I am Prajakta Gharat. I have completed Post Graduation in physics in 2020. Currently I am working as a Subject Matter Expert in Physics for Lambdageeks. I try to explain Physics subject easily understandable in simple way.
By knowing the concept of damping, we must understand the difference between overdamped vs critically damped oscillations.
To understand overdamped vs critically damped, one can say that a system that is overdamped goes slowly toward equilibrium, whereas a system that is critically damped moves as swiftly as possible toward equilibrium without fluctuating about it.
Now let us see a table below where all information has summarize to make a comparative analysis of overdamped vs critically damped oscillations.
Overdamped vs Critically Damped Oscillation:
Overdamped
Critically damped
Overdamping occurs when oscillations come to a halt after a significant period of time has passed since the resistive force was applied.
In oscillatory system, the oscillations come to a halt as soon as critical damping is reached.
If a system responds to a step-change input by taking up a new position, it can either fluctuate around the final position before settling to the new value, or it can gradually approach the new value over time.
At a given level of damping, the system does not actually oscillate; however, it may slightly exceed before returning to the final value.
By solving damped harmonic oscillator, the case of overdamping is given by, b2>4mk
By solving damped harmonic oscillator, the case of critical damping is given by, b2=4mk
In the case of Overdamping b is comparatively large than m and k
In the case of Critical damping b is just between over and underdamping
The roots of overdamping are real and distinct. Because the roots are real, overdamping is the simplest situation to solve mathematically.
The roots of critically damped oscillator are real and same.
The characteristic roots can be given as, -b+√(b2-4mk)/2m r2=-b-√(b2-4mk)/2m
The characteristic roots of critical damping are given as, -b/2m, -b/2m.
The general solution for a critically damped oscillation can be given as follows:
Where:
is the displacement at time ( t ).
and are constants determined by the initial conditions of the system.
is the damping coefficient.
( e ) is the base of the natural logarithm.
This is the detailed comparative analysis of overdamped vs critically damped oscillation.
Best example of swing illustrating Overdamped Vs Critically damped
If a system responds to a step-change input by taking up a new position, it can either fluctuate around the final position, finally settling to the new value, or it can steadily approach the new value, taking its time.
The system doesn’t truly oscillate at a certain level of damping; however, it may slightly overshoot before immediately returning to the final value. This is critical dampening, and it’s typically the goal.
Overdamped Vs Critically Damped
Damped Oscillator:
We know the damped harmonic oscillator equation can be given as:
…..(1)
With m > 0, b ≥ 0 and k > 0. It has characteristic equation
ms2+bs+k=0………. (2)
With characteristic roots
Depending on the sign of the term under the square root, there are three possibilities:
b2 < 4mk (This is the case of Underdamping as b is comparatively small than m and k)
b2 > 4mk (This is the case of Overdamping as b is comparatively large than m and k)
b2 = 4mk(This is the case of Critical damping as b is just between over and underdamping)
Overdamping is the simplest situation to solve mathematically since the roots are real. Most people, however, perceive the oscillatory behaviour of a damped oscillator.
Here we will see the case of Overdamping and critical damping as we have to do comparative analysis of overdamped vs critically damped oscillation.
Overdamping (real and distinct roots):
When b2 > 4mk , then the value under the square root will be positive and the characteristic roots will be real and distinct. In case of b2 > 4mk the damping constant b should be comparatively large.
One thing to remember is that in this situation, the roots are both negative. You can know this by looking at equation (2). Because the quantity under the square root is assumed to be positive, the roots are real.
By using these roots to solve the equation (1),
The characteristic roots are:
Exponential solutions are:
Therefore, the general solution can be given as:
Let’s take a look at this from a physical point of view. When the damping is high, the frictional force is so high that the system cannot oscillate. Unusually, an unforced overdamped harmonic oscillator does not oscillate. Because both exponents are negative, any solution in this situation approaches x = 0 asymptotically.
Many doors have a spring at the top that closes them automatically. The spring is damped to control the rate at which the door closes. If the damper is powerful enough to overdampen the spring, the door will simply settle back to its mean position (i.e., closed) without oscillating, which is normally what is desired in this situation.
Critical Damping (real and same roots):
When b2 = 4mk, then the value under the square root becomes 0 and the characteristic polynomials has same roots -b/2m , -b/2m.
Now by using the roots to solve equation (1) in this situation. Because we only have one exponential answer, we must multiply it by t to obtain the second.
Therefore, basic solutions are:
And the general solutions can be given as:
This does not fluctuate like the overdamped situation. It’s worth mentioning that picking b as the critical damping value for a fixed m and k results in the quickest return of the system to its equilibrium state.
This is frequently a desired feature in engineering design. This can be observed by verifying the roots, but we won’t go over the algebra that illustrates it.
Damped oscillation and forced oscillation are two different types of oscillatory motion. In damped oscillation, the amplitude of the oscillation gradually decreases over time due to the presence of damping forces, such as friction or air resistance. This results in the oscillation eventually coming to a stop. On the other hand, forced oscillation occurs when an external force is applied to a system, causing it to oscillate at a frequency determined by the force. The amplitude of the forced oscillation can vary depending on the frequency and magnitude of the applied force.
Key Takeaways
Damped Oscillation
Forced Oscillation
Amplitude decreases over time
Amplitude can vary
Damping forces present
External force applied
Oscillation comes to a stop
Oscillation continues
Frequency determined by system
Frequency determined by force
Understanding Oscillations
Oscillations are a fascinating phenomenon that can be observed in various systems, from mechanical systems to electrical circuits. They involve the repetitive back-and-forth motion of an object or a system around a central position. In simpler terms, oscillations refer to the regular swinging or vibrating motion of an object.
Definition of Oscillations
Oscillations can be defined as the periodic motion of an object or a system between two extreme points or positions. This motion is characterized by the presence of a restoring force that brings the object back to its equilibrium position. The restoring force acts in the opposite direction to the displacement of the object, causing it to oscillate around the equilibrium point.
In the context of oscillations, several key terms are important to understand:
Amplitude: The maximum displacement of an oscillating object from its equilibrium position.
Periodic Force: An external force that is applied periodically to an oscillating system, causing it to oscillate.
Restoring Force: The force that acts on an object or a system, bringing it back to its equilibrium position.
Oscillation Frequency: The number of complete oscillations or cycles that occur in a given time period.
Oscillation Period: The time taken for one complete oscillation or cycle to occur.
Phase Difference: The difference in phase between two oscillating objects or systems.
Damping Force: The force that opposes the motion of an oscillating object, leading to energy dissipation and a decrease in amplitude.
Damping Coefficient: A measure of the damping force in an oscillating system.
Damping Ratio: The ratio of the actual damping coefficient to the critical damping coefficient.
Critical Damping: The damping condition where the oscillating system returns to its equilibrium position without any oscillation.
Underdamping: The damping condition where the oscillating system experiences oscillations that gradually decrease in amplitude.
Overdamping: The damping condition where the oscillating system returns to its equilibrium position without oscillating, but with a slower rate of convergence.
Transient State: The initial phase of an oscillation where the system’s behavior is influenced by its initial conditions.
Steady State Oscillation: The long-term behavior of an oscillating system after the transient state has passed.
Natural Frequency: The frequency at which an oscillating system tends to oscillate in the absence of any external force.
Resonance: The phenomenon where an oscillating system is forced to oscillate at its natural frequency by an external force.
Resonance Frequency: The frequency at which resonance occurs in an oscillating system.
Harmonic Oscillator: A system that exhibits simple harmonic motion, where the restoring force is directly proportional to the displacement.
Types of Oscillations
Oscillations can be classified into different types based on various factors. Some common types of oscillations include:
Free Oscillation: Also known as natural or unforced oscillation, it occurs when an oscillating system is left to oscillate on its own without any external force.
Driven Oscillation: This type of oscillation occurs when an external force is continuously applied to an oscillating system, causing it to oscillate at a frequency different from its natural frequency.
