Free Forced and Damped Oscillations

Free Oscillations: In free oscillations, a system oscillates without any external force acting on it. The system’s natural frequency and damping determine the oscillation’s frequency and amplitude.

Forced Oscillations: In forced oscillations, an external force drives the system, causing it to oscillate at the driving force’s frequency. The system’s natural frequency and damping influence the oscillation’s amplitude and phase.

Damped Oscillations: Damped oscillations occur when a system loses energy due to friction or other resistive forces. The oscillations gradually decrease in amplitude until the system eventually stops oscillating.

Relation between Free, Forced, and Damped Oscillations: Free oscillations are the natural oscillations of a system, while forced oscillations are driven by an external force. Damped oscillations occur when energy is lost from the system, causing the oscillations to decrease in amplitude.

Applications: Free, forced, and damped oscillations have numerous applications in various fields, including physics, engineering, and music. They are essential in understanding phenomena such as the motion of springs, pendulums, and sound waves.

Definition of Oscillation

Definition of Oscillation

Oscillation is the repetitive motion of a body or system about a central point or position. It is a periodic motion that occurs when a system is disturbed from its equilibrium position and then returns to it. Oscillations can be simple or complex, and they can occur in a variety of systems, including mechanical, electrical, and biological systems.

Examples of Oscillations

• Simple harmonic motion: This is the simplest type of oscillation, in which a body moves back and forth along a straight line. Examples of simple harmonic motion include the motion of a pendulum, the vibration of a spring, and the oscillation of a mass on a spring.
• Damped oscillations: These oscillations gradually decrease in amplitude over time due to the presence of friction or other resistive forces. Examples of damped oscillations include the motion of a pendulum in air, the vibration of a spring with a damper, and the oscillation of a mass on a spring with a damper.
• Forced oscillations: These oscillations are caused by an external force that is applied to the system. Examples of forced oscillations include the motion of a pendulum that is driven by a clock, the vibration of a spring that is driven by a motor, and the oscillation of a mass on a spring that is driven by a force.
• Resonance: This occurs when the frequency of the external force that is applied to the system is equal to the natural frequency of the system. At resonance, the amplitude of the oscillations is maximum. Examples of resonance include the swinging of a pendulum when the frequency of the driving force is equal to the natural frequency of the pendulum, the vibration of a spring when the frequency of the driving force is equal to the natural frequency of the spring, and the oscillation of a mass on a spring when the frequency of the driving force is equal to the natural frequency of the mass-spring system.

Applications of Oscillations

Oscillations have a wide variety of applications in science, engineering, and everyday life. Some examples include:

• Pendulums: Pendulums are used to measure time, to study the motion of objects, and to calibrate other instruments.
• Springs: Springs are used to store energy, to absorb shock, and to provide tension in a variety of devices.
• Mass-spring systems: Mass-spring systems are used to study the motion of objects, to design shock absorbers, and to build musical instruments.
• Resonance: Resonance is used to amplify signals, to tune musical instruments, and to design antennas.

Oscillations are a fundamental part of the physical world, and they play an important role in a wide variety of applications.

How Is Oscillation Calculated?

How Is Oscillation Calculated?

Oscillation is the periodic variation of a quantity about a central value. It can be calculated using a variety of methods, depending on the specific application.

1. Simple Harmonic Motion

The simplest type of oscillation is simple harmonic motion (SHM). This occurs when a mass is attached to a spring and set in motion. The motion of the mass is described by the following equation:

x = A cos(ωt + φ)

where:

• x is the displacement of the mass from its equilibrium position
• A is the amplitude of the oscillation
• ω is the angular frequency of the oscillation
• t is the time
• φ is the phase angle

The amplitude of an oscillation is the maximum displacement of the mass from its equilibrium position. The angular frequency is the rate at which the mass oscillates, and is measured in radians per second. The phase angle is the angle at which the mass starts its oscillation.

2. Damped Oscillation

Damped oscillation is a type of oscillation in which the amplitude of the oscillation decreases over time. This is due to the presence of friction or other forces that oppose the motion of the mass. The equation for damped oscillation is:

x = Ae^(-bt) cos(ωt + φ)

where:

• b is the damping coefficient

The damping coefficient is a measure of the strength of the damping force. The larger the damping coefficient, the faster the amplitude of the oscillation will decrease.

