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Questions and Answers
What effect does damping have on harmonic motion?
What effect does damping have on harmonic motion?
- It causes the motion to become less periodic over time. (correct)
- It increases the oscillation amplitude.
- It has no effect on the equilibrium point.
- It results in a constant oscillation frequency.
In which situation is damping considered unwanted?
In which situation is damping considered unwanted?
- In clocks that require precise timekeeping. (correct)
- In audio equipment to reduce noise.
- In automobiles with shock absorbers.
- In heavy machinery for stability.
What characterizes underdamping in an oscillator?
What characterizes underdamping in an oscillator?
- The system does not undergo any oscillations.
- The motion is a simple linear decay without oscillation.
- The system reaches equilibrium instantaneously.
- The system oscillates several times with decreasing amplitude. (correct)
What is the equation for the retarding force in a damped harmonic motion?
What is the equation for the retarding force in a damped harmonic motion?
What is the primary characteristic of critically damped motion?
What is the primary characteristic of critically damped motion?
What type of damping causes the system to experience a slower than exponential decay without oscillation?
What type of damping causes the system to experience a slower than exponential decay without oscillation?
What behavior is expected when damping is large in a harmonic oscillator?
What behavior is expected when damping is large in a harmonic oscillator?
How does the restoring force in an oscillator typically relate to displacement?
How does the restoring force in an oscillator typically relate to displacement?
What is the relationship between amplitude relaxation time and energy relaxation time?
What is the relationship between amplitude relaxation time and energy relaxation time?
What factor is represented by 'r' in the context of relaxation times?
What factor is represented by 'r' in the context of relaxation times?
How do eddy currents affect the performance of braking systems in trains?
How do eddy currents affect the performance of braking systems in trains?
What is one of the main advantages of induction stoves compared to traditional cooking methods?
What is one of the main advantages of induction stoves compared to traditional cooking methods?
In the context of eddy currents, what happens when a conductor moves through a magnetic field?
In the context of eddy currents, what happens when a conductor moves through a magnetic field?
What characterizes a damped oscillator?
What characterizes a damped oscillator?
What happens when the resistance in an electrical oscillator exceeds the critical resistance value?
What happens when the resistance in an electrical oscillator exceeds the critical resistance value?
Which term describes the rate at which the amplitude of oscillatory motion decays?
Which term describes the rate at which the amplitude of oscillatory motion decays?
What is the natural frequency of a system with no retarding force called?
What is the natural frequency of a system with no retarding force called?
In the damped harmonic motion, what occurs when the retarding force is minor?
In the damped harmonic motion, what occurs when the retarding force is minor?
What is termed as critically damped in an oscillator?
What is termed as critically damped in an oscillator?
What does the quality factor of an oscillator indicate?
What does the quality factor of an oscillator indicate?
What does the relaxation time in a damped system refer to?
What does the relaxation time in a damped system refer to?
What is the expression for the potential energy stored in a stretched spring?
What is the expression for the potential energy stored in a stretched spring?
In a frictionless simple harmonic oscillator, when is the energy entirely kinetic?
In a frictionless simple harmonic oscillator, when is the energy entirely kinetic?
What type of circuit is formed when a capacitor is connected to an inductor?
What type of circuit is formed when a capacitor is connected to an inductor?
What is analogous to the kinetic energy stored in a moving block in a simple harmonic motion?
What is analogous to the kinetic energy stored in a moving block in a simple harmonic motion?
What condition is assumed for energy conservation in an LC circuit?
What condition is assumed for energy conservation in an LC circuit?
What energy is stored in the electric field of a capacitor?
What energy is stored in the electric field of a capacitor?
At what position in simple harmonic motion is potential energy maximum?
At what position in simple harmonic motion is potential energy maximum?
In an LC circuit, what does the inductor's energy depend on?
In an LC circuit, what does the inductor's energy depend on?
What defines the amplitude relaxation time?
What defines the amplitude relaxation time?
How is the energy relaxation time (te) related to the amplitude relaxation time (ta)?
How is the energy relaxation time (te) related to the amplitude relaxation time (ta)?
What does a higher quality factor (Q) indicate about an oscillator?
What does a higher quality factor (Q) indicate about an oscillator?
What is the formula for the quality factor (Q) based on energy considerations?
What is the formula for the quality factor (Q) based on energy considerations?