Forced Vibration: When an oscillating system is subjected to an external force that matches its natural frequency, it undergoes forced vibration, resulting in large amplitude oscillations.
Mechanical Resonance: The phenomenon where an oscillating system vibrates with maximum amplitude at its natural frequency due to the resonance effect.
Oscillation System: A system that exhibits oscillatory motion, such as a pendulum, a mass-spring system, or an electrical LC circuit.
Understanding oscillations is crucial in various fields, including physics, engineering, and even music. By studying the behavior of oscillating systems, we can gain insights into the fundamental principles that govern the motion of objects and systems in our world.
Damped Oscillations
Definition and Explanation of Damped Oscillations
Damped oscillations refer to a type of oscillatory motion where the amplitude of the oscillations gradually decreases over time due to the presence of a damping force. In simple terms, it is the motion of a system that experiences energy dissipation, causing the oscillations to gradually come to a stop.
To understand damped oscillations better, let’s consider a harmonic oscillator, which is a mechanical system that exhibits oscillatory motion. In a harmonic oscillator, there are two main forces at play: the restoring force and the damping force. The restoring force acts to bring the system back to its equilibrium position, while the damping force opposes the motion and dissipates energy.
The behavior of damped oscillations is influenced by various factors, including the damping coefficient, the mass of the system, and the external forces acting on it. The damping coefficient determines the strength of the damping force, while the mass affects the natural frequency of the system. When the damping force is relatively weak compared to the restoring force, the system exhibits underdamping. On the other hand, if the damping force is too strong, the system shows overdamping. Critical damping occurs when the damping force is just enough to prevent oscillations from continuing indefinitely.
Factors affecting Damped Oscillations
Several factors can affect the behavior of damped oscillations:
Damping coefficient: The damping coefficient determines the strength of the damping force. A higher damping coefficient leads to faster energy dissipation and a quicker decay of the oscillations.
Mass of the system: The mass of the system affects the natural frequency of the oscillations. A higher mass results in a lower natural frequency, which in turn affects the rate at which the oscillations decay.
External forces: The presence of external forces can influence the behavior of damped oscillations. Periodic forces with frequencies close to the natural frequency of the system can cause resonance, leading to larger oscillations.
Real-world examples of Damped Oscillations
Damped oscillations can be observed in various real-world phenomena. Here are a few examples:
Pendulum: A swinging pendulum experiences damping due to air resistance. Over time, the pendulum‘s oscillations gradually decrease in amplitude until it comes to a stop.
Car suspension system: The suspension system of a car undergoes damped oscillations when encountering bumps or uneven road surfaces. The damping force helps absorb the energy and prevents excessive bouncing.
Musical instruments: Instruments like pianos, guitars, and drums exhibit damped oscillations when their strings or membranes are struck. The damping force helps control the decay of sound and prevents prolonged vibrations.
Forced oscillations refer to the phenomenon where an oscillating system is subjected to an external periodic force, causing it to deviate from its natural frequency and amplitude. In simple terms, it is the forced vibration of a system that is driven by an external force.
When a mechanical oscillation system is subjected to a periodic force, it undergoes oscillatory motion known as forced oscillations. This external force can be of any frequency and amplitude, and it can either be in phase or out of phase with the system’s natural frequency. The system responds to this external force by oscillating with a frequency equal to the frequency of the applied force.
The behavior of forced oscillations is influenced by various factors, including the damping force, energy dissipation, and the natural frequency of the system. Let’s explore these factors in more detail.
Factors affecting Forced Oscillations
Damping Force: The damping force in a system plays a crucial role in forced oscillations. It determines the rate at which energy is dissipated from the system. The damping force can be classified into three categories: underdamping, overdamping, and critical damping. Underdamping occurs when the damping force is less than the critical damping, resulting in oscillations with a decreasing amplitude. Overdamping occurs when the damping force is greater than the critical damping, leading to slow decay of oscillations. Critical damping occurs when the damping force is equal to the critical damping, resulting in the fastest decay of oscillations.
Natural Frequency: The natural frequency of an oscillating system is the frequency at which it vibrates in the absence of any external force. When a periodic force is applied to the system, it can either be in resonance or out of resonance with the natural frequency. Resonance occurs when the frequency of the external force matches the natural frequency of the system, leading to a significant increase in the amplitude of oscillations. Out of resonance occurs when the frequency of the external force is different from the natural frequency, resulting in smaller amplitudes.
Amplitude and Phase Difference: The amplitude of forced oscillations depends on the amplitude of the external force. If the external force has a large amplitude, the oscillations will also have a large amplitude. The phase difference between the external force and the system’s response also affects the behavior of forced oscillations. In-phase forces result in maximum energy transfer, while out-of-phase forces result in energy cancellation.
Real-world examples of Forced Oscillations
Forced oscillations can be observed in various real-world scenarios. Here are a few examples:
Pendulum Clock: The swinging motion of a pendulum clock is an example of forced oscillations. The periodic force applied by the clock mechanism keeps the pendulum oscillating at a constant frequency.
Musical Instruments: When a musician plays a musical instrument, the strings or air columns in the instrument are forced to vibrate at specific frequencies, producing different notes. The musician controls the external force applied to the instrument to create the desired sound.
Suspension System in Vehicles: The suspension system in vehicles is designed to dampen the oscillations caused by uneven road surfaces. The system uses springs and dampers to absorb the external forces and minimize the impact on the vehicle’s body.
Difference between Damped Oscillation and Forced Oscillation
Damped oscillation and forced oscillation are two types of mechanical oscillations that exhibit different behaviors and characteristics.
Comparative Analysis of Damped and Forced Oscillations
Damped oscillation refers to the oscillatory motion of a system that experiences energy dissipation due to the presence of a damping force. This damping force causes the amplitude of the oscillation to decrease over time, eventually bringing the system to a rest. In contrast, forced oscillation occurs when an external force is applied to a system, causing it to oscillate at a frequency different from its natural frequency.
One key difference between damped and forced oscillations lies in their energy behavior. In damped oscillation, energy is gradually dissipated due to the damping force, resulting in a decrease in the amplitude of the oscillation. On the other hand, in forced oscillation, energy is continuously supplied to the system by the external force, allowing the oscillation to persist.
Another difference is observed in the response of the system to the applied force. In damped oscillation, the system’s response is influenced by both the damping force and the external force. The amplitude of the oscillation is determined by the balance between these two forces. In forced oscillation, the amplitude of the oscillation is primarily determined by the characteristics of the external force, such as its frequency and magnitude.
How Damping affects Forced Oscillations
The presence of damping in a forced oscillation system can significantly affect its behavior. The damping force can modify the amplitude, phase, and frequency response of the system. When the damping force is small, the system exhibits underdamping, where the amplitude of the oscillation is reduced but the frequency remains close to the natural frequency. In the case of overdamping, the system takes a longer time to return to its equilibrium position after being displaced.
The damping ratio, which represents the ratio of the actual damping to the critical damping, plays a crucial role in determining the response of the system. A higher damping ratio leads to a faster decay of the amplitude and a wider frequency response. Conversely, a lower damping ratio results in a slower decay of the amplitude and a narrower frequency response.
The role of external force in Damped and Forced Oscillations
In damped oscillation, the external force is not required for the system to oscillate. The system can undergo undriven oscillation, where it oscillates naturally at its own frequency. However, the presence of an external force can still affect the behavior of the system, altering its amplitude and phase.
In forced oscillation, the external force is essential for the system to oscillate. The system responds to the periodic force by oscillating at the frequency of the applied force. The amplitude of the forced oscillation depends on the frequency of the external force and the resonance frequency of the system. When the frequency of the external force matches the resonance frequency, the system exhibits resonance, resulting in a significant increase in the amplitude.