3. Forced Oscillation

Forced oscillation is a type of oscillation in which the mass is driven by an external force. The equation for forced oscillation is:

x = A cos(ωt + φ) + F(t)

where:

• F(t) is the external force

The external force can be any type of function, but it is often a sinusoidal function. The amplitude of the forced oscillation is determined by the amplitude of the external force and the damping coefficient.

4. Resonance

Resonance is a phenomenon that occurs when the frequency of the external force is equal to the natural frequency of the mass-spring system. At resonance, the amplitude of the forced oscillation is maximum.

Examples of Oscillation

Oscillation is a common phenomenon in nature and engineering. Some examples of oscillation include:

• The motion of a pendulum
• The vibration of a guitar string
• The oscillation of a spring
• The rotation of the Earth

Oscillation is also used in a variety of devices, such as clocks, watches, and radios.

Simple Harmonic Motion

Simple Harmonic Motion (SHM) is a periodic motion where the restoring force is directly proportional to the negative displacement from the equilibrium position. It is a special case of periodic motion and is characterized by its sinusoidal nature.

Characteristics of SHM:

1. Restoring Force: The restoring force in SHM is always directed towards the equilibrium position and is proportional to the negative displacement. This means that the force acts to bring the oscillating object back to its equilibrium position.

2. Sinusoidal Motion: The displacement of an object undergoing SHM is a sinusoidal function of time. This means that the object moves back and forth along a straight line, with its position varying smoothly and periodically.

3. Amplitude: The amplitude of SHM is the maximum displacement of the object from its equilibrium position. It represents the extent of the object’s oscillation.

4. Period: The period of SHM is the time taken for the object to complete one full oscillation. It is the time taken for the object to move from its equilibrium position, to the maximum displacement in one direction, back to the equilibrium position, to the maximum displacement in the opposite direction, and finally back to the equilibrium position.

5. Frequency: The frequency of SHM is the number of oscillations completed in one second. It is the reciprocal of the period and is measured in Hertz (Hz).

Examples of SHM:

1. Mass-Spring System: A mass attached to a spring is a classic example of SHM. When the mass is pulled away from its equilibrium position and released, it will oscillate back and forth with a sinusoidal motion. The restoring force in this case is provided by the spring.

2. Pendulum: A pendulum swinging back and forth also undergoes SHM. The restoring force in this case is provided by gravity.

3. Sound Waves: Sound waves are mechanical waves that consist of oscillations in pressure. These oscillations can be represented as SHM, with the displacement being the variation in pressure.

4. Alternating Current (AC) Circuits: In AC circuits, the voltage and current vary sinusoidally with time. This sinusoidal variation can be represented as SHM, with the displacement being the voltage or current.

SHM is a fundamental concept in physics and has applications in various fields, including mechanics, acoustics, and electrical engineering. Understanding SHM is essential for analyzing and predicting the behavior of oscillating systems.

Different Types of Oscillation

Types of Oscillation

Oscillation is the repetitive motion of a body or system about a central point or position. There are many different types of oscillation, each with its own unique characteristics. Some of the most common types of oscillation include:

• Simple harmonic oscillation is the simplest type of oscillation, and it occurs when a body moves back and forth along a straight line. The motion of a pendulum is a simple harmonic oscillation.
• Damped oscillation is a type of oscillation in which the amplitude of the motion decreases over time. This is due to the presence of friction or other forces that oppose the motion. The motion of a spring-mass system is a damped oscillation.
• Forced oscillation is a type of oscillation in which the motion of the body is driven by an external force. The motion of a child on a swing is a forced oscillation.
• Resonance is a type of oscillation in which the amplitude of the motion increases dramatically when the frequency of the driving force matches the natural frequency of the system. The Tacoma Narrows Bridge collapse is an example of resonance.

Examples of Oscillation

There are many examples of oscillation in the world around us. Some of the most common examples include:

• The motion of a pendulum
• The motion of a spring-mass system
• The motion of a child on a swing
• The motion of a guitar string
• The motion of a wave

Applications of Oscillation

Oscillation has many applications in science, engineering, and everyday life. Some of the most common applications include:

• Clocks and watches
• Tuning forks
• Seismographs
• Sonar
• Radar
• Radio waves
• Microwaves
• Lasers

Oscillation is a fundamental phenomenon that plays an important role in many aspects of our world. By understanding the different types of oscillation and their applications, we can better understand the world around us and use it to our advantage.

