonance occurs; the large amplitude is its characteristic response. Merely observing that a body vibrates is insufficient. It is necessary to connect the enhanced response with the relation between the driving and natural frequencies.

Compare responses with a similar strength of driving force. A stronger push can itself produce a larger motion, so a comparison that changes both the force and its frequency does not isolate the effect of frequency matching.

Damping limits the amplitude reached in a real resonant system. The driver supplies energy, but some is also dissipated. Do not claim that resonance gives an unlimited amplitude in a practical experiment.

Note: Every resonant vibration is a forced vibration, but every forced vibration is not resonant. Resonance requires the appropriate frequency relation as well as a continuing periodic drive.

The distinction joins the chapter together: natural vibration supplies the reference frequency, forcing supplies the repeated input, damping accounts for energy loss, and resonance describes the enhanced response near a natural frequency.

How can matching tuning forks demonstrate resonance?

A standard demonstration uses two tuning forks with the same natural frequency, mounted on resonance boxes. These are hollow boxes that help make the forks' sound audible. Sound from the vibrating fork provides a periodic disturbance that can drive the other fork.

What is the procedure and observation?

1. Place the matching forks on their boxes near each other, with the box openings facing each other.
2. Strike one fork so that it vibrates. Leave the second fork unstruck.
3. Allow the first fork's sound to act on the second fork. The second fork receives a periodic driving disturbance.
4. Stop the first fork by touching its prongs. Sound from the second fork can still be heard briefly, showing that it has been set into vibration.

The first fork is the driver. The second fork is the driven body. Their matching natural frequencies allow the second fork to respond strongly to the first fork's periodic disturbance. The effect is explained by resonance.

The second fork was not struck directly. Its vibration demonstrates transfer of energy from the vibrating source through the intervening air. The explanation should identify the energy source rather than imply that an initially stationary fork begins to vibrate without any input.

What does stopping the first fork establish?

While both forks are sounding, the sound heard may include the first fork's own contribution. Stopping the driver helps reveal that the second fork has acquired its own vibration. It provides evidence beyond simply hearing a louder combined sound.

Once the first fork is stopped, the second no longer receives that drive. It then vibrates freely for a time and its amplitude decreases through damping. The same demonstration therefore connects resonance, free vibration and damping at different stages.

For comparison, a fork with a sufficiently different natural frequency does not show the same strong resonant response to that driver. This does not justify saying that a mismatched body cannot be forced to vibrate at all. The comparison concerns the strength of the response.

Keep the source, separation and arrangement comparable when comparing forks. Otherwise, a difference in the strength of the disturbance reaching the second fork could be confused with a difference caused by the frequency relation.

How are vibration and resonance used in simple applications?

How does periodic pushing maintain a swing?

A swing provides a familiar application of the same ideas. When displaced and released, it makes free vibrations. Resistance gradually reduces their amplitude. Repeated pushes supply energy, so they can maintain the motion against these losses.

Pushes timed appropriately with the swing's natural rhythm can build up a large motion. This illustrates resonance. A push merely repeated at an arbitrary rate should not automatically be called resonant; the timing must relate to the natural motion.

The swing example also distinguishes amplitude from frequency. Building up the size of the excursion describes an increase in amplitude. The resonance explanation concerns the matching of the repeated pushes to the natural vibration, rather than an assertion that the swing must move through more cycles each second.

How do strings and air columns respond?

A stretched string and an air column, a length of air confined in a tube, have natural frequencies. Both can undergo forced vibrations. Their response becomes resonant when the external frequency is close to one of those natural frequencies.

In a glass tube partly filled with water, the air above the water forms an air column. Changing the water level changes its length and therefore its natural frequencies. A suitable vibrating source can produce a strong response when the air column is adjusted appropriately.

Here the adjustable part is the air column, while the source can retain the same frequency. Resonance can therefore be reached by changing the receiving system, not just by changing the driver. In either case, the relevant requirement is the frequency relation.

Why can damping be useful?