Special Case: Free Damped vs Forced Oscillations
Understanding Free Damped Oscillations
In the realm of mechanical oscillations, we encounter two fascinating phenomena: free damped oscillations and forced oscillations. Let’s delve into the intricacies of free damped oscillations first.
Free damped oscillations occur when a mechanical system, such as a harmonic oscillator, undergoes oscillatory motion in the absence of any external force. The motion is influenced by a damping force, which leads to energy dissipation over time. This damping force arises due to various factors like friction, air resistance, or other dissipative forces present in the system.
The behavior of free damped oscillations is characterized by the system’s natural frequency, damping ratio, and initial conditions. The natural frequency represents the frequency at which the system oscillates in the absence of damping. It is determined by the mass and stiffness of the system.
The amplitude of the oscillation gradually decreases over time due to the energy dissipation caused by the damping force. Eventually, the system reaches a state of equilibrium known as the steady state oscillation. In this state, the amplitude remains constant, and the system exhibits periodic motion.
The damping ratio plays a crucial role in free damped oscillations. It determines the type of damping present in the system: underdamping, overdamping, or critical damping. Underdamping occurs when the damping ratio is less than 1, resulting in oscillations with a gradually decreasing amplitude. Overdamping, on the other hand, occurs when the damping ratio is greater than 1, leading to slower and smoother oscillations. Critical damping occurs when the damping ratio is exactly 1, resulting in the fastest return to equilibrium without any oscillations.
Comparing Free Damped Oscillations with Forced Oscillations
Now that we have a good understanding of free damped oscillations, let’s compare them with forced oscillations.
Forced oscillations occur when a periodic force is applied to a mechanical system, causing it to oscillate at a frequency different from its natural frequency. This external force can be of various forms, such as vibrations, sound waves, or any other form of disturbance.
In forced oscillations, the system responds to the applied force by oscillating at the frequency of the external force. The amplitude of the oscillation depends on the resonance frequency, which is the frequency at which the system responds most strongly to the external force. When the resonance frequency matches the frequency of the external force, the system exhibits resonance, resulting in a significant increase in the amplitude of the oscillation.
One key difference between free damped oscillations and forced oscillations is the presence of the external force in the latter. While free damped oscillations occur naturally in the absence of any external force, forced oscillations require an external force to induce the oscillatory motion.
Damped oscillation occurs when an oscillating system gradually loses energy due to the presence of a damping force. This results in the amplitude of the oscillation decreasing over time until it eventually comes to rest. Damped oscillation is commonly observed in systems such as a swinging pendulum or a vibrating spring with friction.
On the other hand, forced oscillation occurs when an external force is applied to an oscillating system. This external force drives the system to oscillate at a specific frequency, known as the driving frequency. The amplitude of the forced oscillation depends on the frequency and magnitude of the applied force.
While both damped and forced oscillations involve the motion of an object back and forth, they differ in terms of the energy loss and the presence of an external driving force. Understanding these differences is crucial in various fields, including physics, engineering, and even music.
What is the difference between damped oscillation and forced oscillation, and how does it relate to the concept of overdamped and critically damped?
Damped oscillation refers to the phenomenon where the amplitude of an oscillating system gradually decreases over time due to energy dissipation. On the other hand, forced oscillation occurs when an external force causes a system to oscillate at a specific frequency. The concepts of overdamped and critically damped are related to damped oscillation and describe different behavior patterns. Difference between overdamped and critically damped. In overdamped systems, the damping force is greater than necessary to bring the system to equilibrium, resulting in slower decay and no oscillation. Critically damped systems reach equilibrium in the shortest possible time without any oscillation. Both these concepts illustrate different ways in which damping affects the behavior of oscillating systems.
Frequently Asked Questions
What is the difference between damped oscillation and forced oscillation?
Damped oscillation refers to the oscillatory motion where the amplitude of oscillation decreases over time due to the presence of a damping force, which leads to energy dissipation. On the other hand, forced oscillation is when an external force drives the oscillation at a frequency that may be different from the system’s natural frequency.
How do the differences between damped and forced oscillation affect the amplitude of the oscillation?
In damped oscillation, the amplitude decreases over time due to energy dissipation caused by the damping force. However, in forced oscillation, the amplitude is determined by the balance between the driving force and the damping force. If the driving force’s frequency matches the system’s natural frequency, the amplitude can increase significantly, a phenomenon known as resonance.
What is the difference between free damped and forced oscillations?
Free damped oscillation is a type of oscillatory motion where there is no external force acting on the system, and the amplitude decreases over time due to the damping force. On the contrary, forced oscillation occurs when an external force drives the system, and the amplitude does not necessarily decrease over time.
How does the damping ratio affect the type of damping in a mechanical oscillation?
The damping ratio determines the type of damping in a mechanical oscillation. If the damping ratio is less than 1, it’s underdamping, and the system oscillates with a gradually decreasing amplitude. If the damping ratio equals 1, it’s critical damping, and the system returns to equilibrium as quickly as possible without oscillating. If the damping ratio is greater than 1, it’s overdamping, and the system returns to equilibrium without oscillating but slower than in critical damping.
What is the difference between undriven and driven oscillation?
Undriven oscillation, also known as free oscillation, occurs when no external force is applied to the system after it is displaced from its equilibrium position. The frequency of this oscillation is the natural frequency of the system. Driven oscillation, also known as forced oscillation, occurs when an external force drives the system at a frequency that can be different from its natural frequency.
How does the oscillation period relate to the natural frequency and amplitude in simple harmonic motion?
In simple harmonic motion, the oscillation period is the time it takes for one complete cycle of oscillation. It is inversely proportional to the natural frequency of the system and is independent of the amplitude.
How does the restoring force contribute to the oscillatory motion?
The restoring force is the force that brings a system back to its equilibrium position. In oscillatory motion, it is proportional to the displacement from the equilibrium position and acts in the opposite direction. This force is responsible for the system’s tendency to oscillate around its equilibrium position.
What is the role of the damping coefficient in the oscillation equation?
The damping coefficient is a parameter in the oscillation equation that represents the amount of damping in the system. It determines how quickly the oscillations die out. A larger damping coefficient means more rapid energy dissipation and quicker damping of oscillations.
How does resonance occur in a forced vibration system?
Resonance in a forced vibration system occurs when the frequency of the external force matches the natural frequency of the system. This causes the amplitude of the oscillation to increase significantly, leading to large oscillations.
What is the significance of phase difference in steady state oscillation?
The phase difference in steady state oscillation refers to the difference in phase between the driving force and the response of the system. It provides information about how much the response of the system lags or leads the driving force. This phase difference depends on the damping and the difference between the driving frequency and the system’s natural frequency.
In the actual world, oscillations do not always follow the proper SHM pattern. In most cases, friction of some kind results in damping oscillations. Let us see some damped oscillation examples as follows:
One can witness in any common science laboratory that the oscillations occur when some mass m is coupled to a spring with a force constant ‘k’. When the spring is compressed or released from some distance then one can observe the oscillations taking place. These oscillations take place as a result of energy stored in the spring.
Eventually, the oscillations decays and finally the spring stops oscillating at some point as a result of air friction. This decay in the oscillations is nothing but a damping of oscillations. This is the most common damped oscillation example.
Kids on spring horse:
The oscillations produced by children seated on spring horses in the park are something we see on a regular basis. Once the horse has been brought back and freed, it is possible to witness the child sitting on the horse moving back and forth, which is equivalent to performing oscillations. Eventually, it slows down and finally comes to a complete stop, which is nothing but the damping of oscillations in the spring horse.
If we swing a pendulum with a specific length of string, we will see that it achieves its greatest height during the first oscillation and then steadily drops in height as the number of oscillations increases. This is owing to the presence of opposing forces such as air drag. The moment arrives when the pendulum comes to a complete halt. In this case, the vibration is being dampened, or we may say that it is losing energy.