Frequently Asked Questions – FAQs

Can a motion be oscillatory but not simple harmonic? Explain with valid reason.

Yes, a motion can be oscillatory but not simple harmonic. Simple harmonic motion is a special type of oscillatory motion in which the restoring force is directly proportional to the displacement from the equilibrium position and acts in the opposite direction. This means that the motion is periodic and the acceleration is always directed towards the equilibrium position.

On the other hand, oscillatory motion refers to any motion that repeats itself at regular intervals. This means that the position, velocity, and acceleration of the object undergoing oscillatory motion repeat themselves at regular intervals. However, oscillatory motion does not necessarily have to be simple harmonic.

For example, consider the motion of a pendulum. The pendulum swings back and forth, but the restoring force is not directly proportional to the displacement from the equilibrium position. Instead, the restoring force is proportional to the sine of the angle of displacement. This means that the motion of the pendulum is not simple harmonic, but it is still oscillatory.

Another example of oscillatory motion that is not simple harmonic is the motion of a spring-mass system. When a mass is attached to a spring and set into motion, the mass will oscillate back and forth. However, the restoring force is not directly proportional to the displacement from the equilibrium position. Instead, the restoring force is proportional to the amount of stretch or compression of the spring. This means that the motion of the spring-mass system is not simple harmonic, but it is still oscillatory.

In general, any motion that repeats itself at regular intervals is oscillatory motion. However, only those oscillatory motions in which the restoring force is directly proportional to the displacement from the equilibrium position and acts in the opposite direction are simple harmonic motions.

What is the basic condition for the motion of a particle to be a simple harmonic motion?

Simple harmonic motion (SHM) is a periodic motion where the restoring force is directly proportional to the negative displacement from the equilibrium position. The basic condition for the motion of a particle to be SHM is that the force acting on the particle must be a linear restoring force. This means that the force must be proportional to the negative displacement from the equilibrium position.

Mathematically, this can be expressed as:

$$F = -kx$$

Where:

• F is the force acting on the particle
• k is the spring constant
• x is the displacement from the equilibrium position

The negative sign indicates that the force is always directed towards the equilibrium position.

Some examples of SHM include:

• A mass-spring system, where the spring provides the restoring force.
• A pendulum, where the gravitational force provides the restoring force.
• A vibrating string, where the tension in the string provides the restoring force.

In each of these cases, the force acting on the particle is proportional to the negative displacement from the equilibrium position, and therefore the motion is SHM.

What happens to the oscillation of the body in damped oscillation?

Damped Oscillation

In damped oscillation, the amplitude of the oscillation decreases over time due to the dissipation of energy. This is in contrast to undamped oscillation, in which the amplitude remains constant.

The damping force is a force that opposes the motion of the oscillating object. It can be caused by friction, air resistance, or other factors. The greater the damping force, the faster the amplitude of the oscillation will decrease.

The equation of motion for a damped oscillator is:

$$m\frac{d^2x}{dt^2} + c\frac{dx}{dt} + kx = 0$$

where:

• m is the mass of the oscillating object
• c is the damping coefficient
• k is the spring constant

The solution to this equation is:

$$x(t) = e^{-\frac{ct}{2m}} A\cos(\omega t + \phi)$$

where:

• A is the amplitude of the oscillation
• ω is the angular frequency of the oscillation
• φ is the phase angle

The amplitude of the oscillation decreases exponentially with time, with a time constant of:

$$\tau = \frac{2m}{c}$$

The angular frequency of the oscillation is also affected by the damping force, and decreases with increasing damping.

Examples of Damped Oscillation

• A pendulum swinging in air will eventually come to a stop due to air resistance.
• A spring-mass system will eventually stop oscillating due to friction.
• A sound wave will eventually dissipate due to the absorption of sound energy by the air.

Damped oscillation is a common phenomenon in nature and engineering. It is important to understand the effects of damping in order to design systems that oscillate at the desired frequency and amplitude.

What is free oscillation?

Free oscillation is a type of periodic motion that occurs when a system is displaced from its equilibrium position and then released. The system will then oscillate back and forth around its equilibrium position, with the frequency of oscillation determined by the system’s natural frequency.

