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Forced oscillations and resonance are the two different cases of a body performing oscillatory motion.
- Oscillation is referred to the recurrent movement of an item between two different positions or states.
- Simply, oscillation is the back-and-forth motion of a body with regard to a fixed point.
- The motion of the pendulum is an example of oscillatory motion.
- Another illustration of oscillation is the string's vibration in instruments like the guitar.
- The oscillation of a body is said to execute forced oscillations when it is compelled to oscillate with a frequency other than its natural frequency.
- Resonance is a particular case of forced oscillation in which the frequency of the outside driving force is equal to the natural frequency of the system.
| Table of Content |
Key Terms: Resonance, Damped oscillation, Forced oscillation, Frequency, Free oscillation, Amplitude, Oscillatory motion, Force
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What is Resonance?
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The phenomenon of resonance occurs when an external oscillating system having a particular frequency causes another system to oscillate with the same frequency and with a higher amplitude. The frequency with which the resonance occurs is known as resonant frequency.
- This oscillation at a large amplitude might lead to the failure of the bridge is an example of resonance.
- The amplitude of the string's vibration is highest at the resonance frequency while tuning a guitar (one system) using a different guitar (another system).
- The system's ability to store vibrational energy accounts for the huge amplitude oscillations that are generated at the resonant frequencies.
Damped, Free, and Forced Oscillation
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Oscillation is the regular fluctuation in position or magnitude about a fixed point known as mean position. Some examples of oscillatory motion are free, forced, and damped oscillations.
Damped Oscillation
The oscillation in which the amplitude decreases gradually with time is called Damped oscillation.
- Without any outside influences, the amplitude and time duration of an oscillating body stays constant.
- Damping is zero when the amplitude of the oscillating body remains constant.
- Damping occurs when the amplitude of the oscillating body gradually decreases (system energy loss), due to the factors like air drag and friction.
Damped oscillations
Free Oscillation
A system is said to execute free oscillations, if on being displaced or disturbed from its mean position, it oscillates itself with its natural frequency.
- This natural frequency of the free oscillating body dies eventually due to the damping process.
- However, they can be maintained by an external agency. These are called forced or driven oscillations.
Forced Oscillation
The oscillation of a body is said to be forced oscillation when it is compelled to oscillate with a frequency other than its natural frequency.
- The forced oscillation occurs when an external periodic force affects a body's oscillation.
- The oscillation's amplitude is dampened in this case, but the system's external energy keeps it from changing.
Forced oscillation
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What Distinguishes Free from Forced Oscillations?
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The following characteristics set free oscillations apart from forced oscillations:
- When there is no outside force acting on an object, it will vibrate or oscillate freely. Nevertheless, forced oscillations happen as a result of an outside driving force.
- The frequency of forced oscillations is influenced by both the source of vibration and the frequency of the applied driving force, whereas the frequency of free oscillations simply depends on the source of vibrations.
- Free oscillations have a steady vibrational frequency throughout the whole process. Yet, by altering the frequency of the driving force, the frequency of forced oscillations may be changed.
- With free oscillations, the vibrations' amplitude is constant. Yet, the amplitude of forced oscillations may rise, fall, or remain constant.
Different Types of Resonances
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There are several forms of resonance that include:
Mechanical Resonance
When a mechanical system's oscillation frequency and natural vibration frequency match, the mechanical system tends to absorb more energy. Mechanical resonance is a phenomenon that can cause intense vibrations, erratic swaying motions, and frequently the collapse of objects including buildings, bridges, trains, and aeroplanes.
Acoustic resonance
Acoustic resonance is the term used to describe mechanical vibrations that fall within the audible range of the human ear.
- It is a subset of mechanical resonance that deals with vibrations that occur between 20 Hz and 20 kHz in frequency.
- Building resonator-based instruments, such as the body and strings of a violin, the length of a fluted tube, or the tension of a drum membrane, requires consideration of acoustic resonance.
Electrical Resonance
Electrical circuits exhibit the phenomena of electrical resonance. It is utilized for wireless communication, such as that used in radio, television, and cell phones.
Optical Resonance
Lasers frequently employ optical resonators or resonant optical cavities. It consists of a configuration of optical elements that allows a beam of light to go through a closed route.
Orbital Resonance
An idea connected to celestial mechanics is called orbital resonance. A periodic and regular gravitational force is jointly exerted by two orbiting bodies in the event of orbital resonance. The outcome is a significant enhancement of the bodies' mutual gravitational influence.