When a person hops off a bridge or a platform, a long, elastic rope is tied to the ankles of the individual, which causes a series of vertical oscillations to be generated on the bridge or platform. These vertical oscillations will continue to occur as long as the elastic rope has energy. And, once it has used all of its energy, it causes oscillations to be dampened. Bungee jumping is one of the best damped oscillations examples.
Have you ever observed the person who is standing on the diving board in the swimming pool? When a person is standing on the diving board, ready to jump into the pool, you must have seen that the diving board bends downward. The bending of the board indicates that the energy is being stored in the board itself.
When the individual jumps off the diving board, he or she flies a little high in the air before diving into the water as an act of the force. After that, we can see that the diving board is still oscillating a little bit after it has taken off. Damping oscillation is a phenomenon in which stored energy in a diving board gradually diminishes and eventually ceases, as demonstrated by the diving board.
Have you ever taken a trip to an amusement park or a water park? Yes. Of course, we all did. The pirate ship trip is something we always see and appreciate. In some parts of the world, a pirate ship is referred to as a dragon boat, and vice versa.
The engine causes the ride to oscillate back and forth motion when it is turned on. Eventually, these oscillations come to a halt, which is nothing more than a damping of oscillations.
In electronics, damping-driven oscillation is a common phenomenon. An electronic damped driven oscillator is a fundamental component in a variety of applications. Let’s have a look at a series RLC circuit to see how damped driven oscillator works in electronics.
If the capacitor is charged to its maximum capacity and the voltage source is withdrawn from the circuit, a capacitor will discharge and current will flow through the inductor in a closed circuit. According to Lenz Law, which opposes current flow, the inductor will accumulate stored magnetic energy.
Once the capacitor has been entirely discharged, the inductor’s magnetic field causes a reversal of the current flowing through the capacitor, which charges it in the opposite way. This course of action will be repeated, but it will be dampened by the resistive parts in the circuit.
To compensate for the energy loss, a sinusoidal signal source is required to maintain the RLC circuit’s oscillation. A damped driven oscillator is formed as a result of the addition of the signal source to the RLC circuit.
String instruments:
There are so many string instruments from which we are very much familiar with guitar and violin. When we pluck a string of a guitar or rub a bow on the string of the violin, then we can hear melodious sounds.
This sound is caused by the up and down vibrations of the string of the respective instrument. After some time has passed, it is found that the strings cease to vibrate, demonstrating the phenomena of damping of oscillation.
Image Credits: eyes355, with elements byː Lardyfatboy, Wayne Rogers, Rama, Martin Möller, Gringer, Musik- och teatermuseet, Stringed Instruments, CC BY-SA 3.0
Swinging on a playground swing:
We all have been swinging since we were able to walk, whether in a baby swing at home or a kiddie one on the playground, or any other swing in the garden. To take a swing, we move a little backward with the help of our legs and then release or set free our legs.
This process acts as a force and stores energy which then results in initializing the oscillations. These to and fro oscillations eventually slow down as the effect of air resistance and hence the oscillations get dampened.
When we simply press and release the spring, it will eventually return to its original position after some compression and relaxation have occurred. This is also the damping of oscillations.
Understanding pendulum physics is important in order to understand motion, gravity, inertia, and centripetal force. Let us see the examples of pendulum uses in our surroundings.
The majority of children have been on swings since they were able to walk, whether it was in an infant swing at home, a toddler swing at a playground, or on a swing set in their garden. Here, in this case, the child acts as a mass suspended by the ropes on both sides of the attached seat of the swing, which can move freely once the motion is set. This motion can be initiated by taking a swing a little backward with the help of the leg and then releasing it.
Depending on the clock’s age and style, a huge pendulum or a quartz crystal vibrates to keep time. A pendulum is used to maintain the accuracy of a mechanical clock. The length of the pendulum and the force of gravity both influence the time it takes for the pendulum to swing, which is referred to as the period.
In order to operate a gear system, the top end of the pendulum’s arm is attached to a mechanism, while the bottom end is connected to the ground. The gears are responsible for driving the hands of the clock. Friction causes a little amount of the pendulum’s motion to be lost; this is compensated for by a wind-up spring or weights. Because they are in Simple Harmonic Motion, the oscillators have a constant period of oscillation.
This enables them to maintain an accurate track of time. This precise moment is important for more than simply convenience.
Now, the pendulum uses can be seen in the sea dragon or the huge boat swing at the amusement park. Keep in mind that a pendulum is made up of an object with a bob that is hanging from the end of a rod or string, allowing the object to freely swing in the opposite direction. Newton’s first rule states that an element at rest will remain at rest (if there is no outside interference), which means that a motor must be used to lift the amusement park ride into the air.
Then gravity takes over and brings the ride back to the ground. The ride is pushed forward by inertia, which keeps it moving. With the aid of inertia and gravity, the ride rises and falls in elevation. Friction, which is provided by the brakes, is the only item that can bring the ride to a complete stop.
To tell the time, a Foucault pendulum is used independently of any other device. It’s commonly constructed with a large metal ball that’s linked to a lengthy cable. In order for the pendulum to be able to swing freely in any vertical plane, the wire must be suspended from a vantage point on the ceiling. When the ball is safely released, it swings back and forth, but as time passes, the Earth’s rotation alters the direction of the swing.
In one day, the pendulum will swing around the poles and complete a full round on the ground. At the equator, the Earth has no effect on it; it will continue to swing in the same direction indefinitely. At intermediate locations, it will cover a portion of a circle in a single day, with the amount of time covering the circle increasing with latitude. Given enough information about the latitude, the location of the pendulum may be used to determine the current time of day.
A metronome is a device that creates rhythmic ticks in order to assist musicians in maintaining a consistent tempo when playing a composition. In this case, the structure is a variant of the pendulum. The oscillating arm is fastened at the bottom of the frame in this instance. Within the metronome, a fixed counterweight is used to counterbalance a second weight that is attached to the oscillating arm.
The period may be adjusted by moving the counterweight up and down the arm, so altering how quickly the metronome oscillates. Gravity, acting on the fixed counterweight, serves as the restoring force in this instance. The closer the weight is to the bottom of the arm, the faster the arm will swing and create ticks, and the more ticks it will generate.
In this technique, a long, elastic rope is linked to the ankles of a person, who then hops off a bridge or a platform, causing a number of vertical oscillations to be generated. A very precise calculation is made to determine the amplitude of the oscillations since a calculation error might result in the loss of a life. To avoid harm, this sport should only be practised with the utmost safety measures possible.
This oscillating system is referred to as a cantilever, which itself is described as a rigid structure that is only fixed at one end. SHM is performed on the diving board and the person on the edge of it as he or she hops up and down on the end of the diving board’s end cape.
Several factors influence this motion, including the force constant of the board (a stronger board would not oscillate as much) and the weight of the person on the board (The amplitude of oscillation will be increased in proportion to the weight of the individual.)
A wrecking ball, which is used to demolish structures, is yet another example of pendulum motion. After securing the wrecking ball with a sturdy cable and guiding it toward the structure to be demolished, a skilled crane operator swings and releases the wrecking ball. Energy is accumulated throughout the upswing and released when the ball makes contact with an object.
Even your own legs respond in a similar manner to pendulums. In fact, allowing your legs to swing at their natural velocity when walking is the most effective method of transportation. How long it takes your leg to accomplish its back and forth movement is determined by the length of your legs. That’s why people with long legs sometimes appear to meander around, while people with short legs appear to walk quickly.
A ballistic pendulum is a huge block of wood that is suspended from ropes and has been in use by police agencies for many years. The weight of the wood may be calculated with precision. A technician shoots a bullet through the wall of the building. The bullet embeds itself in it, causing it to go into action. The bullet’s momentum and energy are represented by the point at which it swings backwards the furthest. Using the bullet’s mass and velocity, the technician may then calculate the bullet’s velocity.
