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is a crucial concept in physics, describing how real-world oscillations lose energy over time. Unlike ideal systems, damped oscillators experience forces that oppose motion, causing their to decrease exponentially.

Understanding damped motion helps explain everyday phenomena, from swinging pendulums to shock absorbers in cars. By exploring the interplay between restoring forces, damping, and external drivers, we gain insight into the behavior of countless physical systems.

Damped Harmonic Motion

Concept of damped harmonic motion

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  • Oscillatory motion where decreases over time due to
  • Damping force opposes motion and dissipates energy from system (friction, air resistance, fluid drag)
  • Amplitude decreases exponentially with time ()
  • remains constant, but may change slightly due to damping force
  • System eventually comes to rest at (pendulum, spring-mass system)

Equation for damped oscillators

  • Derived by applying to system with and damping force
  • : Fr=−kxF_r = -kx, kk is , xx is displacement from equilibrium
  • Damping force: Fd=−bvF_d = -bv, bb is , vv is velocity
  • : md2xdt2+bdxdt+kx=0m\frac{d^2x}{dt^2} + b\frac{dx}{dt} + kx = 0, mm is mass, d2xdt2\frac{d^2x}{dt^2} is acceleration, dxdt\frac{dx}{dt} is velocity
  • Solution depends on relative values of damping coefficient and , bc=2mkb_c = 2\sqrt{mk}
    1. (b<bcb < b_c): Oscillations decay exponentially with time
    2. (b=bcb = b_c): Returns to equilibrium in shortest possible time without oscillating
    3. (b>bcb > b_c): Returns to equilibrium slowly without oscillating

Free vs damped vs forced oscillations

  • :
    • No external forces acting on system, except restoring force
    • Energy conserved, amplitude remains constant (ideal pendulum, spring-mass system)
  • Damped oscillations:
    • Damping force dissipates energy from system
    • Amplitude decreases exponentially with time (real pendulum, spring-mass system with friction)
  • :
    • External periodic driving force acts on system
    • Energy continuously supplied to system
    • Amplitude depends on frequency and strength of driving force, , and damping (driven pendulum, AC circuits)

Effects of driving force and damping

  • Driving force:
    • occurs when frequency of driving force matches natural frequency of system
    • At , amplitude of oscillations is maximum (tuning fork, resonance disaster)
    • Amplitude of forced oscillations depends on strength of driving force and damping in system
  • Damping:
    • Increasing damping reduces amplitude of oscillations
    • Damping affects resonance behavior by broadening resonance peak and reducing maximum amplitude at resonance
    • Critical damping is minimum amount of damping required to prevent oscillations
    • Overdamping causes system to return to equilibrium slowly without oscillating (door closer, shock absorber)

Characterizing Damped Oscillations

  • : Rate at which energy is lost from the system due to damping forces
  • : Measure of how an oscillator is, related to the rate of energy loss
  • : Natural logarithm of the ratio of any two successive amplitudes, used to quantify damping in a system

Key Terms to Review (34)

