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f = 1 2 Ï k m f = 1 2 Ï k m | https://openstax.org/books/university-physics-volume-1/pages/15-key-equations |
E Total = 1 2 k x 2 + 1 2 m v 2 = 1 2 k A 2 E Total = 1 2 k x 2 + 1 2 m v 2 = 1 2 k A 2 | https://openstax.org/books/university-physics-volume-1/pages/15-key-equations |
v = ± k m ( A 2 â x 2 ) v = ± k m ( A 2 â x 2 ) | https://openstax.org/books/university-physics-volume-1/pages/15-key-equations |
x ( t ) = A cos ( Ï t + Ï ) x ( t ) = A cos ( Ï t + Ï ) | https://openstax.org/books/university-physics-volume-1/pages/15-key-equations |
v ( t ) = â v max sin ( Ï t + Ï ) v ( t ) = â v max sin ( Ï t + Ï ) | https://openstax.org/books/university-physics-volume-1/pages/15-key-equations |
a ( t ) = â a max cos ( Ï t + Ï ) a ( t ) = â a max cos ( Ï t + Ï ) | https://openstax.org/books/university-physics-volume-1/pages/15-key-equations |
d 2 θ d t 2 = â g L θ d 2 θ d t 2 = â g L θ | https://openstax.org/books/university-physics-volume-1/pages/15-key-equations |
Ï = g L Ï = g L | https://openstax.org/books/university-physics-volume-1/pages/15-key-equations |
T = 2 Ï L g T = 2 Ï L g | https://openstax.org/books/university-physics-volume-1/pages/15-key-equations |
Ï = m g L I Ï = m g L I | https://openstax.org/books/university-physics-volume-1/pages/15-key-equations |
T = 2 Ï I m g L T = 2 Ï I m g L | https://openstax.org/books/university-physics-volume-1/pages/15-key-equations |
T = 2 Ï I κ T = 2 Ï I κ | https://openstax.org/books/university-physics-volume-1/pages/15-key-equations |
m d 2 x d t 2 + b d x d t + k x = 0 m d 2 x d t 2 + b d x d t + k x = 0 | https://openstax.org/books/university-physics-volume-1/pages/15-key-equations |
x ( t ) = A 0 e â b 2 m t cos ( Ï t + Ï ) x ( t ) = A 0 e â b 2 m t cos ( Ï t + Ï ) | https://openstax.org/books/university-physics-volume-1/pages/15-key-equations |
Ï 0 = k m Ï 0 = k m | https://openstax.org/books/university-physics-volume-1/pages/15-key-equations |
Ï = Ï 0 2 â ( b 2 m ) 2 Ï = Ï 0 2 â ( b 2 m ) 2 | https://openstax.org/books/university-physics-volume-1/pages/15-key-equations |
â k x â b d x d t + F o sin ( Ï t ) = m d 2 x d t 2 â k x â b d x d t + F o sin ( Ï t ) = m d 2 x d t 2 | https://openstax.org/books/university-physics-volume-1/pages/15-key-equations |
x ( t ) = A cos ( Ï t + Ï ) x ( t ) = A cos ( Ï t + Ï ) | https://openstax.org/books/university-physics-volume-1/pages/15-key-equations |
A = F o m 2 ( Ï 2 â Ï o 2 ) 2 + b 2 Ï 2 A = F o m 2 ( Ï 2 â Ï o 2 ) 2 + b 2 Ï 2 | https://openstax.org/books/university-physics-volume-1/pages/15-key-equations |
amplitude (A) : maximum displacement from the equilibrium position of an object oscillating around the equilibrium position | https://openstax.org/books/university-physics-volume-1/pages/15-key-terms |
critically damped : condition in which the damping of an oscillator causes it to return as quickly as possible to its equilibrium position without oscillating back and forth about this position | https://openstax.org/books/university-physics-volume-1/pages/15-key-terms |
elastic potential energy : potential energy stored as a result of deformation of an elastic object, such as the stretching of a spring | https://openstax.org/books/university-physics-volume-1/pages/15-key-terms |
equilibrium position : position where the spring is neither stretched nor compressed | https://openstax.org/books/university-physics-volume-1/pages/15-key-terms |
force constant (k) : characteristic of a spring which is defined as the ratio of the force applied to the spring to the displacement caused by the force | https://openstax.org/books/university-physics-volume-1/pages/15-key-terms |
frequency (f) : number of events per unit of time | https://openstax.org/books/university-physics-volume-1/pages/15-key-terms |
natural angular frequency : angular frequency of a system oscillating in SHM | https://openstax.org/books/university-physics-volume-1/pages/15-key-terms |
oscillation : single fluctuation of a quantity, or repeated and regular fluctuations of a quantity, between two extreme values around an equilibrium or average value | https://openstax.org/books/university-physics-volume-1/pages/15-key-terms |
overdamped : condition in which damping of an oscillator causes it to return to equilibrium without oscillating; oscillator moves more slowly toward equilibrium than in the critically damped system | https://openstax.org/books/university-physics-volume-1/pages/15-key-terms |