To summarize, A pendulum is simply a weight that is suspended from a pivot and allows it to swing freely. When a pendulum is pulled sideways from its resting, equilibrium position, gravity exerts a restoring force on it, forcing it to swing back to its resting, equilibrium position. The restoring force acting on the pendulum’s mass causes it to oscillate about the equilibrium position, swinging in both directions, once it is liberated.
The period is the length of time it takes for a whole cycle to finish, which includes both a left and right swing. The length of a pendulum and, to a lesser degree, the amplitude, or width of the pendulum’s swing, define the period of the pendulum. Above examples of pendulum uses explains how the science of pendulum is being used in real life.
When you want to alter the direction of a control line (rope), or when you want to modify the mechanical force of the line and movement necessary to move a connected item, a pulley is the tool you want to employ.
When working on big sailboats, compound pulleys are more commonly utilised in association with other ropes and pulleys. They are employed in a variety of situations when it is necessary to alter the direction of a tug on a rope as well as to provide mechanical advantage.
A foresail (the rope used to lift the sail) will pass through a block at the top of the mast while hoisting a sail, allowing you to pull down on the rope while standing on deck, and the sail will be hoisted.
Using a system of pulleys can be somewhat more difficult, but it can give a significant mechanical benefit by significantly lowering the amount of force used to move an object. It is possible to reduce the amount of force necessary to elevate an object attached to a moveable pulley in half by using only one movable pulley. But it shifts the essential force’s direction.
An elevator is a contemporary engineering device that utilises a pulley system to accomplish a function similar to that of elevating a huge stone for pyramid construction.
An elevator without pulleys would require a large motor to straighten the rope. Rather than relying on a powerful motor, some elevators rely on a heavy weight that utilises gravity to assist in raising the elevator car.
It is typical on construction sites, where cranes are frequently used to lift large steel and concrete structures. A combination of pulleys (also known as compound pulleys) or ropes may have more than two pulleys or ropes in order to get the desired result. The more pulleys connected to the system, the easier it is to lift the weight.
A compound pulley may consist of more than two pulleys or ropes, depending on the application. The greater the number of pulleys used, the longer it may take to raise an object, but the weight will be considerably easier to lift as a result of the increased number of pulleys.
With the help of the compound pulley, it will be much simpler to lift the load, though it may take a little longer to do so.
One of the common compound pulley examples which we observe in our surroundings is the garage door. Four pulleys are located on the garage door: one on each side at the top corner where the vertical and horizontal tracks connect, and one at the end of each spring.
These pulleys are equipped with ball bearings, which allow them to move smoothly and silently as the door is raised and lowered.
Pulling down on a pulley rather than pulling up on a rope to lift a heavy object saves time and energy since the rope is not lifted. As an illustration, consider another compound pulley example of flagpole.
A flagpole is raised into the air when its rope is pulled down, and the flag is sent flying up into the air. This is due to the fact that a flagpole is equipped with a compound pulley.
Gym equipment like machines with pulley, lat pulling down and many other chest exercise equipment are the most common compound pulley examples . The more pulleys attached to the load, the easier it is to raise. A lot less work is required because of the way it is designed.
So these all are the common and easily available compound pulley examples in our surroundings.
FAQ’s
Q. What do you mean by pulley?
Ans: Pulleys are simply called the ‘simple machines’.
A pulley is nothing but simply a collection of one or more wheels around which a rope is wrapped.
These are referred to as “simple machines” by scientists since they enable people to double the forces required to carry a heavy object. Utilizing a pulley significantly increases the force generated by your physical efforts.
Fixed: In a fixed pulley, the axle is supported by bearings. A stationary pulley reverses the force on a rope or belt moving around it. Accompanying a fixed pulley with a moveable pulley or a variable diameter fixed pulley provides mechanical benefit.
Movable: A moveable pulley is a pulley with an axle that is contained within a movable block. Two segments of the same rope are used to support a single moving pulley, giving it a mechanical advantage of two over the other.
Compound: Compound pulleys, commonly known as block and tackle pulleys, are used to create a block and tackle system. Multiple pulleys can be installed on the fixed and movable axles of a block and tackle, significantly improving the mechanical advantage. The compound pulley examples are sailboats, elevator, flagpole etc.
Q. What are the different advantages of using a pulley?
Ans: There are many advantages we get by using a pulley system.
It is one of the simplest alternatives for heavy lifting and installation. The amount of force required to move (lift) a heavy object decreases considerably.
It offers excellent structural support for the thing.
Force may be exerted in any direction at any point in time. It contributes to the shift in the direction of a force or movement.
The pulley mechanism does not store energy during operation.
Q. Is there any disadvantage of using a pulley?
Ans: There are some disadvantages of using the pulley.
The pulley operates by friction. It may slide, resulting in energy loss in the form of heat.
The weight travels a greater distance when employing a combination pulley system (increase lifting distance). It takes longer to achieve a desired position with a pulley than it was before.
The constant stress on the driving parts creates the stretches. It may cause rope creep and finally break.
Angular velocity and linear velocity are two fundamental concepts in physics that describe the motion of objects. Angular velocity is the rate of change of angular displacement, while linear velocity is the rate of change of linear displacement. Understanding the relationship between these two types of velocities is crucial for analyzing the motion of objects, particularly in circular motion. This comprehensive guide will delve into the technical details, formulas, examples, and numerical problems related to angular velocity and linear velocity, providing a valuable resource for science students and enthusiasts.
Understanding Angular Velocity
Angular velocity, denoted by the Greek letter omega (ω), is a measure of the rate of change of angular displacement. It is expressed in radians per second (rad/s) and can be calculated using the formula:
ω = Δθ / Δt
Where:
– ω is the angular velocity (in rad/s)
– Δθ is the change in angular displacement (in radians)
– Δt is the change in time (in seconds)
For example, if an object covers an angle of 2π radians (360 degrees) in 10 seconds, its angular velocity would be:
ω = 2π rad / 10 s = π/5 rad/s
Angular velocity is a vector quantity, meaning it has both magnitude and direction. The direction of angular velocity is determined by the right-hand rule, where the thumb points in the direction of the rotation axis, and the fingers curl in the direction of rotation.
Factors Affecting Angular Velocity
Several factors can influence the angular velocity of an object:
Rotational Inertia: The object’s moment of inertia, which is a measure of its resistance to changes in rotational motion, can affect its angular velocity. Objects with a higher moment of inertia will have a lower angular velocity for the same applied torque.
Applied Torque: The torque applied to an object can change its angular velocity. Applying a larger torque will result in a greater change in angular velocity, as described by the equation:
τ = Iα
Where:
– τ is the applied torque (in N·m)
– I is the moment of inertia (in kg·m²)
– α is the angular acceleration (in rad/s²)
Radius of Rotation: For objects moving in a circular path, the radius of the circular path can affect the angular velocity. As the radius increases, the angular velocity decreases, as described by the relationship:
v = ωr
Where:
– v is the linear velocity (in m/s)
– ω is the angular velocity (in rad/s)
– r is the radius of the circular path (in m)
Understanding Linear Velocity
Linear velocity, denoted by the symbol v, is a measure of the rate of change of linear displacement. It is expressed in meters per second (m/s) and can be calculated using the formula:
v = Δs / Δt
Where:
– v is the linear velocity (in m/s)
– Δs is the change in linear displacement (in meters)
– Δt is the change in time (in seconds)
For example, if an object covers a distance of 10 meters in 5 seconds, its linear velocity would be:
v = 10 m / 5 s = 2 m/s
Linear velocity is a vector quantity, meaning it has both magnitude and direction. The direction of linear velocity is determined by the direction of the object’s motion.
Factors Affecting Linear Velocity
Several factors can influence the linear velocity of an object:
Applied Force: The force applied to an object can change its linear velocity. Applying a larger force will result in a greater change in linear velocity, as described by Newton’s second law of motion:
F = ma
Where:
– F is the applied force (in N)
– m is the mass of the object (in kg)
– a is the linear acceleration (in m/s²)
Mass: The mass of an object can affect its linear velocity. For the same applied force, an object with a lower mass will experience a greater change in linear velocity.