Free oscillations can occur in a variety of systems, including mechanical systems, electrical systems, and even biological systems. Some examples of free oscillations include:

• A pendulum swinging back and forth
• A mass on a spring bouncing up and down
• An electrical circuit with a capacitor and inductor oscillating between charging and discharging
• A population of animals fluctuating between periods of growth and decline

The frequency of a free oscillation is determined by the system’s natural frequency, which is a property of the system that depends on its physical characteristics. For example, the natural frequency of a pendulum is determined by the length of the pendulum, while the natural frequency of a mass on a spring is determined by the mass and stiffness of the spring.

Free oscillations can be damped, which means that the amplitude of the oscillations decreases over time. This can be caused by a variety of factors, such as friction, air resistance, or other forms of energy dissipation.

Free oscillations are important in a variety of applications, such as clocks, watches, and musical instruments. They are also used in a variety of scientific and engineering applications, such as measuring the properties of materials and designing control systems.

In free oscillation, what happens to the amplitude, frequency and energy of the oscillating body?

Amplitude:

In free oscillation, the amplitude of the oscillating body gradually decreases over time due to energy loss. This is because some of the energy is dissipated as heat due to friction and other resistive forces. As a result, the oscillations become smaller and smaller until they eventually stop.

Frequency:

The frequency of the oscillating body remains constant throughout the oscillation. This is because the frequency is determined by the physical properties of the system, such as the mass and stiffness of the spring.

Energy:

The total energy of the oscillating body also decreases over time due to energy loss. This is because some of the energy is dissipated as heat due to friction and other resistive forces. As a result, the oscillations become smaller and smaller until they eventually stop.

Examples:

• A pendulum swinging back and forth: The amplitude of the pendulum decreases over time due to air resistance and friction at the pivot point. The frequency of the pendulum remains constant, determined by the length of the pendulum and the acceleration due to gravity. The total energy of the pendulum decreases over time due to energy loss.
• A mass-spring system: The amplitude of the mass-spring system decreases over time due to friction between the mass and the surface it is oscillating on. The frequency of the mass-spring system remains constant, determined by the mass and the stiffness of the spring. The total energy of the mass-spring system decreases over time due to energy loss.

Why is the frequency of free oscillation called natural frequency?

Natural Frequency: The Inherent Rhythm of Oscillating Systems

When a physical system is disturbed from its equilibrium position and allowed to oscillate freely, it exhibits a characteristic frequency at which it vibrates. This inherent frequency is known as the natural frequency of the system. It is a fundamental property that arises from the system’s physical characteristics, such as mass, stiffness, and damping.

Understanding Natural Frequency through Examples:

1. Pendulum: Consider a simple pendulum consisting of a mass suspended from a string. When displaced from its equilibrium position and released, the pendulum swings back and forth at a specific frequency. This frequency depends on the length of the string and the mass of the bob. The longer the string or the heavier the bob, the slower the oscillations, resulting in a lower natural frequency.

2. Mass-Spring System: Imagine a mass attached to a spring. When pulled and released, the mass oscillates back and forth around its equilibrium position. The natural frequency of this system depends on the mass and the stiffness of the spring. A heavier mass or a stiffer spring leads to a higher natural frequency.

3. Musical Instruments: The natural frequencies of vibrating strings, air columns, or membranes determine the pitch of musical notes. For instance, in a guitar, each string has a specific natural frequency, producing a distinct musical note when plucked.

Significance of Natural Frequency:

1. Resonance: When an external force is applied to a system at its natural frequency, the system resonates, amplifying the oscillations. This phenomenon is crucial in various applications, such as tuning musical instruments, designing bridges to withstand earthquakes, and avoiding structural failures in buildings.

2. Energy Transfer: Systems oscillating at their natural frequency transfer energy efficiently. This principle is utilized in energy harvesting devices, where ambient vibrations are converted into electrical energy.

3. Stability Analysis: Natural frequency plays a vital role in stability analysis of engineering structures. Engineers ensure that the natural frequencies of structures are well below the frequencies of potential external disturbances to prevent resonance and potential catastrophic failures.

In summary, the natural frequency of a system represents its inherent tendency to oscillate at a specific frequency when disturbed. Understanding and considering natural frequencies is essential in various fields, including physics, engineering, and music, to design and analyze systems that perform optimally and safely.