period (T) : time taken to complete one oscillation | https://openstax.org/books/university-physics-volume-1/pages/15-key-terms |
periodic motion : motion that repeats itself at regular time intervals | https://openstax.org/books/university-physics-volume-1/pages/15-key-terms |
phase shift : angle, in radians, that is used in a cosine or sine function to shift the function left or right, used to match up the function with the initial conditions of data | https://openstax.org/books/university-physics-volume-1/pages/15-key-terms |
physical pendulum : any extended object that swings like a pendulum | https://openstax.org/books/university-physics-volume-1/pages/15-key-terms |
resonance : large amplitude oscillations in a system produced by a small amplitude driving force, which has a frequency equal to the natural frequency | https://openstax.org/books/university-physics-volume-1/pages/15-key-terms |
restoring force : force acting in opposition to the force caused by a deformation | https://openstax.org/books/university-physics-volume-1/pages/15-key-terms |
simple harmonic motion (SHM) : oscillatory motion in a system where the restoring force is proportional to the displacement, which acts in the direction opposite to the displacement | https://openstax.org/books/university-physics-volume-1/pages/15-key-terms |
simple harmonic oscillator : a device that oscillates in SHM where the restoring force is proportional to the displacement and acts in the direction opposite to the displacement | https://openstax.org/books/university-physics-volume-1/pages/15-key-terms |
simple pendulum : point mass, called a pendulum bob, attached to a near massless string | https://openstax.org/books/university-physics-volume-1/pages/15-key-terms |
stable equilibrium point : point where the net force on a system is zero, but a small displacement of the mass will cause a restoring force that points toward the equilibrium point | https://openstax.org/books/university-physics-volume-1/pages/15-key-terms |
torsional pendulum : any suspended object that oscillates by twisting its suspension | https://openstax.org/books/university-physics-volume-1/pages/15-key-terms |
underdamped : condition in which damping of an oscillator causes the amplitude of oscillations of a damped harmonic oscillator to decrease over time, eventually approaching zero | https://openstax.org/books/university-physics-volume-1/pages/15-key-terms |
Periodic motion is a repeating oscillation. The time for one oscillation is the periodTand the number of oscillations per unit time is the frequencyf. These quantities are related byf=1Tf=1T. | https://openstax.org/books/university-physics-volume-1/pages/15-summary |
Simple harmonic motion (SHM) is oscillatory motion for a system where the restoring force is proportional to the displacement and acts in the direction opposite to the displacement. | https://openstax.org/books/university-physics-volume-1/pages/15-summary |
Maximum displacement is the amplitudeA. The angular frequencyÏÏ, periodT, and frequencyfof a simple harmonic oscillator are given byÏ=kmÏ=km,T=2Ïmk,andf=12ÏkmT=2Ïmk,andf=12Ïkm, wheremis the mass of the system andkis the force constant. | https://openstax.org/books/university-physics-volume-1/pages/15-summary |
Displacement as a function of time in SHM is given byx(t)=Acos(2ÏTt+Ï)=Acos(Ït+Ï)x(t)=Acos(2ÏTt+Ï)=Acos(Ït+Ï). | https://openstax.org/books/university-physics-volume-1/pages/15-summary |
The velocity is given byv(t)=âAÏsin(Ït+Ï)=âvmaxsin(Ït+Ï),v(t)=âAÏsin(Ït+Ï)=âvmaxsin(Ït+Ï),wherevmax=AÏ=Akmvmax=AÏ=Akm. | https://openstax.org/books/university-physics-volume-1/pages/15-summary |
The acceleration isa(t)=âAÏ2cos(Ït+Ï)=âamaxcos(Ït+Ï)a(t)=âAÏ2cos(Ït+Ï)=âamaxcos(Ït+Ï), whereamax=AÏ2=Akmamax=AÏ2=Akm. | https://openstax.org/books/university-physics-volume-1/pages/15-summary |
The simplest type of oscillations are related to systems that can be described by Hookeâs law,F= âkx, whereFis the restoring force,xis the displacement from equilibrium or deformation, andkis the force constant of the system. | https://openstax.org/books/university-physics-volume-1/pages/15-summary |
Elastic potential energyUstored in the deformation of a system that can be described by Hookeâs law is given byU=12kx2.U=12kx2. | https://openstax.org/books/university-physics-volume-1/pages/15-summary |
Energy in the simple harmonic oscillator is shared between elastic potential energy and kinetic energy, with the total being constant:ETotal=12mv2+12kx2=12kA2=constant.ETotal=12mv2+12kx2=12kA2=constant. | https://openstax.org/books/university-physics-volume-1/pages/15-summary |