Damping is useful where continued oscillation is unwanted. Shock absorbers are devices used in vehicle suspension systems to reduce repeated bouncing through energy dissipation. The desired effect is a reduction of oscillation after a disturbance.

This application does not require the vibration to be maintained. It uses energy loss deliberately. Compare it with a swing being pushed: there the driver supplies energy to continue the motion, whereas damping removes energy from the mechanical vibration.

These examples serve different purposes. The swing illustrates repeated driving and suitable timing; the air column illustrates frequency matching; the shock absorber illustrates the practical value of reducing amplitude. Name the physical process before discussing the outcome.

How should vibration diagrams and graphs be read?

A diagram shows the arrangement of a vibrating system. A displacement-time graph instead shows how the displacement of a chosen body changes as time passes. Its curve is a record of motion, not a drawing of the physical path travelled by that body.

What does the spring and block diagram show?

Use x for the block's displacement from equilibrium. A positive or negative value indicates the side of the chosen origin. The diagram compares an equilibrium position with a displaced position; it does not show two independent blocks vibrating.

A horizontal spring joins a wall on the left to a block on a horizontal surface. A coloured reference position and a displaced block are shown. An arrow marked x indicates the displacement between the reference and displaced positions.

See Fig. 13.2(a) in your NCERT textbook

The ideal arrangement has a frictionless surface. When the block is displaced and released, the spring provides a restoring force. “Frictionless” is a stated simplifying condition, so constant-amplitude ideal motion must not be confused with a claim about every real spring arrangement.

What do the height and spacing of a graph mean?

Use t for time and A for amplitude. A graph's vertical distance from the central line to a peak gives the amplitude. The horizontal time interval between successive equivalent points, such as successive positive peaks, gives the period.

Simple harmonic motion is oscillatory motion in which the restoring force is proportional to displacement and directed towards the mean position. Its ideal displacement-time curve has a smooth repeating wave shape.

The vertical axis is labelled Displacement and marks A, 0 and −A. The horizontal axis is marked t. A smooth curve starts at positive maximum displacement and repeats between equal positive and negative limits.

See Fig. 13.5 in your NCERT textbook

In that graph, equal peak heights represent unchanged amplitude. For a damped vibration, successive peaks would move closer to the central line. Do not confuse shrinking height with shrinking horizontal spacing: the former describes amplitude, while the latter concerns period.

Axes and labels are therefore part of the explanation. Before interpreting any curve, check whether the vertical quantity is displacement and whether the horizontal quantity is time. Without that check, a description of “higher” or “closer” peaks is incomplete.

How can the four vibration ideas be distinguished?

The four terms answer related but different questions. Natural vibration describes the body's free response, damping describes energy loss, forcing identifies a continuing periodic input, and resonance identifies an enhanced response near a natural frequency.

Which feature should be compared?

Damped vibration is not a separate choice that excludes all free or forced motion. A freely vibrating body can lose energy. A driven body can also lose energy while receiving fresh energy from its driver. The observed amplitude depends on the combined effects.

How should an unfamiliar description be analysed?

1. Identify the body whose vibration is being described and its mean position.
2. Decide whether it was simply disturbed and released or continues to receive a periodic driving force.
3. Look for evidence that amplitude decreases, remains steady or becomes particularly large under the stated conditions.
4. For resonance, connect the enhanced response to the relation between the driving frequency and a natural frequency.

A complete explanation links the observation to its cause. “The amplitude decreases” needs energy loss as the reason. “The body vibrates continuously” needs a stated drive if losses are present. “Resonance occurs” needs the relevant frequency condition.

Keep observations and conclusions separate. A loud sound can accompany forced vibration, and a large motion can follow a strong disturbance. Neither observation alone supplies the frequency comparison needed to establish resonance.

How does simple harmonic motion model free vibrations?

Simple harmonic motion, abbreviated SHM, is an ideal oscillation in which the restoring force is proportional to displacement and acts towards equilibrium. Its mathematics describes an undamped vibration. It does not describe the decreasing amplitude of a damped vibration exactly.

