Periodic Motion, Waves, and Sound Flashcards
What are the two characteristics of a linear restoring force?
- The force is toward the equilibrium position.
- The magnitude (and acceleration) is proportional to the displacement.
F = -kx a = -xω^2 ω = (k/m)^1/2
What type of systems exhibit continuous repetitive movement?
Oscillating Systems
Simple Harmonic Motion
An object’s oscillation around an equilibrium point due to an elastic linear restoring force. If the path of a particle moving with uniform circular motion were on a line, the particle would oscillate between the points of maximum displacement. Examples are a simple pendulum and a mass-spring.
What is the measure of a spring’s stiffness and what does it mean if the number is large?
A large spring constant indicates a stiffer spring.
Hooke’s Law
F = -kx for a mass-spring
Angular Frequency
ω = (k/m)^1/2 = 2πf = 2π/T
v = fλ = ω/k = λ/T
Period
The number of seconds it takes to complete one cycle. T = 1/f
Amplitude
It is the point of maximum displacement.
Where do the points of maximum potential and kinetic energy occur in oscillating systems?
At the equilibrium point, potential energy is zero and kinetic energy is at its maximum. At the points of maximum displacement, kinetic energy is zero and potential energy is at its maximum.
Where does maximum force occur in an oscillating system?
At maximum displacement
Equations for a mass-spring
T = 2π(m/k)^1/2 ω = (k/m)^1/2 KE = 1/2mv^2 U = 1/2kx^2
Equations for simple pendulum (θ < 10°)
k = mg/L T = 2π(L/g)^1/2 ω = (g/L)^1/2 KE = 1/2mv^2 U = mgh
Transverse Waves
Like light, where the particles oscillate perpendicular to the direction of motion.
Longitudinal Waves
Like sound, the particles oscillate parallel to the direction of motion.
Displacement in a wave
y = Y sin(kx - ωt)
k is the wave number = 2π/λ
Wavelength
Distance between two equivalent consecutive points on a wave.
Frequency
Number of cycles per second (Hz)
Phase Difference
Angle that a sine curve leads or lags another, meaning that the waves’ crests and troughs occur at different points in time.
Node
Point of zero displacement in a standing wave
Anti-node
Point of maximum displacement in a standing wave
Constructive Interference
In phase overlapping waves’ amplitudes add together.
Destructive Interference
Out of phase overlapping waves’ amplitudes subtract.
Wave Speed
v = fλ = ω/k = λ/T
Traveling Wave
A propagating wave that reflects and inverts upon reaching it’s fixed boundary. The two waves interfere with each other.
Standing Waves
Between two fixed nodes only certain wave frequencies can occur. Examples: strings and pipes
λ = 2L/n
Higher harmonics have shorter wavelengths and higher frequencies, but the same wave speed.
Resonance
Without external forces, the system oscillates at a natural frequency, where the amplitude will reach its maximum.
Free swinging pendulum: f = (1/2π)(g/L)^1/2 ➡️ only one natural frequency
Mass-spring: f = (1/2π)(k/m)^1/2 ➡️ infinite natural frequencies
Forced Oscillation
Application of periodically varying force that is usually small unless close to the natural frequency of the system.
How does sound travel?
In a longitudinal wave that causes a mechanical disturbance in a deformable medium, with a relative speed that depends on the particle spacing. It cannot travel through a vacuum and it travels faster through a solid than through a liquid or gas.
In air, at 0°C, sound travels at 331m/s
Audible Waves for Humans
20 Hz to 20000 Hz
Below infrasonic
Above ultrasonic
Intensity
P = IA
Units: W/m^2
Sound Level
β = 10 log (I/Io) Io = 10^-12 W/m^2
Beats
The absolute differences in frequencies of two waves
Doppler Effect
f’ = f (V +/- VD) / (V +/- VS)
The frequency detected us less than the actual frequency when the source and detector move toward each other. No frequency shift occurs when the source and detector are moving in the same directions at the same speed.
Harmonic Series
All the possible frequencies a standing wave can support.
Fundamental Frequency
First harmonic of standing waves
f = nv/2L for strings fixed and pipes open at both ends
f = nv/4L for pipes closed at one end (odd integer only)
