Oscillations Physics Lesson 15 by Owen Borville 12.9.2025
Periodic motion is a type of repeating oscillation, or back and forth motion of an object between two points of deformation. The time for one oscillation is the period (T) and the number of oscillations per unit time is the frequency (f).
f (frequency) = 1/T
An oscillation may create a wave, a disturbance that travels from where it was created.
Simple harmonic motion (SHM) is oscillatory motion for a system, described by Hooke's Law, where the restoring force is proportional to the displacement and acts in the direction opposite to the displacement. Maximum displacement is the amplitude A. This system is also called a simple harmonic oscillator.
Angular frequency ω, period T, and frequency f of a simple harmonic oscillator (mass-spring) is:
ω = √k/m
T = 2π√m/k
f = 1/2π√k/m
where m is the mass of the system and k is the force constant.
Maximum displacement is the amplitude X.
Displacement as a function of time in SHM is: x(t) = Xcos(2πt/T) or Acos(2π/T*t + φ) = Acos (ωt + φ), where φ is phase shift measured in radians.
Velocity is v(t) = -Aωsin(ωt + φ) = -vmax*sin (ωt + φ), where vmax = Aω = A√k/m or -vmaxsin(2πt/T). v.max = √k/mX
Acceleration is a(t) = -Aω^2cos(ωt + φ) = -amax*cos(ωt + φ), where amax = Aω^2 = A*k/m. Also, a(t) = -kX/m cos 2πt/T
The maximum displacement (amplitude) of SHM is xmax = A
Hooke's Law (simplest oscillations) F = -kx, where F is the restoring force, x is the displacement from equilibrium or deformation, and k is the force constant of the system.
Elastic Potential energy (U) stored in the deformation of a system that can be described by Hooke's law is = PEel or U = 1/2kx^2
Energy in the simple harmonic oscillator is shared between elastic potential energy and kinetic energy, with the total being constant:
Etotal = 1/2mv^2 + 1/2kx^2 = 1/2kA^2 = constant.
The magnitude of the velocity as a function of position for the simple harmonic oscillator is = |v| = √k/m*(A^2-x^2)
An an example or model of simple harmonic motion is uniform circular motion. In the circle is radius A and constant angular speed ω. A point on the edge of the circle moves at a constant tangential speed of vmax = Aω. The projection of the radius onto the x-axis is x(t) = Acos(ωt+φ), where (φ) is the phase shift. The x-component of the tangential velocity is v(t) = -Aωsin(ωt + φ).
Maximum velocity depends on three factors: (1) it is directly proportional to amplitude (2) it is greater for stiffer systems, (3) and it is smaller or objects that have larger masses: v(max) = √k/mX
Maximum velocity of SHM = |vmax| = Aω
Maximum acceleration of SHM = |amax| = Aω^2
A Simple Pendulum contains a mass m suspended by a wire of length L and a negligible mass. This simple pendulum undergoes SHM for amplitudes less than about 15 degrees. The period of a simple pendulum is T = 2π√L/g, where L is the length of the string and g is the acceleration due to gravity.
Angular frequency for a simple pendulum is ω = √g/L
The period of a physical pendulum T = 2π√I/mgL where I is the moment of inertia. The length between the point of rotation and the center of mass is L. Angular frequency is ω = √mgL/I
The period of a torsional pendulum T = 2π√I/κ where I is the moment of inertia and κ is the torsion constant.
Damping in physics is any effect that reduces the amplitude of oscillations in an oscillating system, causing it to lose energy over time until it comes to rest.
Damped harmonic oscillators have non-conservative forces that dissipate their energy. Critical damping returns the system to equilibrium as fast as possible without overshooting.
Underdamped systems will oscillate through the equilibrium position. Overdamped systems move more slowly toward equilibrium than one that is critically damped.
Forced oscillations occur when an oscillating system is continuously driven by an external, periodic force, causing it to vibrate at the driving force's frequency, not its own natural frequency.
A system's natural frequency is the frequency at which the system oscillates if not affected by driving or damping forces. A periodic force driving a harmonic oscillator at its natural frequency produces resonance and the system resonates.
The less damping a system has, the higher the amplitude of the forced oscillations near resonance. The more damping a system has, the broader response it has to varying driving frequencies.