Write displacement as $x=A\cos(\omega t+\phi)$. Here $A$ is amplitude, $t$ is time, $\omega$ is angular frequency and $\phi$ is the phase constant, which specifies the initial stage of the cycle.

The angular frequency and period are related by $\omega=2\pi/T$. Velocity is $v=-\omega A\sin(\omega t+\phi)$, and acceleration is $a=-\omega^2x$. Thus the maximum speed is $v_{\mathrm{max}}=\omega A$, while the maximum acceleration magnitude is $a_{\mathrm{max}}=\omega^2A$.

Use metres for displacement and amplitude, seconds for time, radians per second for angular frequency, metres per second for velocity and metres per second squared for acceleration. The negative sign in the acceleration relation indicates its direction towards the mean position.

Derivation: Period of a mass attached to a spring

Consider a mass $m$ on an ideal spring of spring constant $k$, with friction neglected. The spring constant measures restoring force per unit extension and is expressed in newtons per metre.

1. The restoring force obeys $F=-kx$, where $x$ is measured from equilibrium.
2. Newton's second law gives $F=ma$, so $a=-kx/m$.
3. Compare this with the SHM relation $a=-\omega^2x$. The coefficients give $\omega^2=k/m$, hence $\omega=\sqrt{k/m}$.
4. Substitute this angular frequency into $T=2\pi/\omega$ to obtain the period.

Result: $T=2\pi\sqrt{m/k}$. A larger mass gives a longer period, while a stiffer spring gives a shorter period.

Derivation: Total energy of an undamped harmonic oscillator

Let $K$ denote kinetic energy, $U$ potential energy and $E$ total mechanical energy. Take the spring's potential energy to be zero at equilibrium. Use $k=m\omega^2$ to express both forms of energy in terms of the spring constant.

1. Substitute the velocity into $K=\frac12mv^2$, giving $K=\frac12kA^2\sin^2(\omega t+\phi)$.
2. Substitute the displacement into $U=\frac12kx^2$, giving $U=\frac12kA^2\cos^2(\omega t+\phi)$.
3. Add the energies: $E=K+U$, so $E=\frac12kA^2[\sin^2(\omega t+\phi)+\cos^2(\omega t+\phi)]$.
4. The sine squared and cosine squared of the same angle add to one, leaving a constant total energy.

Result: $E=\frac12kA^2$. Energy changes between kinetic and potential forms during each cycle, but their sum stays constant when no energy is dissipated.

How can periods, frequencies and energies be calculated?

For a simple pendulum making small oscillations, $T=2\pi\sqrt{L/g}$, where $L$ is the pendulum length and $g$ is acceleration due to gravity. Length is measured in metres and acceleration due to gravity in metres per second squared.

This relation assumes a small bob on a massless, unstretchable string. It explains why the same pendulum has different periods where gravity differs. A pendulum that ticks seconds takes two seconds for one complete oscillation, since each tick corresponds to half a cycle.

Worked example 1. A human heart beats 75 times in one minute. Calculate its frequency and period.

Formula: $f=N/t$, where $N$ counts beats during time $t$; $T=1/f$.

Substitute: One minute is 60 s, so $f=75/60=1.25\,\mathrm{Hz}$. Then $T=1/1.25=0.80\,\mathrm{s}$.

Answer: The frequency is 1.25 Hz and the period is 0.80 s.

Worked example 2. A 1 kg block attached to a spring of constant 50 N/m is pulled 10 cm from equilibrium and released from rest on a frictionless surface. Calculate its kinetic, potential and total energies at a displacement of 5 cm.

Formula: $E=\frac12kA^2$; $U=\frac12kx^2$; $K=E-U$.

Substitute: The amplitude is $A=0.10\,\mathrm{m}$ and the displacement is $x=0.05\,\mathrm{m}$. Thus $E=\frac12(50)(0.10)^2=0.25\,\mathrm{J}$ and $U=\frac12(50)(0.05)^2=0.0625\,\mathrm{J}$.