Name a few damping forces.

Damping forces are forces that oppose the motion of an object. They can be caused by a variety of factors, including friction, air resistance, and viscosity.

Friction is a force that opposes the motion of two surfaces in contact with each other. It is caused by the interaction of the microscopic irregularities on the surfaces. Friction can be reduced by lubricating the surfaces or by using materials with a low coefficient of friction.

Air resistance is a force that opposes the motion of an object through the air. It is caused by the collision of the object with air molecules. Air resistance increases with the speed of the object and the density of the air.

Viscosity is a force that opposes the flow of a fluid. It is caused by the interaction of the molecules of the fluid. Viscosity increases with the temperature of the fluid and the size of the molecules.

Here are some examples of damping forces:

• The friction between a car’s tires and the road slows the car down.
• The air resistance of a skydiver slows them down as they fall.
• The viscosity of oil slows down the flow of oil through a pipe.

Damping forces are important in a variety of applications. They can be used to control the motion of objects, to reduce noise and vibration, and to improve the efficiency of machines.

In damping oscillation, why does the oscillation decrease exponentially?

In damping oscillation, the oscillation decreases exponentially due to the presence of a damping force. This force opposes the motion of the oscillating system and causes its energy to dissipate over time. The rate at which the oscillation decreases depends on the strength of the damping force.

Mathematically, the equation of motion for a damped harmonic oscillator is given by:

m(d^2x/dt^2) + c(dx/dt) + kx = 0

where:

• m is the mass of the oscillating object
• c is the damping coefficient
• k is the spring constant
• x is the displacement of the object from its equilibrium position

The solution to this equation is:

x(t) = e^(-ct/2m) * A * cos(ωt + φ)

where:

• A is the amplitude of the oscillation
• ω is the angular frequency of the oscillation
• φ is the phase angle

The exponential term e^(-ct/2m) represents the damping factor. It shows that the amplitude of the oscillation decreases exponentially with time. The rate of decay is determined by the damping coefficient c. A larger damping coefficient corresponds to a faster decay.

Here are some examples of damping oscillations:

• A pendulum swinging in air will eventually stop due to air resistance.
• A spring-mass system will eventually stop oscillating due to internal friction.
• A sound wave will eventually dissipate due to absorption by the medium through which it is traveling.

In each of these cases, the oscillation decreases exponentially due to the presence of a damping force.

What is the damping force?

The damping force is a force that opposes the motion of an object. It is caused by the interaction of the object with its surroundings, such as friction, air resistance, or the viscosity of a fluid. The damping force is always proportional to the velocity of the object, and it acts in the opposite direction to the motion.

The damping force can be expressed mathematically as follows:

F_d = -bv

where:

• F_d is the damping force
• b is the damping coefficient
• v is the velocity of the object

The damping coefficient is a measure of the strength of the damping force. The larger the damping coefficient, the greater the damping force.

The damping force can have a significant effect on the motion of an object. For example, a car with a high damping coefficient will come to a stop more quickly than a car with a low damping coefficient. This is because the damping force will oppose the motion of the car, causing it to slow down.

The damping force can also be used to control the motion of an object. For example, a shock absorber is a device that uses damping force to control the motion of a vehicle’s suspension. Shock absorbers help to keep the vehicle from bouncing up and down too much, which can make the ride more comfortable and safe.

Here are some examples of damping force in everyday life:

• The friction between a car’s tires and the road is a damping force that slows the car down.
• The air resistance that a cyclist experiences is a damping force that slows the cyclist down.
• The viscosity of water is a damping force that slows down a swimmer.

The damping force is an important force that can have a significant effect on the motion of objects. It is a force that is always present, and it can be used to control the motion of objects in a variety of ways.

What is forced oscillation?

Forced oscillation is a phenomenon in which an external force or disturbance causes a system to oscillate at a specific frequency. This is in contrast to free oscillation, where the system oscillates at its natural frequency without any external influence.

Forced oscillations occur when a system is subjected to a periodic force or disturbance. The frequency of the oscillation is determined by the frequency of the external force, and the amplitude of the oscillation depends on the strength of the force and the damping in the system.

Examples of forced oscillations include:

• A child on a swing being pushed by a parent
• A pendulum being driven by a clock mechanism
• A guitar string vibrating when plucked

In each of these cases, the external force (the parent pushing the swing, the clock mechanism driving the pendulum, or the finger plucking the guitar string) causes the system to oscillate at a specific frequency.