What does the parameter 'r' represent in the equation d = rT?
What does the parameter 'r' represent in the equation d = rT?
If ta = 1/r, what will be the relationship when t = t0 + ta?
If ta = 1/r, what will be the relationship when t = t0 + ta?
In the context of a damped harmonic oscillator, what outcome occurs when energy is lost rapidly?
In the context of a damped harmonic oscillator, what outcome occurs when energy is lost rapidly?
What happens during one complete cycle (dt = T) of a damped harmonic oscillator?
What happens during one complete cycle (dt = T) of a damped harmonic oscillator?
What is the term for the maximum displacement from the equilibrium position in simple harmonic motion?
What is the term for the maximum displacement from the equilibrium position in simple harmonic motion?
What does the spring constant (k) represent in the context of spring oscillations?
What does the spring constant (k) represent in the context of spring oscillations?
In simple harmonic motion, what causes the acceleration of the object?
In simple harmonic motion, what causes the acceleration of the object?
What is the relationship between frequency and period in simple harmonic motion?
What is the relationship between frequency and period in simple harmonic motion?
What happens to the total mechanical energy in an ideal simple harmonic oscillator over time?
What happens to the total mechanical energy in an ideal simple harmonic oscillator over time?
Which of the following best describes a cycle in the context of simple harmonic motion?
Which of the following best describes a cycle in the context of simple harmonic motion?
What characterizes the restoring force of a spring in simple harmonic motion?
What characterizes the restoring force of a spring in simple harmonic motion?
In a simple pendulum, which aspect does 'L' refer to in the equation $s = L\theta$?
In a simple pendulum, which aspect does 'L' refer to in the equation $s = L\theta$?
Flashcards
Simple Harmonic Motion (SHM)
Simple Harmonic Motion (SHM)
A type of motion where the restoring force is directly proportional to the negative of the displacement from equilibrium.
Equilibrium Position
Equilibrium Position
The point where the spring is neither stretched nor compressed.
Amplitude
Amplitude
The maximum displacement from the equilibrium position.
Period (T)
Period (T)
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Frequency (f)
Frequency (f)
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Restoring Force
Restoring Force
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Spring Constant (k)
Spring Constant (k)
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Simple Pendulum
Simple Pendulum
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Total Mechanical Energy in SHM
Total Mechanical Energy in SHM
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Potential Energy in SHM (U)
Potential Energy in SHM (U)
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Kinetic Energy in SHM (K)
Kinetic Energy in SHM (K)
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LC Circuit
LC Circuit
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Electric Potential Energy in LC
Electric Potential Energy in LC
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Magnetic Energy in LC
Magnetic Energy in LC
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Energy Conservation in LC Circuit
Energy Conservation in LC Circuit
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Natural Resonant Frequency of LC
Natural Resonant Frequency of LC
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Energy in LC Circuit
Energy in LC Circuit
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Energy Loss in Oscillators
Energy Loss in Oscillators
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Damped Harmonic Motion
Damped Harmonic Motion
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Underdamping
Underdamping
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Critical Damping
Critical Damping
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Overdamping
Overdamping
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Differential Equation of Damped Harmonic Motion
Differential Equation of Damped Harmonic Motion
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Damped Oscillator
Damped Oscillator
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Exponential Decay
Exponential Decay
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Natural Frequency
Natural Frequency
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Logarithmic Decrement
Logarithmic Decrement
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Quality Factor (Q)
Quality Factor (Q)
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Amplitude Relaxation Time
Amplitude Relaxation Time
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Energy Relaxation Time
Energy Relaxation Time
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Eddy Currents
Eddy Currents
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Amplitude Relaxation Time (ta)
Amplitude Relaxation Time (ta)
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Energy Relaxation Time (te)
Energy Relaxation Time (te)
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Relationship between ta and te
Relationship between ta and te
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How is Q related to energy loss?
How is Q related to energy loss?
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How to Interpret Q practically
How to Interpret Q practically
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How does damping affect Q?
How does damping affect Q?
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What is the relationship between damping and relaxation time?
What is the relationship between damping and relaxation time?