Subtract to find $K=0.25-0.0625=0.1875\,\mathrm{J}$. Keeping these digits until the final step preserves the energy sum.

Answer: The kinetic energy is 0.1875 J, approximately 0.19 J; the potential energy is 0.0625 J and the total energy is 0.25 J.

Worked example 3. Find the length of a simple pendulum that ticks seconds, using an acceleration due to gravity of 9.8 m/s².

Formula: $T=2\pi\sqrt{L/g}$. Squaring and rearranging gives $L=gT^2/(4\pi^2)$.

Substitute: A complete oscillation takes $T=2\,\mathrm{s}$. Hence $L=9.8\times2^2/(4\pi^2)\approx0.993\,\mathrm{m}$, to three significant figures.

Answer: The length is approximately 0.993 m, or about 1 m.

Worked example 4. A spring of constant 1200 N/m carries a mass of 3 kg on a horizontal table. The mass is displaced 2.0 cm and released. Find its frequency, maximum acceleration and maximum speed, treating the motion as undamped SHM.

Formula: $\omega=\sqrt{k/m}$; $f=\omega/(2\pi)$; $a_{\mathrm{max}}=\omega^2A$; $v_{\mathrm{max}}=\omega A$.

Substitute: Convert the amplitude to $A=0.020\,\mathrm{m}$. Then $\omega=\sqrt{1200/3}=20\,\mathrm{rad\,s^{-1}}$ and $f=20/(2\pi)\approx3.18\,\mathrm{Hz}$.

The acceleration magnitude is $a_{\mathrm{max}}=20^2\times0.020=8.0\,\mathrm{m\,s^{-2}}$, and the speed is $v_{\mathrm{max}}=20\times0.020=0.40\,\mathrm{m\,s^{-1}}$.

Answer: The frequency is approximately 3.18 Hz, the maximum acceleration magnitude is 8.0 m/s² and the maximum speed is 0.40 m/s.

Worked example 5. A locomotive piston has a stroke of 1.0 m and moves with SHM at an angular frequency of 200 rad/min. Find its maximum speed.

Formula: The stroke is twice the amplitude, so $A=\mathrm{stroke}/2$; $v_{\mathrm{max}}=\omega A$.

Substitute: $A=1.0/2=0.50\,\mathrm{m}$. Convert angular frequency to $\omega=200/60\,\mathrm{rad\,s^{-1}}$. Then $v_{\mathrm{max}}=(200/60)\times0.50\approx1.67\,\mathrm{m\,s^{-1}}$.

Answer: The maximum speed is approximately 1.67 m/s, equivalent to 100 m/min.

Worked example 6. A simple pendulum has a period of 3.5 s on Earth. Find its period on the Moon, where acceleration due to gravity is 1.7 m/s², compared with 9.8 m/s² on Earth.

Formula: For the unchanged pendulum length, $T_{\mathrm{E}}=2\pi\sqrt{L/g_{\mathrm{E}}}$ and $T_{\mathrm{M}}=2\pi\sqrt{L/g_{\mathrm{M}}}$. Dividing gives $T_{\mathrm{M}}/T_{\mathrm{E}}=\sqrt{g_{\mathrm{E}}/g_{\mathrm{M}}}$.

Substitute: $T_{\mathrm{M}}=3.5\sqrt{9.8/1.7}\approx8.40\,\mathrm{s}$. The smaller acceleration due to gravity gives a longer period.

Answer: The period on the Moon is approximately 8.4 s.