Radius of Circular Motion: For objects moving in a circular path, the radius of the circular path can affect the linear velocity. As the radius increases, the linear velocity increases, as described by the relationship:
v = ωr
Where:
– v is the linear velocity (in m/s)
– ω is the angular velocity (in rad/s)
– r is the radius of the circular path (in m)
Relationship between Angular Velocity and Linear Velocity
The relationship between angular velocity and linear velocity is given by the formula:
v = ωr
Where:
– v is the linear velocity (in m/s)
– ω is the angular velocity (in rad/s)
– r is the radius of the circular path (in m)
This formula shows that the linear velocity of an object moving in a circular path is directly proportional to its angular velocity and the radius of the circular path. In other words, as the angular velocity or the radius of the circular path increases, the linear velocity also increases.
Examples
Wheel with Constant Angular Velocity:
Radius of the wheel: 0.5 meters
Angular velocity: 10 rad/s
Linear velocity: v = ωr = 10 rad/s × 0.5 m = 5 m/s
Car in Circular Motion:
Linear velocity: 60 km/h (16.67 m/s)
Radius of the circular path: 100 meters
Angular velocity: ω = v/r = 16.67 m/s / 100 m = 0.16 rad/s
Satellite in Circular Orbit:
Radius of the circular orbit: 6,731 kilometers (6,731,000 meters)
Angular velocity: approximately 0.000015 rad/s
Linear velocity: v = ωr = 0.000015 rad/s × 6,731,000 m = 28,000 km/h
Numerical Problems
A wheel has a radius of 0.2 meters and an angular velocity of 25 rad/s. Calculate the linear velocity of a point on the rim of the wheel.
Solution:
– Radius of the wheel, r = 0.2 m
– Angular velocity, ω = 25 rad/s
– Linear velocity, v = ωr = 25 rad/s × 0.2 m = 5 m/s
A car is traveling at a speed of 72 km/h. If the car is moving in a circular path with a radius of 50 meters, calculate the angular velocity of the car.
Solution:
– Linear velocity, v = 72 km/h = 20 m/s
– Radius of the circular path, r = 50 m
– Angular velocity, ω = v/r = 20 m/s / 50 m = 0.4 rad/s
A satellite is orbiting the Earth in a circular path with a radius of 42,164 kilometers. If the satellite’s linear velocity is 7.9 km/s, calculate its angular velocity.
Solution:
– Radius of the circular orbit, r = 42,164 km = 42,164,000 m
– Linear velocity, v = 7.9 km/s = 7,900 m/s
– Angular velocity, ω = v/r = 7,900 m/s / 42,164,000 m = 0.000187 rad/s
Conclusion
Angular velocity and linear velocity are two fundamental concepts in physics that describe the motion of objects. Understanding the relationship between these two types of velocities is crucial for analyzing the motion of objects, particularly in circular motion. This comprehensive guide has provided a detailed overview of angular velocity and linear velocity, including their definitions, formulas, examples, and numerical problems. By mastering these concepts, science students and enthusiasts can deepen their understanding of the physical world and apply these principles to various real-world scenarios.
There are so many constant angular velocity examples taking place in our surroundings and our day-to-day life. Following is the list of examples of constant angular velocity.
One of the best constant angular velocity example is our own living planet earth. The planet earth continues to revolve around itself with the center axis passing through it. When the earth is traveling in a circle at a constant speed, it is experiencing a constant linear acceleration in order to maintain its circular motion. Because it continuously sweeps out a fixed arc length per unit time, the angle of rotation of the object is not affected by this. Uniform circular motion is the term used to describe constant angular velocity in a circle.
This is the most common example we see in our home. When we switched on the fan, we see that the blades of the ceiling fan travel along a circular path with a constant central angle corresponding to their position from the center, thus having constant angular displacement with respect to time. If we increase the speed of the fan by one level then it will continue to perform in a circular path with constant angular velocity.
A gramophone, like any other mechanical device, operates on the concept of constant angular velocity. The needle of the gramophone travels over the surface of the record disc, covering an equal angular distance in a given interval of time, and as a result, the relationship between angular displacement and that of a given unit of time is constant. And hence it is one the example of constant angular velocity.
In the old-fashioned automation technology, vehicle tyres used to be found in varying speeds depending on the acceleration paddle. And today, as a result of the most recent advancements in technology, a new invention has been introduced: the speed lock function. This feature allows the driver to lock the speed wherever required which enables the vehicle to run continuously with the constant speed as a result of constant angular velocity achieved from the tyre of the vehicle. As long as the driver does not disable the feature of speed lock, tyre of the vehicle will continue to achieve the constant angular velocity.
Time is represented by the hands of a clock traveling along a circular route with a constant central angle equal to their distance from the center, and therefore with constant angular displacement with respect to time. The time it takes for each of the three hands (hour, minute, and second) to complete one rotation is different, even though they all move with the same constant angular velocity.
To travel an angle of θ (theta) = 90° time taken by three hands respectively will be as follows:
1) Seconds hand will take 15 sec
2) Minutes hand will take 15 mins (900 sec)
3) Hour hand will take 3 hours (180 mins / 10800 sec)
A satellite orbiting the planet moves along a circular route as a result of the gravitational pull of the earth and the circular motion of the satellite. We know that the satellite will never cease orbiting or will orbit at various speeds. This happens as a result of the constant angular velocity. In a given unit of time, a satellite travels an equal angular distance (or sweeps an equal area of an arc) and continues to do so, resulting in a constant angular velocity for the satellite.
In the industrial sector, we frequently see the usage of an electric cutting blade for cutting heavy and thick items such as marbles. This heavy electronic equipment always comes with the ability to adjust the power levels, which are used to adjust the cutting strength of the blade in relation to the size of the object being cut through. When a certain power level is selected, the cutting blade continues to revolve at a constant speed, which is nothing more than a phenomenon of continuous angular velocity that is utilized to cut the items.
Every kitchen contains a grinder, which is the most generally accessible and most widely used electrical equipment. Because the blades of the grinder revolve in a circular manner, they are able to grind the material that is contained inside it. During the time that the grinder’s blade is moving, it covers an equal area in equal intervals of time, resulting in an angular velocity that remains constant throughout the operation.
This speed remains constant until the power level is changed. And, after the level is changed, the blades of the grinder operate at a constant angular velocity instead of the prior velocity at which they performed. Fine chopping of the material depends on the level of constant angular velocity that occurs as a result of the application of power.
These all are the common constant angular velocity examples in our daily life.
FAQ’s
Q. What is the meaning of angular velocity?
Ans:Angular velocity is an important component in an object’s rotational motion.
The angular velocity of an item or particle is the speed at which it rotates about a center or a specified location in a certain time period. Additionally, it is referred to as rotational velocity. Angle per unit time or radians per second (rad/s) are used to express angular velocity. And the rate of change in angular velocity is called angular acceleration.
Q. What is constant angular velocity?
Ans: The axis around which it revolves and the rate at which it rotates are both constant.
An item’s or particle’s angular velocity is the rate at which it revolves about a center or a defined place in a given time period. And constant means not changing or remains the same. So the constant angular velocity implies neither its rate of rotation and nor the axis around which it revolves are changing.
Q. What is the value of the angular velocity of the earth?
Ans: It can be calculated by using the formula of angular velocity (⍵)
The angular velocity of the earth (⍵)= 2???? radians / Time of the day in seconds
⍵ = 2???? radians / 84600 s
⍵ = 7.25 ×10⁻⁵ radians / s
Therefore the angular velocity of the earth is 7.25 ×10⁻⁵ radians / s
Heat transfer by radiation is a fundamental process in which energy is exchanged through the emission and absorption of electromagnetic waves. This mode of heat transfer is distinct from conduction and convection, as it does not require the presence of a physical medium for the energy to be transmitted. Understanding the principles and mechanisms of radiative heat transfer is crucial in various fields, from astrophysics and climate science to engineering and everyday life.