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Study Notes
Oscillations and Waves
- Oscillations are repeating motions of an object returning to a given position after a fixed time interval
- Waves are oscillations that transfer energy
- Sound waves and water waves are examples of oscillations
- Simple harmonic motion (SHM) is a special type of periodic motion with a single frequency and well-defined amplitude
- SHM is a basic building block for more complex periodic motions
- All periodic motions can be modeled as combinations of simple harmonic motions
- SHM is fundamental to understanding mechanical waves.
Simple Harmonic Motion (SHM) - Spring Oscillations
- A spring-mass system is a model of a periodic system
- Periodic motion is when an object vibrates or oscillates over the same path, each cycle taking the same amount of time
- Displacement is measured from the equilibrium point and amplitude is the maximum displacement
- A cycle is a full back-and-forth motion
- Period is the time required to complete one cycle
- Frequency is the number of cycles completed per second
Simple Harmonic Motion - Spring Oscillations (cont.)
- The force exerted by a spring is proportional to the negative of its displacement: F = -kx
- The minus sign indicates the force is a restoring force (pulls the mass towards its equilibrium position)
- k is the spring constant
Simple Harmonic Motion
- An object moves with simple harmonic motion when acceleration is proportional to its position and is oppositely directed to the displacement from equilibrium
- x(t) = Acos(wt + φ) is a solution
- T = 2π√(m/k)
- f = 1/T = 1/(2π√(m/k))
Simple Harmonic Motion – Spring Oscillations (detailed)
- Displacement is measured from equilibrium point. Amplitude is maximum displacement
- A cycle is a complete to-and-fro motion
- Period is the time to complete one cycle.
- Frequency is the number of cycles per second
Simple Pendulum
- A simple pendulum consists of a mass on a lightweight cord
- The cord does not stretch, and its mass is negligible
- d²θ/dt² = -(g/L)sinθ
- For small angles, sinθ≈θ
- d²θ/dt² ≈ -(g/L)θ ≈ -ω²θ
- ω² = g/L
Energy Conservation in Oscillatory Motion
- In an ideal system, with no energy loss, total mechanical energy is conserved (E = K + U)
- Potential energy (U) of a spring: U = ½kx²
- Kinetic energy (K) of a mass: K = ½mv²
- Total energy (E) is the sum of potential and kinetic energy
Energy in Simple Harmonic Motion
- If the mass is at its maximum displacement (limits of motion), energy is all potential
- If the mass is at the equilibrium point, energy is all kinetic
- Total energy is constant and equal to ½kA²
- Energy is transferred between potential and kinetic forms throughout the cycle.
Damped Harmonic Motion
- Damped harmonic motion occurs with a frictional or drag force
- The mechanical energy of the system diminishes with time
- Depending on the damping force, the oscillations may or may not be present
- Underdamping: Several small oscillations before coming to rest
- Critical damping: Fastest way to equilibrium
- Overdamping: Oscillations are absent, slow return to equilibrium
Damped Harmonic Motion (cont.)
- Retarding force: R = -bv
- Restoring force: F = -kx
- d²x/dt² = -kx - bv; simplified as second order linear differential equation.
Damped Harmonic Motion(cont.)
- x(t) = Be⁻ªᵗ is a solution form
- Different scenarios exist
Electrical Oscillator: LC circuit
- Combining a capacitor and an inductor creates an LC circuit
- The energy exchanged between electrical (stored in capacitor) and magnetic (stored in the inductor) forms.
- The total energy is constant.
- Q(t) = Qmaxcos(ωt + φ)
Electrical Oscillator: LCR circuit
- Introducing a resistor adds damping to the LC circuit
- Q(t) = Qmaxe⁻rtSin(ωt + φ) describes damped oscillations
- ω varies due to damping
Electrical Oscillator: Damped Harmonic Motions
- Energy is still conserved; however, losses cause oscillations to decrease over time
- Q(t) = Qmax e⁻rt Sin(ωt + φ) describes oscillations with decreasing amplitude.
- R is the Resistance, ω is the frequency
Characterization of Damping
- Logarithmic decrement ( δ ) measures the rate at which the amplitude decays
- Relaxation time ( τ ) is the duration for the amplitude to decrease to 1/e of its original value
Characterization of Damping (cont.)
- Quality factor (Q) is a figure of merit, measures how little energy is lost per cycle
- Eddy currents from changing magnetic fields cause resistance and dampen oscillations
Applications
- Braking systems in transit systems use eddy currents for braking
- Induction stoves use eddy currents to heat cookware
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