Glossary

• Vibration — A to-and-fro motion of a body about its mean position.
• Equilibrium position — A position where a body left at rest remains at rest.
• Restoring force — A force tending to return a displaced body towards its equilibrium position.
• Displacement — The directed change in position measured from a chosen reference position.
• Amplitude — The magnitude of the greatest displacement from the mean position during a vibration.
• Time period — The time taken by a vibrating body to complete one full cycle.
• Frequency — The number of complete vibrations made per unit time by a vibrating body.
• Natural frequency — A frequency at which a system vibrates freely after an initial disturbance.
• Natural vibrations — Vibrations following an initial disturbance without a continuing external periodic driving force.
• Damped vibrations — Vibrations whose amplitude decreases with time as mechanical energy is lost.
• Forced vibrations — Vibrations maintained by energy supplied through an external periodic driving force.
• Driving frequency — The frequency of the external periodic force acting on the vibrating body.
• Resonance — A large response to periodic driving at or near a natural frequency.
• Dissipation — Transfer of mechanical energy into other forms through effects such as friction.
• Steady state — The settled response of a driven system after its initial changes have died away.

Common errors and misconceptions

• Misconception: Amplitude and frequency describe the same property. Correct: Amplitude describes maximum displacement; frequency describes the number of complete vibrations per unit time.
• Misconception: Free vibrations require repeated external pushes. Correct: They follow an initial disturbance without a continuing periodic driver; the restoring force acts during the motion.
• Misconception: A real pendulum maintains its amplitude indefinitely after release. Correct: Resistance dissipates mechanical energy, so its free vibrations are damped and eventually cease.
• Misconception: Damping means energy has been destroyed. Correct: Energy has left the organised mechanical vibration and been transferred into other forms or to the surroundings.
• Misconception: Every forced vibration is resonance. Correct: Resonance requires a driving frequency matching or close to a natural frequency and is associated with an enhanced response.
• Misconception: Resonance requires the driver and driven body to have equal amplitudes. Correct: The relevant matching condition concerns frequencies, not amplitudes.
• Misconception: Zero displacement means a vibration has stopped. Correct: A vibrating body passes through its mean position during its motion; its amplitude can remain non-zero.
• Misconception: Louder sound from a tabletop proves resonance. Correct: A tuning fork can force the tabletop to vibrate; frequency matching must be established before calling the response resonant.

Exam-style questions with model answers

Q1. Define natural vibrations and natural frequency. [2 marks]

1. Natural vibrations are vibrations following an initial disturbance, without a continuing external periodic driving force.
2. Natural frequency is a frequency at which the system vibrates freely after that disturbance.

Q2. A pendulum is displaced slightly and released in air without further pushes. Its successive swings become smaller. Name the kind of vibration shown by the decreasing amplitude, explain the decrease, and state its eventual condition. [3 marks]

1. The decreasing amplitude shows that the pendulum undergoes damped vibrations. Its maximum displacement from the mean position becomes smaller in successive swings.
2. Resistance, including that of the surrounding air, removes mechanical energy from the vibration and transfers it into other forms.
3. The pendulum eventually comes to rest at its equilibrium position because no continuing external drive replaces the lost energy.

Q3. Define forced vibrations. State their steady-state frequency and explain whether a driven body can have constant amplitude despite damping. [3 marks]

1. Forced vibrations are maintained by an external periodic force, which repeatedly supplies energy to the vibrating body.
2. After the initial changes have died away, the steady-state vibration has the frequency of the driving force, rather than necessarily the body's natural frequency.
3. Constant amplitude is possible despite damping when the driver supplies, over each cycle, the energy lost through resistance.

Q4. Define resonance, state its elementary frequency condition, identify its characteristic amplitude response and explain why every forced vibration is not resonant. [4 marks]

1. Resonance is a special case of forced vibration in which periodic driving produces a particularly large response.
2. The elementary condition is that the driving frequency equals a natural frequency of the body.
3. The amplitude is large compared with the response to a comparable drive at a frequency far from resonance.
4. Forced vibration can occur without the required frequency matching, so a driven response does not by itself establish resonance.