Amplitude: Amplitude is the maximum displacement of a point on a wave from its equilibrium position. It is a measure of the energy carried by the wave.
Amplitude: Amplitude is the maximum displacement or extent of a periodic motion, such as a wave or an oscillation, from its equilibrium position. It represents the magnitude or size of the motion and is a fundamental characteristic of various physical phenomena described in the topics of 1.7 Solving Problems in Physics, 8.4 Potential Energy Diagrams and Stability, 15.1 Simple Harmonic Motion, and beyond.
Critical Damping Coefficient: The critical damping coefficient is the minimum amount of damping required to prevent an oscillating system from exhibiting any oscillations. It represents the threshold between an overdamped and an underdamped system, where the system returns to its equilibrium position as quickly as possible without any oscillations.
Critically damped: A critically damped system returns to equilibrium as quickly as possible without oscillating. It occurs when the damping coefficient is equal to the critical damping value.
Critically Damped: Critically damped refers to a specific condition in a damped oscillating system where the damping force is precisely balanced to allow the system to return to equilibrium in the shortest time without oscillating. This condition is significant as it represents the transition point between underdamped (oscillating) and overdamped (slower return) behaviors, ensuring a quick stabilization with minimal overshoot.
Damped harmonic motion: Damped harmonic motion describes an oscillatory motion where the amplitude of the oscillations decreases over time due to an external force like friction or resistance. It is a type of motion observed in many physical systems that lose energy over time.
Damped Harmonic Motion: Damped harmonic motion refers to the oscillatory motion of a system that experiences a dissipative force, such as friction or air resistance, which causes the amplitude of the oscillations to decrease over time. This type of motion is characterized by a gradual loss of energy, resulting in the system eventually coming to rest.
Damping coefficient: The damping coefficient is a parameter that quantifies the extent to which oscillations decrease in amplitude over time due to energy loss from the system. It indicates how much resistance is present in a system, impacting the rate of decay of oscillatory motion and influencing the behavior of both damped and forced oscillations. This coefficient plays a crucial role in determining the system's response to external forces and its tendency to return to equilibrium after being disturbed.
Damping Force: Damping force is a force that opposes the motion of an oscillating system, causing the amplitude of the oscillations to decrease over time. It is a crucial concept in the understanding of damped oscillations, which describe the behavior of systems that experience energy dissipation during their motion.
Energy Dissipation: Energy dissipation is the process by which energy is lost or converted into a less useful form, often as heat, within a system. It is a fundamental concept in physics that describes the inevitable loss of energy due to various mechanisms, such as friction, resistance, or damping.
Equation of Motion: The equation of motion is a fundamental concept in classical mechanics that describes the relationship between the position, velocity, acceleration, and time of an object undergoing motion. It is a mathematical expression that allows for the prediction and analysis of an object's movement under the influence of various forces.
Equilibrium position: The equilibrium position is the point at which the net force acting on an oscillating system is zero. At this position, the system experiences no acceleration and remains at rest if undisturbed.
Equilibrium Position: The equilibrium position is the point at which a system is in a state of balance, where the net force or torque acting on the system is zero. This concept is fundamental in understanding the behavior of various physical systems, including those related to simple harmonic motion, circular motion, damped oscillations, and wave propagation.
Exponential Decay: Exponential decay is a mathematical model that describes the decrease of a quantity over time. It is characterized by an initial value that diminishes at a rate proportional to its current value, resulting in a continuous, non-linear decline.
Forced Oscillations: Forced oscillations refer to the oscillatory motion of a system that is driven by an external force or periodic input, rather than by the system's own natural frequency. This type of oscillation occurs when a system is subjected to a continuous, time-varying force that causes it to vibrate at a frequency different from its natural frequency.
Free Oscillations: Free oscillations refer to the natural, unforced vibrations of a system that occur when it is displaced from its equilibrium position and allowed to oscillate without any external driving force. These oscillations are determined by the inherent properties of the system, such as its mass, stiffness, and damping characteristics.
Frequency: Frequency is a fundamental concept in physics that describes the number of occurrences of a repeating event or phenomenon per unit of time. It is a crucial parameter in various areas of physics, including wave behavior, oscillations, and sound propagation.
Hooke's Law: Hooke's law is a fundamental principle in physics that describes the linear relationship between the force applied to an elastic object and the resulting deformation or displacement of that object. It is a crucial concept that underpins the understanding of various physical phenomena, including work, conservative and non-conservative forces, potential energy diagrams and stability, stress, strain, and elasticity, as well as simple harmonic motion.
Logarithmic Decrement: The logarithmic decrement is a measure of the rate of decay of an oscillating system. It quantifies the exponential decrease in the amplitude of successive oscillations in a damped system.
Natural angular frequency: Natural angular frequency is the rate at which an undamped system oscillates when not subjected to any external force. It is denoted by $\omega_0$ and is measured in radians per second.
Natural Frequency: Natural frequency is the inherent frequency at which a system tends to oscillate when it is not affected by external forces. It is a fundamental property of a system that depends on its physical characteristics and determines how the system will respond to various inputs or disturbances.
Newton's Second Law: Newton's Second Law states that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass. It describes the relationship between an object's motion and the forces acting upon it, providing a quantitative framework for understanding the dynamics of physical systems.
Orbital period: The orbital period is the time taken for a satellite or celestial body to complete one full orbit around another object. It is typically measured in seconds, minutes, hours, or years.
Overdamped: An overdamped system is one in which the damping force is so strong that it prevents oscillations and the system returns to equilibrium without oscillating. The motion is slow and exponential as the system approaches equilibrium.
Overdamped: Overdamped refers to a system that exhibits damped oscillations, where the oscillations decay rapidly and the system returns to its equilibrium position without any oscillations. This term is particularly relevant in the context of 15.5 Damped Oscillations, where it describes a specific type of damped motion.
Period: The period of a periodic phenomenon is the time taken for one complete cycle or repetition of the event. This concept is fundamental in understanding various physics topics, including uniform circular motion, simple harmonic motion, and wave phenomena.
Quality Factor: The quality factor, or Q-factor, is a dimensionless parameter that describes the quality or performance of a resonant system. It quantifies the ratio of a system's stored energy to its dissipated energy, and is an important concept in the analysis of damped and forced oscillations.
Resonance: Resonance occurs when a system is driven at its natural frequency, leading to a significant increase in amplitude. It is a crucial concept in oscillations and wave phenomena.
Resonance: Resonance is a phenomenon that occurs when a system is driven by a force that matches the system's natural frequency of oscillation, leading to a significant increase in the amplitude of the system's response. This concept is fundamental across various fields in physics, including mechanics, acoustics, and electromagnetism.
Restoring force: A restoring force is a force that gives rise to an equilibrium in a physical system. It acts in the direction opposite to the displacement of the object, aiming to bring it back to its equilibrium position.
Restoring Force: The restoring force is the force that acts to return an object or system to its equilibrium or original state after it has been displaced or disturbed from that state. This force plays a crucial role in understanding various physical phenomena, including the behavior of oscillating systems, the stability of structures, and the energy changes associated with different types of motion.
Spring Constant: The spring constant, often denoted as 'k', is a measure of the stiffness of a spring. It quantifies the force required to stretch or compress a spring by a unit distance, and it is a fundamental property of a spring that is crucial in understanding its behavior in various physical contexts.
Underdamped: An underdamped system is one where the damping force is not strong enough to prevent oscillations. It results in oscillatory motion with gradually decreasing amplitude over time.
Underdamped: Underdamped refers to a system that oscillates with a decreasing amplitude after being disturbed from its equilibrium position. In this type of damping, the system exhibits oscillatory behavior as it returns to its equilibrium state.
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15.6 Forced Oscillations