Warmup: Good Vibrations Physics Warmup #114 The source of all waves is a vibrating object.
********************************************************************************************* Complete the table below by identifying the source of each wave described.
Wave Source
A tuning fork is struck with a rubber hammer, producing a sound wave. A motorboat moves through the water, leaving its wake behind. A performer sings a high note.
A light bulb gives off light.
the vibrating tuning fork
the propeller blades
vocal cords
• The motion of an oscillating object depends on the restoring forces that make it go back and forth.
• The simplest type of restoring force is a spring force.
Hooke’s Law: Fs = -kx where k is the spring constant and x is the displacement
• The negative sign indicates that the force is opposite to the displacement from the springs relaxed position.
• Motion under the influence of the type of force described by Hooke’s Law is called:
simple harmonic motion (SHM)
A block on a spring undergoes simple harmonic motion.
a) The block is at the equilibrium position, x = 0.
a) The force of a hand, Fh, pulls the block for a displacement of x = A. The force of the spring is Fs.
• At the time of the release, t = 0.
• The time it takes to complete one period of oscillation is T.
c) At t = T/4, the block is back at the equilibrium position.
d) at t = T/2, the block is at x = -A.
e) During the next half of the cycle, the motion is to the right.
displacement - the distance of an object, including direction ( x), from its equilibrium position.
amplitude (A) - the magnitude of the maximum displacement of a mass from its equilibrium position.
period (T) - the time needed to complete one cycle of oscillation.
frequency (f) - the number of cycles per second.
• frequency and period are related by:
f = 1
T
Observe how sinusoidal curve is traced out on the moving paper.
Since the object’s initial displacement is +A, the equation can be written as
damped harmonic motion - without a driving force, the amplitude or energy of an oscillating body will decrease with time.
A simple pendulum consists of a small, heavy object on a string.
For small angles of oscillation (Ɵ < 10°), a good approximation for period is:
T = 2
L
Period of a simple pendulum On Gold Sheet
g
where L is the length of the string g is the acceleration due to gravity
13.2: Equations of Motion
Example 13.3: An object with a mass of 1.0 kg is attached to a spring with a spring constant of 10 N/m. The object is displaces by 3.0 cm from the
equilibrium position and let go.
13.2: Equations of Motion
Example 13.4: The pendulum of a grandfather clock is 1.0 m long.
a) What is its period on the Earth?
Wave
Transverse Wave :Wave that travels perpendicular to the
disturbance that cause the wave.
Longitudinal Wave that travels parallel to the disturbance that
caused the wave.
Wave Source
A sound wave
A water wave caused by a boat moving
A wave in a rope caused by one end being moved up and down
A wave in a coiled spring caused by pushing one end in and out repeatedly
A light wave
transverse
longitudinal longitudinal
transverse
13.3: Wave Motion
wave motion - the propagation of a disturbance (energy and momentum) through a material.
• Only energy is transferred, not matter.
periodic wave - requires a disturbance from an oscillation source. • If the driving source maintains constant amplitude of the wave, the result is SHM.
A periodic wave can be characterized by the following:
amplitude - the magnitude of displacement of the particles of the material from their equilibrium position.
wavelength - the distance between two successive crests or troughs.
frequency - the number of wavelengths that passes by a given point in a second.
wave speed - the speed of wave motion (speed of a crest or trough) given by:
v =
f =
/T
13.3: Wave Motion
Example 13.6: A student reading her physics book on a lake dock notices that the distance between two incoming wave crests is about 2.4 m, and she then measures the time of arrival between wave crests to be 1.6 s. What is the approximate speed of the waves?
Answer:
= 2.4 m T = 1.6 s
13.3: Wave Motion
transverse wave - the particle motion is perpendicular to the direction of the wave velocity.
ex: guitar string; electromagnetic wave
longitudinal wave - the particle oscillation is parallel to the direction of the wave velocity.
• also called a compressional wave
• can propagate in solids, liquids, or gases
ex: sound waves
Combination of transverse and longitudinal waves: ex: seismic, water
View Wave Motion:
http://paws.kettering.edu/~drussell/Demos/w aves/wavemotion.html
13.4: Wave Phenomena
interference - when waves meet or overlap
principle of superposition - at any time, the waveform of two or more interfering waves is given by the sum of the displacements of the individual waves at each point in the medium.
constructive interference - if the amplitude of the combined wave is greater than that of any of the individual waves.
destructive interference - if the amplitude of the combined wave is smaller than that of any of the individual waves.