Q5. Two tuning forks with equal natural frequencies stand on resonance boxes near each other, with the openings facing. One fork is struck; the other is left unstruck. After a short time, the first fork is stopped and sound is heard briefly from the second. Identify the driver, explain the second fork's vibration and the role of equal frequencies, interpret the observation after stopping the first, and explain why the second eventually stops. [5 marks]

1. The struck first fork is the driver. Its vibration produces a periodic disturbance that reaches the second fork through the air.
2. The second fork is the driven body. It gains energy from this periodic disturbance even though it has not been directly struck.
3. Its natural frequency equals the driving frequency supplied by the first fork, so the second fork develops a strong resonant response.
4. Hearing it after the first fork is stopped shows that the second fork itself has been set into vibration.
5. After the driver stops, the second fork vibrates freely with damping. Energy losses reduce its amplitude until the vibration dies away.

Q6. The stem of a vibrating tuning fork is pressed against a tabletop. The sound becomes louder. No information about the tabletop's natural frequencies is given. Identify the driver and driven body, name the vibration, explain the louder sound and assess whether resonance has been established. [5 marks]

1. The vibrating tuning fork is the driver because its motion supplies a repeating external force through its stem.
2. The tabletop is the driven body. It receives energy from the fork and is set into motion by that repeated action.
3. The tabletop undergoes forced vibrations because an external periodic source maintains its response while the fork acts on it.
4. The larger vibrating surface of the tabletop sets more surrounding air into vibration, making the sound louder.
5. Resonance has not been established. The observation supplies no evidence that the driving frequency matches or lies close to a natural frequency of the tabletop.

Q7. A displacement-time graph crosses the central line repeatedly, while successive positive and negative peaks approach that line. Explain what the crossings mean, what the changing peak heights show and the energy change responsible. [3 marks]

1. The central-line crossings represent instants when the body passes through its mean position. Zero displacement at a crossing does not establish that the body has stopped.
2. The decreasing distances of successive peaks from that line show that the amplitude is decreasing, identifying damped vibration.
3. Mechanical energy is being removed from the vibration through dissipative effects such as resistance, producing progressively smaller excursions.

Q8. State the relation between frequency and time period, defining both symbols. Then distinguish what frequency and amplitude measure. [2 marks]

1. The relation is f = 1/T, where f is frequency in hertz and T is the time period in seconds.
2. Frequency measures complete vibrations per unit time; amplitude measures the magnitude of the maximum displacement from the mean position.

Key takeaways

• Amplitude measures maximum displacement, while frequency measures how many complete vibrations occur per unit time.
• Natural vibrations follow an initial disturbance without a continuing external periodic force maintaining the motion.
• Damped vibrations lose mechanical energy, so their amplitude becomes smaller as time passes after release.
• Forced vibrations receive energy from a periodic driver and follow its frequency in the steady state.
• Resonance is a special case of forcing, with a large response near a natural frequency.
• Damping limits real resonant responses; a continuing input of energy does not imply unlimited amplitude.
• A tabletop driven by a tuning fork illustrates forced vibration; loudness alone does not establish resonance.
• A real body can vibrate freely while damping reduces its amplitude, so the descriptions can overlap.

Test yourself

Which quantity measures how far a vibrating body moves from its mean position at its greatest displacement?

Amplitude is the magnitude of the maximum displacement from the mean position.

What distinguishes a single release from a continuing periodic drive?

A single release starts free vibration; a continuing periodic drive supplies energy repeatedly and maintains forced vibration.

Why does a pendulum released in air eventually stop?

Resistance dissipates its mechanical energy, reducing its amplitude until it comes to rest at equilibrium.

Can a free vibration also be damped?

Yes. It can have no continuing driver while resistance removes energy and decreases its amplitude.

What frequency does a forced vibration follow after the initial changes die away?

Its steady-state frequency is the frequency of the external periodic driving force.

What frequency condition gives the elementary description of resonance?

The driving frequency equals a natural frequency of the vibrating body.

Why does hearing the second tuning fork after stopping the first help the demonstration?

It shows that the second fork itself has acquired vibration, rather than merely hearing sound from the first.

What is the difference between the height and horizontal spacing of successive peaks on a displacement-time graph?

Peak height from the mean line gives amplitude; the time between successive positive peaks gives the period.