The Stefan-Boltzmann Law: Quantifying Radiative Heat Transfer
The rate at which an object radiates energy is governed by the Stefan-Boltzmann law, which states that the total energy radiated per unit surface area of a body is proportional to the fourth power of its absolute temperature. Mathematically, this relationship is expressed as:
Radiation rate = k × T^4
Where:
– Radiation rate is the total energy radiated per unit surface area (W/m^2)
– k is the Stefan-Boltzmann constant, with a value of 5.67 × 10^-8 W/m^2·K^4
– T is the absolute temperature of the object (in Kelvin)
This law provides a quantitative framework for understanding the relationship between an object’s temperature and its radiative heat transfer. For example, a body at 300 K (27°C) will radiate approximately 450 W/m^2, while a body at 600 K (327°C) will radiate around 3,600 W/m^2, a nearly eightfold increase.
Wavelength and Frequency Dependence of Thermal Radiation
The wavelength and frequency of the radiated electromagnetic waves are also closely linked to the temperature of the emitting object. This relationship is described by Wien’s displacement law, which states that the wavelength at which the maximum intensity of radiation occurs is inversely proportional to the absolute temperature of the object:
λ_max = b / T
Where:
– λ_max is the wavelength at which the maximum intensity of radiation occurs (in meters)
– b is Wien’s displacement constant, with a value of 2.898 × 10^-3 m·K
– T is the absolute temperature of the object (in Kelvin)
As the temperature of an object increases, the wavelength of the peak intensity of the emitted radiation decreases, and the frequency increases. This phenomenon is responsible for the visible glow of hot objects, such as the coils of an electric toaster or the filament of an incandescent light bulb.
At typical room temperatures, most objects radiate energy in the infrared region of the electromagnetic spectrum, which is invisible to the human eye. However, as the temperature rises, the emitted radiation shifts towards shorter wavelengths, eventually reaching the visible spectrum and appearing as a characteristic color.
Blackbody Radiation and Emissivity
The concept of a “blackbody” is central to understanding the principles of radiative heat transfer. A blackbody is an idealized object that absorbs all incident radiation, regardless of the wavelength or angle of incidence, and emits radiation in a characteristic way determined solely by its temperature.
The radiation emitted by a blackbody is known as blackbody radiation, and it is described by Planck’s law, which relates the spectral radiance (power per unit area per unit solid angle per unit wavelength) of a blackbody to its temperature and the wavelength of the radiation:
B_λ(λ, T) = (2hc^2 / λ^5) / (e^(hc / λkT) – 1)
Where:
– B_λ(λ, T) is the spectral radiance of the blackbody (W/m^2·sr·μm)
– h is Planck’s constant (6.626 × 10^-34 J·s)
– c is the speed of light (3.00 × 10^8 m/s)
– λ is the wavelength of the radiation (in meters)
– k is the Boltzmann constant (1.38 × 10^-23 J/K)
– T is the absolute temperature of the blackbody (in Kelvin)
Real-world objects, however, are not perfect blackbodies, and their ability to emit and absorb radiation is characterized by their emissivity, a dimensionless quantity between 0 and 1 that represents the efficiency of the object’s radiation compared to a blackbody at the same temperature.
The emissivity of an object depends on various factors, such as its material, surface roughness, and temperature. For example, a polished metal surface typically has a low emissivity, while a matte black surface has a high emissivity. Understanding and accounting for the emissivity of materials is crucial in many engineering applications, such as the design of thermal insulation systems and the analysis of radiative heat transfer in industrial processes.
Radiative Heat Transfer in Vacuum and Participating Media
One of the unique features of radiative heat transfer is its ability to occur in the absence of a physical medium, such as in the vacuum of space. This is in contrast to conduction and convection, which require the presence of a material medium for the transfer of energy.
The heat received on Earth from the Sun is a prime example of radiative heat transfer in a vacuum. The electromagnetic waves emitted by the Sun’s surface travel through the void of space and are absorbed by the Earth’s atmosphere and surface, providing the energy that sustains life on our planet.
In situations where the medium between the heat source and the receiver is not a vacuum, the presence of participating media, such as gases, liquids, or solids, can significantly affect the radiative heat transfer process. Participating media can absorb, emit, and scatter the radiant energy, leading to complex interactions and the need for more sophisticated models to describe the heat transfer.
For instance, the Earth’s atmosphere, with its various gases and suspended particles, plays a crucial role in the greenhouse effect, where certain atmospheric components absorb and re-emit infrared radiation, trapping heat and influencing the planet’s climate. Similarly, the design of insulation materials for buildings or spacecraft must consider the radiative properties of the participating media to optimize thermal management.
Applications and Importance of Radiative Heat Transfer
Radiative heat transfer is a fundamental process that underpins numerous applications and phenomena in science and engineering. Some key examples include:
Solar Energy Conversion: The absorption and conversion of solar radiation into useful energy, such as in photovoltaic cells and solar thermal collectors, rely on the principles of radiative heat transfer.
Astrophysics and Cosmology: The study of the universe, from the formation of stars and galaxies to the evolution of the cosmos, heavily depends on the understanding of radiative heat transfer in the vacuum of space.
Thermal Imaging and Remote Sensing: Infrared cameras and other remote sensing technologies utilize the principles of radiative heat transfer to detect and measure the temperature of objects from a distance.
Industrial Processes: Radiative heat transfer plays a crucial role in various industrial applications, such as furnace design, glass manufacturing, and the curing of coatings and paints.
Thermal Management in Electronics: The efficient dissipation of heat in electronic devices, such as computers and smartphones, often involves the optimization of radiative heat transfer mechanisms.
Thermal Insulation and Energy Efficiency: The design of effective thermal insulation systems, both for buildings and spacecraft, relies on the understanding of radiative heat transfer and the mitigation of undesirable radiative heat exchange.
Biomedical Applications: Radiative heat transfer principles are applied in medical imaging techniques, such as infrared thermography, and in the design of medical devices that utilize thermal radiation for therapeutic purposes.
Understanding the fundamental principles of radiative heat transfer and its quantifiable aspects, such as the Stefan-Boltzmann law, Wien’s displacement law, and the concept of emissivity, is essential for advancing scientific research, engineering design, and technological innovation across a wide range of disciplines.
Reference:
Thermal Radiation – an overview | ScienceDirect Topics. (n.d.). Retrieved from https://www.sciencedirect.com/topics/engineering/thermal-radiation
THERMAL RADIATION HEAT TRANSFER. (1971). Retrieved from https://ntrs.nasa.gov/api/citations/19710021465/downloads/19710021465.pdf
Methods of Heat Transfer – The Physics Classroom. (n.d.). Retrieved from https://www.physicsclassroom.com/class/thermalP/Lesson-1/Methods-of-Heat-Transfer
What is radiation heat transfer? | Explained by Thermal Engineers. (n.d.). Retrieved from https://conceptgroupllc.com/glossary/what-is-radiation-heat-transfer/
Thermal radiation – Wikipedia. (n.d.). Retrieved from https://en.wikipedia.org/wiki/Thermal_radiation
You always feel your face getting warmer when you stand in the sunlight. And if you continue standing then you get body tan. It is because of the radiation coming from sunlight. Sun emits different radiation like UV, Visible, Infrared etc. which travels larger distances to reach up to earth. It is because of the fact that the sun is a hotter object and the objects which are hotter, continue to radiate heat energy in radiation form. Hence, radiation heat transfer from the sun makes your skin feel warm and eventually the formation of body tan.