Wave Interference 2:
total constructitve interference - when two waves of the same frequency and amplitude are exactly in phase.
ex: the crest of one wave is aligned with the crest of another.
total destructive interference - when two waves of the same frequency and amplitude are completely 180° out of phase.
13.4: Wave Phenomena
reflection - when a wave strikes and object or comes to a boundary of another medium and is at least partly bounced back.
ex: an echo is a reflected sound wave ex: a mirror reflects light waves
a) When a wave is reflected from a fixed boundary, the reflected wave is inverted, or undergoes a 180° phase shift.
13.4: Wave Phenomena
refraction - when a wave crosses a boundary into another medium and the transmitted wave moves in a different direction.
• When a wave crosses a boundary into another medium, its speed changes.
• When the incident wave enters at an angle, the transmitted wave moves in a different direction.
• Generally, when a wave strikes the boundary, both reflection and refraction occur.
a) Refraction of water waves.
13.4: Wave Phenomena
dispersion - waves of different frequencies spread apart form one another.
• Nondispersive waves travel at the same speed which is determined solely by the properties of the medium. They do not depend on the wavelength (or frequency) or the wave.
ex: a wave on a string, sound
• Dispersion examples: light in water, rainbows
13.4: Wave Phenomena
diffraction - the bending of waves around and object.
ex: A person in a room with an open door can hear sound from outside the room, partially due to diffraction.
13.4: Wave Phenomena: Check for Understanding
1. When waves meet each other and iinterfere, the resultant waveform is determined by
a) reflection
b) refraction
c) diffraction
d) superposition
13.4: Wave Phenomena: Check for Understanding
2. Refraction
a) involves constructive interference
b) refers to a change in direction at the media interfaces
c) is synonymous with diffraction
d) occurs only for mechanical waves, not light
Warmup: Foot Stompin’ Physics Warmup #117
Resonance occurs when the frequency of a forced vibration on an object matches the object’s natural frequency. This causes a great increase in amplitude, which increases the power transmitted by the object. In 1940, the Tacoma Narrows
suspension bridge collapsed when wind-driven oscillations produced resonance in the bridge. Films of its collapse have become favorites among physics teachers an their students. Subsequent designs have incorporated such innovations as
separate parallel roadways as a way to keep this type of disaster from happening again.
********************************************************************************************* In the 1800’s, English soldiers marching across a small suspension bridge caused it to collapse when their marching set it into resonance. Their marching was in rhythm with the bridge’s natural frequency. Since that time, soldiers and marching bands have been told to not march in step when crossing any type of suspension bridge.
Standing Waves and Resonance
standing wave - occur when interfering waves of the same frequency and amplitude traveling in opposite directions, such as in a rope
• Standing waves can be generated in a rope by more than one driving frequency.
• The higher the frequency, the more “loops” in the rope.
nodes - the points on the rope that are always stationary due to destructive interference
• adjacent nodes are separated by a half wavelength, or
antinodes - the points of maximum amplitude, where constructive interference is greatest
Fundamental Frequency and Harmonics
•
Fundamental Frequency: Lowest possible frequency of a standing
wave
•
A wave with two nodes and one antinode would be ½ wave, so the
wavelength of the string would equal 2X the length of the string.
•
Harmonics: integral multiples of the fundamental frequency
Harmonic Series Of Vibrating String
•
𝑓
𝑛
= 𝑛
𝑣
2𝐿
n=1, 2, 3,…….
•
v = speed of wave on the vibrating string
Standing Waves and Resonance
13.5: Standing Waves and Resonance
• The natural frequencies of a stretched string can also be written as:
• Note, the greater the linear mass density of a string, the lower its natural frequencies.
ex: the low note strings on a guitar are thicker, or more massive, than the higher note strings
ex: tightening a string increases all the frequencies of that string
Harmonic Series Of A Pipe Open At Both Ends
•
𝑓
𝑛
= 𝑛
𝑣
Harmonic Series Of A Pipe Closed At One
Ends
•
𝑓
𝑛
= 𝑛
𝑣
4𝐿
n=1, 3, 5,…….