Periodic motion is a type of repeating oscillation, or back and forth motion of an object between two points of deformation. The time for one oscillation is the period (T) and the number of oscillations per unit time is the frequency (f).
f (frequency) = 1/T
An oscillation may create a wave, a disturbance that travels from where it was created.
Simple harmonic motion (SHM) is oscillatory motion for a system, described by Hooke's Law, where the restoring force is proportional to the displacement and acts in the direction opposite to the displacement. Maximum displacement is the amplitude A. This system is also called a simple harmonic oscillator.
Angular frequency ω, period T, and frequency f of a simple harmonic oscillator (mass-spring) is:
ω = √k/m
T = 2π√m/k
f = 1/2π√k/m
where m is the mass of the system and k is the force constant.
Maximum displacement is the amplitude X.
Displacement as a function of time in SHM is: x(t) = Xcos(2πt/T) or Acos(2π/T*t + φ) = Acos (ωt + φ), where φ is phase shift measured in radians.
Velocity is v(t) = -Aωsin(ωt + φ) = -vmax*sin (ωt + φ), where vmax = Aω = A√k/m or -vmaxsin(2πt/T). v.max = √k/mX
Acceleration is a(t) = -Aω^2cos(ωt + φ) = -amax*cos(ωt + φ), where amax = Aω^2 = A*k/m. Also, a(t) = -kX/m cos 2πt/T
The maximum displacement (amplitude) of SHM is xmax = A
Hooke's Law (simplest oscillations) F = -kx, where F is the restoring force, x is the displacement from equilibrium or deformation, and k is the force constant of the system.
Elastic Potential energy (U) stored in the deformation of a system that can be described by Hooke's law is = PEel or U = 1/2kx^2
Energy in the simple harmonic oscillator is shared between elastic potential energy and kinetic energy, with the total being constant:
Etotal = 1/2mv^2 + 1/2kx^2 = 1/2kA^2 = constant.
The magnitude of the velocity as a function of position for the simple harmonic oscillator is = |v| = √k/m*(A^2-x^2)
An an example or model of simple harmonic motion is uniform circular motion. In the circle is radius A and constant angular speed ω. A point on the edge of the circle moves at a constant tangential speed of vmax = Aω. The projection of the radius onto the x-axis is x(t) = Acos(ωt+φ), where (φ) is the phase shift. The x-component of the tangential velocity is v(t) = -Aωsin(ωt + φ).
Maximum velocity depends on three factors: (1) it is directly proportional to amplitude (2) it is greater for stiffer systems, (3) and it is smaller or objects that have larger masses: v(max) = √k/mX
Maximum velocity of SHM = |vmax| = Aω
Maximum acceleration of SHM = |amax| = Aω^2
A Simple Pendulum contains a mass m suspended by a wire of length L and a negligible mass. This simple pendulum undergoes SHM for amplitudes less than about 15 degrees. The period of a simple pendulum is T = 2π√L/g, where L is the length of the string and g is the acceleration due to gravity.
Angular frequency for a simple pendulum is ω = √g/L
The period of a physical pendulum T = 2π√I/mgL where I is the moment of inertia. The length between the point of rotation and the center of mass is L. Angular frequency is ω = √mgL/I
The period of a torsional pendulum T = 2π√I/κ where I is the moment of inertia and κ is the torsion constant.
Damping in physics is any effect that reduces the amplitude of oscillations in an oscillating system, causing it to lose energy over time until it comes to rest.
Damped harmonic oscillators have non-conservative forces that dissipate their energy. Critical damping returns the system to equilibrium as fast as possible without overshooting.
Underdamped systems will oscillate through the equilibrium position. Overdamped systems move more slowly toward equilibrium than one that is critically damped.
Forced oscillations occur when an oscillating system is continuously driven by an external, periodic force, causing it to vibrate at the driving force's frequency, not its own natural frequency.
A system's natural frequency is the frequency at which the system oscillates if not affected by driving or damping forces. A periodic force driving a harmonic oscillator at its natural frequency produces resonance and the system resonates.
The less damping a system has, the higher the amplitude of the forced oscillations near resonance. The more damping a system has, the broader response it has to varying driving frequencies.