The magnitude of the velocity as a function of position for the simple harmonic oscillator can be found by using|v|=km(A2âx2).|v|=km(A2âx2). | https://openstax.org/books/university-physics-volume-1/pages/15-summary |
A projection of uniform circular motion undergoes simple harmonic oscillation. | https://openstax.org/books/university-physics-volume-1/pages/15-summary |
Consider a circle with a radiusA, moving at a constant angular speedÏÏ. A point on the edge of the circle moves at a constant tangential speed ofvmax=AÏvmax=AÏ. The projection of the radius onto thex-axis isx(t)=Acos(Ït+Ï)x(t)=Acos(Ït+Ï), where(Ï)(Ï)is the phase shift. Thex-component of the tangential velocit... | https://openstax.org/books/university-physics-volume-1/pages/15-summary |
A massmsuspended by a wire of lengthLand negligible mass is a simple pendulum and undergoes SHM for amplitudes less than about15°15°. The period of a simple pendulum isT=2ÏLgT=2ÏLg, whereLis the length of the string andgis the acceleration due to gravity. | https://openstax.org/books/university-physics-volume-1/pages/15-summary |
The period of a physical pendulumT=2ÏImgLT=2ÏImgLcan be found if the moment of inertia is known. The length between the point of rotation and the center of mass isL. | https://openstax.org/books/university-physics-volume-1/pages/15-summary |
The period of a torsional pendulumT=2ÏIκT=2ÏIκcan be found if the moment of inertia and torsion constant are known. | https://openstax.org/books/university-physics-volume-1/pages/15-summary |
Damped harmonic oscillators have non-conservative forces that dissipate their energy. | https://openstax.org/books/university-physics-volume-1/pages/15-summary |
Critical damping returns the system to equilibrium as fast as possible without overshooting. | https://openstax.org/books/university-physics-volume-1/pages/15-summary |
An underdamped system will oscillate through the equilibrium position. | https://openstax.org/books/university-physics-volume-1/pages/15-summary |
An overdamped system moves more slowly toward equilibrium than one that is critically damped. | https://openstax.org/books/university-physics-volume-1/pages/15-summary |
A systemâs natural frequency is the frequency at which the system oscillates if not affected by driving or damping forces. | https://openstax.org/books/university-physics-volume-1/pages/15-summary |
A periodic force driving a harmonic oscillator at its natural frequency produces resonance. The system is said to resonate. | https://openstax.org/books/university-physics-volume-1/pages/15-summary |
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. | https://openstax.org/books/university-physics-volume-1/pages/15-summary |
v = λ T = λ f v = λ T = λ f | https://openstax.org/books/university-physics-volume-1/pages/16-key-equations |
μ = mass of the string length of the string μ = mass of the string length of the string | https://openstax.org/books/university-physics-volume-1/pages/16-key-equations |
| v | = F T μ | v | = F T μ | https://openstax.org/books/university-physics-volume-1/pages/16-key-equations |
v = Î Ï v = Î Ï | https://openstax.org/books/university-physics-volume-1/pages/16-key-equations |
y R ( x , t ) = [ 2 A cos ( Ï 2 ) ] sin ( k x â Ï t + Ï 2 ) y R ( x , t ) = [ 2 A cos ( Ï 2 ) ] sin ( k x â Ï t + Ï 2 ) | https://openstax.org/books/university-physics-volume-1/pages/16-key-equations |
v = Ï k v = Ï k | https://openstax.org/books/university-physics-volume-1/pages/16-key-equations |
y ( x , t ) = A sin ( k x â Ï t + Ï ) y ( x , t ) = A sin ( k x â Ï t + Ï ) | https://openstax.org/books/university-physics-volume-1/pages/16-key-equations |
k x â Ï t + Ï k x â Ï t + Ï | https://openstax.org/books/university-physics-volume-1/pages/16-key-equations |
â 2 y ( x , t ) â x 2 = 1 v w 2 â 2 y ( x , t ) â t 2 â 2 y ( x , t ) â x 2 = 1 v w 2 â 2 y ( x , t ) â t 2 | https://openstax.org/books/university-physics-volume-1/pages/16-key-equations |
P ave = E λ T = 1 2 μ A 2 Ï 2 λ T = 1 2 μ A 2 Ï 2 v P ave = E λ T = 1 2 μ A 2 Ï 2 λ T = 1 2 μ A 2 Ï 2 v | https://openstax.org/books/university-physics-volume-1/pages/16-key-equations |
I = P A I = P A | https://openstax.org/books/university-physics-volume-1/pages/16-key-equations |
I = P 4 Ï r 2 I = P 4 Ï r 2 | https://openstax.org/books/university-physics-volume-1/pages/16-key-equations |