Fires frequently start when hot embers from a blazing fire are placed near wood that isn’t currently burning. In that scenario, infrared radiation was responsible for transferring heat from the hot coals to the colder wood, causing the wood to ignite. Those with greater temperatures release more radiation unit area than systems with lower temperatures. Radiation travels in both directions across systems that may exchange radiant energy, although more energy travels from the hotter to the colder item. This net flow of energy is referred to as heat.
Incandescent light bulbs produce light by heating a tiny metal filament contained in a glass bulb filled with inert gas. Electricity is used to heat the filament till it glows. Whenever the lamp is switched on, the light bulb becomes extremely hot. In addition to radiation transmission, some of the heat emitted by the bulb is transmitted to the glass when the bulb heats up. Radiation happens when heat is transmitted between two things that are not in direct contact with one another.
During the winter months, individuals prefer to be in a warm environment, and for this reason, they may opt to sit around a bonfire also called a campfire. In the event of ignition, a campfire begins to warm the environment around it. This occurs as a result of the transmission of heat from the campfire to the surrounding environment through radiation. Radiation from the hot fire causes the surrounding air to get warmer, which in turn leads those who are sitting around it to become warmer.
Recently developing technology of the electronics such as televisions, smartphones, computers, and tablets, among other things, emit a little amount of heat. We aren’t even conscious of the warmth since the amount is so little. Because of the radiation emitted by electronics, this warmth or heat is experienced by the user. Consequently, the electronics examples stated above are radiation heat transfer examples that we can physically feel in our daily lives.
If we leave the pan on the burner for an extended period of time, we will see that the pan will become hot owing to conduction, which is a form of heat transmission that occurs. Maintaining a safe distance above or nearby while heating the pan will allow you to feel the warmth on your hand as the pan heats up further. This is due to the radiation heat transfer that is occurring between the burner and the heated pan, which is responsible for this. The radiation emitted by these two objects continues to heat the air around them, resulting in a warm sensation in the surrounding area.
All forms of solar technology that are capable of reducing power use rely on the radiation heat energy obtained from the sun to operate. Because sunlight from the sun is a form of heat radiation, it may be used to power solar energy equipment, which can be used to both create and save energy.
These days, the microwave oven is rapidly becoming the most often used kitchen equipment in most households. Using this device, you may cook or heat the food that is stored inside it. The fundamental operating concept of a microwave oven is simply the transfer of heat through microwave radiation, as the name implies. These microwave radiation rays from the electromagnetic spectrum transmit heat energy, resulting in the cooking of food in a short time.
If you light a candle and place it in a tiny, closed dark room, you will notice that the temperature of the small room begins to rise after a period of time. This occurs as a result of heat transmission from the candle’s flame, which occurs through radiation. The radiation from the flame heats the air surrounding it, and soon one can feel the warmth coming from it.
So, these all are the radiation heat transfer examples in our surroundings. As per the scientific explanation for this phenomenon of radiation heat transfer, all matter with a temperature higher than absolute zero generates electromagnetic radiation as a result of charged particle oscillations inside it. As a result, every substance in our entire universe emits radiation.
Gravity is a fundamental force in the universe, and understanding its properties is crucial for many areas of physics. One of the key properties of gravity is that it is a conservative force, which means that the work done by gravity is path-independent and only depends on the initial and final positions of the object being moved. This property has important implications for the conservation of energy in a system.
Understanding Conservative Forces
A force field $F_i(x)$ is considered conservative if the following conditions are met:
Path-Independence: For every curve $C$ from a point $y_1$ to a point $y_2$, the integral $\int\limits_C F_i(x)\mathrm{d}x^i$ is the same, so that the energy difference between $y_1$ and $y_2$ is independent of the curve taken from one to the other.
Closed Curve Integral: The integral around a closed curve must be zero, $\oint\limits_C F_i(x)\mathrm{d}x^i=0$ for every closed curve $C$.
These conditions ensure that the work done by a conservative force, such as gravity, is the same regardless of the path taken between two points.
Gravity as a Conservative Force
In the context of gravity, the conditions for a conservative force are met due to the following properties:
Constant Gravitational Force: The force of gravity is always directed towards the center of mass, and the work done by a constant force over a distance is given by $W=Fd$, where $F$ is the force and $d$ is the distance. Since the force of gravity is constant, the work done is proportional to the distance between the two points, regardless of the path taken.
Path-Independent Work: The work done by gravity on an object is the same whether the object moves in a straight line or a curved path between two points. This is because the force of gravity is always directed towards the center of mass.
Closed Curve Integral: The integral around a closed curve of the gravitational force is zero, meaning that the work done by gravity on an object moving in a closed loop is zero.
These properties of gravity ensure that it is a conservative force, which has important implications for the conservation of energy in a system.
Conservation of Energy in a Conservative Force Field
In a conservative force field, such as gravity, the total mechanical energy of a system is conserved, meaning that the sum of the kinetic and potential energy remains constant. This is because the work done by a conservative force is equal to the negative of the change in potential energy, as given by the equation $W=-\Delta U$, where $W$ is the work done and $\Delta U$ is the change in potential energy.
Since the work done by a conservative force is path-independent, the change in potential energy is also path-independent, and the total mechanical energy of the system is conserved. This means that the energy lost in the form of work done by gravity is exactly equal to the change in the object’s potential energy, and the total energy of the system remains constant.
Technical Specifications of Gravity as a Conservative Force
Gravity is a conservative force that obeys the following mathematical conditions:
Path-Independent Work: The work done by gravity is path-independent, meaning that the work done by gravity on an object is the same whether the object moves in a straight line or a curved path between two points.
Closed Curve Integral: The integral around a closed curve of the gravitational force is zero, meaning that the work done by gravity on an object moving in a closed loop is zero.
Constant Gravitational Force: The force of gravity is always directed towards the center of mass, and the work done by a constant force over a distance is given by $W=Fd$, where $F$ is the force and $d$ is the distance.
Potential Energy Relationship: The work done by gravity is equal to the negative of the change in potential energy, as given by the equation $W=-\Delta U$, where $W$ is the work done and $\Delta U$ is the change in potential energy.
Conservation of Mechanical Energy: The total mechanical energy of a system in a conservative force field, such as gravity, is conserved, meaning that the sum of the kinetic and potential energy remains constant.
These technical specifications are important for understanding the behavior of gravity and its role in the conservation of energy in a system.
DIY Experiment to Demonstrate Gravity as a Conservative Force
To demonstrate that gravity is a conservative force, you can perform the following DIY experiment:
Set up a ramp: Create a ramp using a piece of plywood or a long board. The ramp should be at least a few feet long and have a smooth surface to reduce friction.
Add a ball: Place a ball, such as a steel ball or a marble, at the top of the ramp.
Measure the height: Measure the height of the ball from the ground.
Release the ball: Release the ball and let it roll down the ramp.
Measure the speed: Measure the speed of the ball at the bottom of the ramp using a stopwatch or a speed gun.
Calculate the potential and kinetic energy: Calculate the potential energy of the ball at the top of the ramp using the formula $PE=mgh$, where $m$ is the mass of the ball, $g$ is the acceleration due to gravity, and $h$ is the height of the ball. Calculate the kinetic energy of the ball at the bottom of the ramp using the formula $KE=1/2mv^2$, where $v$ is the speed of the ball.
Compare the energies: Compare the potential energy at the top of the ramp to the kinetic energy at the bottom of the ramp. You should find that the total mechanical energy of the system (potential energy + kinetic energy) remains constant, demonstrating that gravity is a conservative force.
By performing this experiment, you can demonstrate the conservation of energy in a system subject to the conservative force of gravity.
Conclusion
Gravity is a conservative force, which means that the work done by gravity is path-independent and only depends on the initial and final positions of the object being moved. This property of gravity has important implications for the conservation of energy in a system, as it ensures that the total mechanical energy of the system remains constant. Understanding the technical specifications and experimental demonstration of gravity as a conservative force is crucial for many areas of physics, from classical mechanics to astrophysics.
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are constants determined by the initial conditions of the system.
is the damping coefficient.