Forced oscillations can also occur in more complex systems, such as electrical circuits and mechanical systems. In these cases, the external force may be a voltage or current in an electrical circuit, or a mechanical force applied to a mechanical system.

The study of forced oscillations is important in many fields, including physics, engineering, and music. By understanding how forced oscillations occur, we can design systems that are resistant to unwanted oscillations or that can be used to produce desired oscillations.

Here are some additional examples of forced oscillations:

• The tides are caused by the gravitational force of the moon and sun on the Earth’s oceans.
• The Earth’s orbit around the sun is a forced oscillation caused by the gravitational force of the sun.
• The rotation of the Earth on its axis is a forced oscillation caused by the gravitational force of the moon.

In each of these cases, the external force (the gravitational force of the moon and sun, or the gravitational force of the sun) causes the system to oscillate at a specific frequency.

How does the damping affect amplitude in forced oscillation?

Damping is a force that opposes the motion of an oscillating system. It can be caused by friction, air resistance, or other factors. The amount of damping in a system is determined by the damping coefficient, which is a measure of the strength of the opposing force.

When damping is present, the amplitude of an oscillation decreases over time. This is because the opposing force does work on the system, which reduces its energy. The rate at which the amplitude decreases depends on the damping coefficient. The higher the damping coefficient, the faster the amplitude will decrease.

The effect of damping on amplitude can be seen in the following graph:

[Image of a graph showing the amplitude of an oscillation decreasing over time]

The blue curve represents the amplitude of an oscillation with no damping. The red curve represents the amplitude of an oscillation with damping. As you can see, the amplitude of the oscillation with damping decreases over time, while the amplitude of the oscillation with no damping remains constant.

Damping can also affect the frequency of an oscillation. In general, damping causes the frequency of an oscillation to decrease. This is because the opposing force slows down the motion of the system, which makes it take longer to complete each oscillation.

The effect of damping on frequency can be seen in the following graph:

[Image of a graph showing the frequency of an oscillation decreasing over time]

The blue curve represents the frequency of an oscillation with no damping. The red curve represents the frequency of an oscillation with damping. As you can see, the frequency of the oscillation with damping decreases over time, while the frequency of the oscillation with no damping remains constant.

Damping is an important factor to consider when designing oscillating systems. The amount of damping in a system can affect the amplitude, frequency, and stability of the oscillation. By carefully choosing the damping coefficient, it is possible to design an oscillating system that meets the desired specifications.

Here are some examples of how damping affects amplitude in forced oscillation:

• A car with shock absorbers has less damping than a car without shock absorbers. This is because the shock absorbers absorb some of the energy of the car’s oscillations, which reduces the amplitude of the oscillations.
• A pendulum with a long string has less damping than a pendulum with a short string. This is because the air resistance on the long string is less than the air resistance on the short string. The reduced air resistance means that the pendulum with the long string loses less energy to damping, which results in a higher amplitude of oscillation.
• A spring-mass system with a heavy mass has less damping than a spring-mass system with a light mass. This is because the heavy mass has more inertia, which means that it is more difficult to stop it from moving. The reduced damping means that the spring-mass system with the heavy mass has a higher amplitude of oscillation.

What is resonance?

Resonance is a phenomenon that occurs when a system is subjected to a periodic force that matches its natural frequency. This causes the system to vibrate with a greater amplitude than it would if the force were not present.

Examples of resonance:

• A swing: When you push a swing, you are applying a periodic force to it. If you push the swing at its natural frequency, it will swing with a greater amplitude than if you push it at a different frequency.
• A tuning fork: When you strike a tuning fork, it vibrates at its natural frequency. This causes the air around the tuning fork to vibrate at the same frequency, which produces a sound.
• A guitar string: When you pluck a guitar string, it vibrates at its natural frequency. This causes the soundboard of the guitar to vibrate at the same frequency, which produces a sound.

Resonance can also be destructive. For example, if a building is subjected to an earthquake, the ground may vibrate at a frequency that matches the building’s natural frequency. This can cause the building to collapse.

Resonance is a fundamental concept in physics and engineering. It has applications in a wide variety of fields, including music, acoustics, and structural engineering.