Atomic Resonance
In particle physics, resonance refers to certain quantum mechanical characteristics that are seen in an atomic nucleus when a magnetic field is supplied externally.
- Many scientific techniques, such as spectroscopy and magnetic resonance imaging (MRI), involve nuclear magnetic resonance (NMR).
- NMR spectroscopy is used to examine molecules, crystals, and non-crystals, whilst MRI is used in medical imaging operations.
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Real-Life Resonance Examples
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Resonance examples include the following:
- When a powerful sound system is switched on all around you or your home, one can always notice the trembling that occurs in the walls and furniture of the building.
- This happens when the resonance frequencies of the furniture resonate with the noise frequency of the music, causing the furniture to vibrate.
- The pendulum and the swing operate on the same principles.
- As we push the pendulum, it will swing backwards and forward.
- If you continue to press at the same moment, the pendulum's movement may increase.
- Giving the pendulum regular push will help it gain more momentum.
- The number of times the pendulum swings forward and backwards in a second is called the resonance frequency.
- We can observe that food is heated in the microwave oven.
- Due to resonance, the electromagnetic radiation produced by microwaves has a certain wavelength and frequency.
- Moreover, the molecules of water and fat have resonance frequencies.
- The molecules absorb the radiation and warm up at particular frequencies.
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Things to Remember
- The Latin word oscillate, which means "to swing," is the source of the English word oscillation.
- A system oscillates at its natural frequency when it is not exposed to any outside stimuli.
- Yet, when a driving force is provided to the system on a regular basis, some energy is added at a frequency other than the system's intrinsic oscillation frequency.
- Forced oscillations will now be produced as the system is "forced" to vibrate at the frequency of the external periodic force.
- The amplitude of the forced vibrations will depend on the difference between the system's natural frequency and the driving force; a greater frequency difference will lead to a lower amplitude.
Sample Questions
Ques. The amplitude at resonance will be larger the higher the damping constant is. Is it real or not? (1 Mark)
a) True
b) False
Ans. The correct option is b.
Explanation: Less forced oscillations will have an impact on raising the amplitude close to the natural frequency. Hence, the resonance amplitude will be lower the greater the damping constant.
Ques. What is the resonant frequency of forced periodic oscillations? (1 Mark)
A) Forced oscillator frequency
B) Their natural frequency
C) There is no set frequency
D) The sum of the forced oscillator's frequency and its natural frequency
Ans. The correct option is a.
Explanation: In the case of forced oscillations, the body's oscillations are not based on its natural frequency but rather oscillate at the frequency of the forced oscillator.
Ques. What are the simple harmonic motion's two fundamental properties? (2 Marks)
Ans. The first two essential qualities of an SHM are as follows:
a) The relationship between acceleration and displacement is a direct proportionality.
b) The acceleration is in the direction of the mean position, whereas the displacement is in the opposite direction of the mean position.
Ques. In a well-known magic trick, the magician sings a note towards a crystal glass until the glass breaks. Discuss how resonance and natural frequency cooperate to make the trick effective. (2 Marks)
Ans. A note that matches the natural frequency of the glass must be sung by the performer. The glass replies by resonating at the same frequency as the sound wave as it is struck by it. As the system receives enough energy, the glass starts to vibrate and finally breaks.
Ques. Which of the instances below best exemplifies periodic motion? (5 Marks)
(A) A swimmer who successfully does one (return) lap from one riverbank to the other and back.
(B) The displacement and release of a bar magnet that is hanging freely.
(C) The rotation of a hydrogen molecule around its mass center.
(D) A bow's discharged arrow.
Ans.
- The motion of the swimmer is not regular. The swimmer moves back and forth between the riverbanks. It does not, however, have a set duration. This is due to the possibility that the swimmer's back-and-forth excursion may not take the same amount of time.
- If a freely suspended magnet is moved away from its N-S direction and released, the motion will be periodic. This is caused by the magnet's periodic oscillations around its location.
- The rotation of a hydrogen molecule around its mass center is periodic. This is due to the fact that a hydrogen molecule rotating about its center of mass always returns to the same place after an equal amount of time.
- When an arrow is discharged from a bow, it only goes forward. It doesn't move in reverse. Its motion is not periodic as a result.
Ques. Which of the following illustrations represents periodic motion but not simple harmonic motion, and which one does? (5 marks)
(a) the earth's rotation around its axis.
(b) the movement of a mercury column in a U-tube that is oscillating.
(c) when a ball bearing is released from a position just above the lowermost point, it moves inside a smooth, curved bowl.
(d) a polyatomic molecule's general oscillations around its equilibrium location.