y ( x , t ) = [ 2 A sin ( k x ) ] cos ( Ï t ) y ( x , t ) = [ 2 A sin ( k x ) ] cos ( Ï t ) | https://openstax.org/books/university-physics-volume-1/pages/16-key-equations |
λ n = 2 n L , n = 1 , 2 , 3 , 4 , 5 ... λ n = 2 n L , n = 1 , 2 , 3 , 4 , 5 ... | https://openstax.org/books/university-physics-volume-1/pages/16-key-equations |
f n = n v 2 L = n f 1 , n = 1 , 2 , 3 , 4 , 5 ... f n = n v 2 L = n f 1 , n = 1 , 2 , 3 , 4 , 5 ... | https://openstax.org/books/university-physics-volume-1/pages/16-key-equations |
antinode : location of maximum amplitude in standing waves | https://openstax.org/books/university-physics-volume-1/pages/16-key-terms |
constructive interference : when two waves arrive at the same point exactly in phase; that is, the crests of the two waves are precisely aligned, as are the troughs | https://openstax.org/books/university-physics-volume-1/pages/16-key-terms |
destructive interference : when two identical waves arrive at the same point exactly out of phase; that is, precisely aligned crest to trough | https://openstax.org/books/university-physics-volume-1/pages/16-key-terms |
fixed boundary condition : when the medium at a boundary is fixed in place so it cannot move | https://openstax.org/books/university-physics-volume-1/pages/16-key-terms |
free boundary condition : exists when the medium at the boundary is free to move | https://openstax.org/books/university-physics-volume-1/pages/16-key-terms |
fundamental frequency : lowest frequency that will produce a standing wave | https://openstax.org/books/university-physics-volume-1/pages/16-key-terms |
intensity (I) : power per unit area | https://openstax.org/books/university-physics-volume-1/pages/16-key-terms |
interference : overlap of two or more waves at the same point and time | https://openstax.org/books/university-physics-volume-1/pages/16-key-terms |
linear wave equation : equation describing waves that result from a linear restoring force of the medium; any function that is a solution to the wave equation describes a wave moving in the positivex-direction or the negativex-direction with a constant wave speedv | https://openstax.org/books/university-physics-volume-1/pages/16-key-terms |
longitudinal wave : wave in which the disturbance is parallel to the direction of propagation | https://openstax.org/books/university-physics-volume-1/pages/16-key-terms |
mechanical wave : wave that is governed by Newtonâs laws and requires a medium | https://openstax.org/books/university-physics-volume-1/pages/16-key-terms |
node : point where the string does not move; more generally, nodes are where the wave disturbance is zero in a standing wave | https://openstax.org/books/university-physics-volume-1/pages/16-key-terms |
normal mode : possible standing wave pattern for a standing wave on a string | https://openstax.org/books/university-physics-volume-1/pages/16-key-terms |
overtone : frequency that produces standing waves and is higher than the fundamental frequency | https://openstax.org/books/university-physics-volume-1/pages/16-key-terms |
pulse : single disturbance that moves through a medium, transferring energy but not mass | https://openstax.org/books/university-physics-volume-1/pages/16-key-terms |
standing wave : wave that can bounce back and forth through a particular region, effectively becoming stationary | https://openstax.org/books/university-physics-volume-1/pages/16-key-terms |
superposition : phenomenon that occurs when two or more waves arrive at the same point | https://openstax.org/books/university-physics-volume-1/pages/16-key-terms |
transverse wave : wave in which the disturbance is perpendicular to the direction of propagation | https://openstax.org/books/university-physics-volume-1/pages/16-key-terms |
wave : disturbance that moves from its source and carries energy | https://openstax.org/books/university-physics-volume-1/pages/16-key-terms |
wave function : mathematical model of the position of particles of the medium | https://openstax.org/books/university-physics-volume-1/pages/16-key-terms |
wave number : 2Ïλ2Ïλ | https://openstax.org/books/university-physics-volume-1/pages/16-key-terms |
wave speed : magnitude of the wave velocity | https://openstax.org/books/university-physics-volume-1/pages/16-key-terms |
wave velocity : velocity at which the disturbance moves; also called the propagation velocity | https://openstax.org/books/university-physics-volume-1/pages/16-key-terms |
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