When can resonance be observed?

Resonance is a phenomenon that occurs when a system is subjected to a periodic force that matches its natural frequency. This can cause the system to vibrate with a greater amplitude than it would if it were subjected to a force of a different frequency.

Examples of resonance:

• A pendulum: When a pendulum is pulled back and released, it will swing back and forth with a natural frequency that depends on its length. If you push the pendulum with a frequency that matches its natural frequency, it will swing with a greater amplitude than if you push it with a different frequency.
• A guitar string: When a guitar string is plucked, it will vibrate with a natural frequency that depends on its length, tension, and mass. If you pluck the string with a frequency that matches its natural frequency, it will vibrate with a greater amplitude than if you pluck it with a different frequency.
• A building: When a building is subjected to an earthquake, it can resonate with the earthquake’s frequency. This can cause the building to collapse.

Resonance can be observed in a variety of systems, including mechanical, electrical, and acoustic systems.

In mechanical systems, resonance can occur when a system is subjected to a periodic force that matches its natural frequency. This can cause the system to vibrate with a greater amplitude than it would if it were subjected to a force of a different frequency.

In electrical systems, resonance can occur when a circuit is subjected to an alternating current (AC) voltage that matches the circuit’s natural frequency. This can cause the current in the circuit to increase to a dangerous level.

In acoustic systems, resonance can occur when a sound wave with a frequency that matches the natural frequency of a room or object is produced. This can cause the sound wave to be amplified, creating a loud and unpleasant sound.

Resonance can be a destructive force, but it can also be used for beneficial purposes. For example, resonance is used in musical instruments to produce sound, and it is also used in some medical devices to treat certain conditions.

What is the nature of the frequency in forced oscillation?

Forced Oscillation

Forced oscillation is a type of oscillation that occurs when an external force is applied to a system. The frequency of a forced oscillation is determined by the frequency of the applied force.

Examples of Forced Oscillation

• A pendulum that is pushed back and forth by a hand
• A child on a swing that is pushed by a parent
• A car that is driven over a bumpy road

Nature of the Frequency in Forced Oscillation

The frequency of a forced oscillation is always the same as the frequency of the applied force. This is because the system is simply responding to the force that is being applied to it.

Resonance

When the frequency of the applied force is close to the natural frequency of the system, the system will experience resonance. Resonance is a condition in which the amplitude of the oscillation is greatly increased.

Damping

Damping is a force that opposes the motion of a system. Damping can reduce the amplitude of an oscillation.

Applications of Forced Oscillation

Forced oscillation is used in a variety of applications, including:

• Tuning musical instruments
• Designing shock absorbers
• Controlling the speed of motors
• Generating electricity

Can a motion be periodic and not oscillatory?

A motion can be periodic but not oscillatory if it repeats itself at regular intervals but does not involve oscillations around a central point. Here are a few examples:

1. Circular Motion: An object moving in a circular path at a constant speed exhibits periodic motion. The object’s position repeats itself every time it completes one revolution, but it does not oscillate back and forth.

2. Elliptical Motion: Similarly, an object moving in an elliptical path also undergoes periodic motion. The object’s position repeats itself after each complete orbit, but it does not oscillate along a straight line.

3. Simple Harmonic Motion with Phase Shift: Simple harmonic motion is typically oscillatory, but it can become periodic and non-oscillatory if there is a phase shift. For instance, consider a mass-spring system where the initial position of the mass is not at the equilibrium point. The motion will be periodic, but it will not oscillate around the equilibrium position.

4. Pendulum with Large Amplitude: A pendulum swinging with a large amplitude can exhibit periodic but non-oscillatory motion. As the pendulum swings, it passes through the equilibrium position with increasing speed, and the motion becomes more circular. At the highest point of its swing, the pendulum momentarily stops before reversing its direction. This motion is periodic but not oscillatory.

5. Rotation of a Rigid Body: The rotation of a rigid body, such as a wheel or a gear, is periodic but not oscillatory. The body’s orientation repeats itself after each complete revolution, but it does not oscillate back and forth.

In summary, periodic motion refers to the repetition of a motion at regular intervals, while oscillatory motion involves oscillations around a central point. Some types of motion can be periodic but not oscillatory, such as circular motion, elliptical motion, and certain cases of simple harmonic motion and pendulum motion.