Ans. (a) The earth does not move back and forth around a fixed point as it rotates. As a result, albeit periodic, it is not S.H.M.
(b) Simple harmonic motion
(c) Simple harmonic motion
(d) A polyatomic molecule's general vibrations about its equilibrium position are periodic but not SHM. There are several inherent frequencies in a polyatomic molecule. As a result, its vibration is a superposition of simple harmonic movements at various frequencies.
Ques. Four x-t charts for a particle's linear motion are shown in Fig. Which plot best exemplifies periodic motion? What is the motion's period, if it is periodic? (5 Marks)
Ans. (a) Because the motion does not recur after a set amount of time, it is not periodic motion.
(b) The provided graph shows a periodic motion that repeats itself every two seconds.
(c) Because the motion is repeated only in one spot, the supplied graph does not show periodic motion. A periodic motion requires sequential repetitions of the full motion throughout one period.
(d) The provided graph shows a periodic motion that repeats every two seconds.
Ques. A particle is moving linearly and harmonically between places A and B, which are 10 cm apart. Consider the direction from A to B as positive and indicate the particle's velocity, acceleration, and force while it is moving in that direction. (5 Marks)
Consider the direction from A to B as positive and indicate the particle's velocity, acceleration, and force while it is moving in that direction.
(a) At the end A
(b) The final B
(c) halfway along AB moving in the direction of A
(d) moving towards A at a distance of 2 cm from B
(e) at a distance of 3 cm from A moving towards B
(f) moving towards A at a distance of 4 cm from B
Ans. (a) A zero, a positive, and a positive.
(b) Zero, negative, and negative
(c) Negative, 0 and 0
(d) Negative, Negative, Negative
(e) Positive, Positive, Positive
(f) Negative, Negative, Negative
Explanation
(a), (b)The next figure depicts the supplied circumstance. Points A and B represent the path's two ends, with AB equalling 10 cm, and 'O' serving as the paths halfway.
Between the endpoints, a particle is moving in a linear, simple harmonic motion. The particle is at rest at point A, which is its extreme. Its velocity is thus zero at this location. Since that it is moving along AO, its acceleration is positive. The particle is moving in a rightward direction in this instance; therefore, force is likewise positive.
The particle is at rest at point B, which is its extreme. Its velocity is thus zero at this location.
(c) The particle is moving in a straightforward harmonic manner. "O" represents the particle's average location. At the mean location O, its velocity is at its highest. As the particle is moving left, the velocity value is negative. In the mean location, a particle performing SHM experiences no acceleration or force.
(d) The particle is traveling from end B towards point O. This motion's direction is the reverse of the typical positive direction, which is in the direction of A and B. The particle's acceleration, velocity, and force are all negative as a result.
(e) The particle is traveling away from endpoint A and towards point O. This motion is going in the traditional positive direction, which is from A to B. The figures for velocity, acceleration, and force are all positive as a result.
(f) This situation is comparable to that described in (d)
Ques. Respond to the following inquiries: (7 marks)
(a) A particle's time period in the SHM relies on its mass m and force constant k:
T = = 2π (√m/√k)
A basic pendulum operates SHM. Hence, why does a pendulum's time period not rely on its mass?
(b) For small-angle oscillations, the motion of a simple pendulum is roughly simple harmonic. A more thorough investigation demonstrates that T is bigger than 2(l/g) for oscillation angles that are larger. To fully comprehend this outcome, consider a qualitative argument.
(c) A man who is holding a wristwatch drops from a tower's top. Does the watch accurately display the time while in free fall?
(d) What is the oscillation frequency of a straightforward pendulum mounted in a cabin that is falling freely with gravity?
Ans. (a) The spring constant k for a straightforward pendulum is inversely proportional to mass. The denominator and the m, which is the numerator, will cancel each other out. Thus, the mass of the bob has no effect on the basic pendulum's time period.
(b) The formula provides the restoring force operating on the bob of a basic pendulum.
F = − mgsinθ
The restoring force is F.
M is the bob's mass.
Gravitational acceleration, or g,
θ is the displacement angle.
When θ is small, sinθ ≈ θ.
Hence, T = 2π(√l/√g) is the formula for a basic pendulum's time period.
When sin, is enormous. The aforementioned equation will not work as a result. The time period T will extend longer.
(c) A wristwatch operates on spring action rather than gravity's acceleration. As a result, the watch will display the accurate time.
(d) The cabin's acceleration owing to gravity will be zero during free fall. As a result, the basic pendulum's frequency of oscillation will also be zero.
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