Title: Vibrations
1Vibrations Waves
211.1 Simple Harmonic Motion
- Consider a mass upon a frictionless surface and
attached to a spring - Define the unstretched position of the mass as
equilibrium position - When the spring is stretched or compressed and
then released, the mass vibrates back forth - Block-spring system simulation
3- Velocity is max at equilibrium (displacement 0)
- Max displacement, force, and acceleration occur
at maximum displacement
4Simple Harmonic Motion
- As mass moves from equilibrium a restoring force
returns it toward equilibrium - SHM is vibration about an equilibrium position in
which the restoring force is proportional to the
displacement from equilibrium
5Hookes Law
- Felastic -kx
- k is the spring constant
- x is displacement
- Recall similarity of elastic potential energy
PEelastic ½ kx2 - A stretched or compressed spring has potential
energy
6Sample problem A
- A mass of 0.55 kg is attached to a vertical
spring, which displaces the spring 2.0 cm from
its equilibrium position. Find the spring
constant. - Since there is no vibration, a static equilibrium
exists, i.e. Fg Felastic - Fg mg 0.55 kg x 9.81m/s2 -5.4N
- Fe -kx
- -k Fe/x
- k -5.4N/-0.02m
- k 270 N/m
7Simple Pendulum
- Definitions assumptions
- Bob is a point mass at the end of a string
- Mass of string air resistance are negligible
(have no effect on operation of pendulum) - For small angles of excursion (15), pendulum
operates as SHM - Pendulum simulation
8Pendulum SHM
- Restoring force is a component of the bobs
weight (Fig 12.6) - x-component of bobs weight pulls bob back toward
equilibrium position - When discounting friction, total mechanical
energy is constant during operation of pendulum - ME KE PE ½ mv2 mgh
- Fig 12.7 Table 12-1
9Conservation of Mechanical Energy in SHM
10Displacement, velocity, acceleration, and force
in SHM
11Graphical Relationship of Position, Velocity, and
Acceleration in SHM
y A cos(2pft f)
Note the relationship of maximum, minimum, and
zero values of these quantities
http//www.unistudyguides.com/wiki/Oscillations
1211.2 Measuring SHM
- Measures of SHM include
- Amplitude
- Maximum displacement from equilibrium
- Period (T)
- Time required for one cycle of SHM
- Frequency (f)
- Number of vibrations or cycles per unit time
13Relationship of Period and Frequency
- Period and frequency are inversely related
14Period of a Simple Pendulum
- Depends upon the length of string and
acceleration of gravity - Where T is the period and L is the length of the
string
15Sample B
- A pendulum extends from the ceiling. If the bob
nearly touches the floor, and its period is 12 s,
how high is the ceiling? - T 12 s g 9.81 m/s2
- Unknown L
16Period of a mass-spring system
- Period of a mass-spring system simulation
17Period of a mass-spring system
- Depends on mass and the spring constant
- m is mass (kg)
- k is the spring constant (N/m)
1811.3 Properties of WavesObjectives
- Distinguish local particle vibrations from
overall wave motion - Differentiate between pulse waves and periodic
waves - Interpret wave forms of transverse and
longitudinal waves - Apply the relationship among wave speed,
frequency, and wavelength to solve problems - Relate energy to amplitude
19Wave motion
- Waves involve the vibration of matter
- Waves, then, are the motion (propagation) of a
disturbance in matter - Medium the material through which the
disturbance travels, e.g. water - Mechanical Wave a wave that requires a physical
medium for propagation
20Wave Types
- Pulse wave
- A single non-periodic disturbance
- Periodic wave
- A wave whose source is some periodic motion
- Phet Wave Simulator
21Wave Types
- Waves can also be classified according to the
direction of the vibration relative to the
direction of the wave - Transverse wave
- Particles vibrate perpendicularly to the
direction of the wave - Longitudinal (compression) wave
- Particles vibrate parallel to the direction of
the wave - Longitudinal and Transverse Wave Motion
22Crest, Trough, Wavelength in a Longitudinal wave
- Correspond to density of the medium during wave
cycles
23Crest, Trough, Wavelength
- Both transverse and longitudinal waves can be
described with a crest, trough, and wavelength - Fig 12.13
- Fig 12.15
24Period, Frequency, Wave Speed
- Period the time for one complete cycle of
vibration - Period time for one wavelength to pass a given
point - Frequency number of wave cycles per second
- Units of s-1 or Hz
25Speed of a wave
26Waves Transfer Energy
- A disturbance is caused by energy
- E.g. the energy from a collision of a stone with
water is transmitted through waves in the water - Energy of a wave is related to its amplitude
- E a2
- Wave damping loss of energy through time
- Wave Simulator
2711.4 Wave InteractionsObjectives
- Apply the superposition principle
- Distinguish constructive and destructive
interference - Predict when reflected waves will be inverted
- Distinguish between traveling waves and standing
waves - Identify nodes and antinodes of a standing wave
28Wave Interference
- Waves on a String (http//phet.colorado.edu/sims/w
ave-on-a-string/wave-on-a-string_en.html - Superposition two waves can occupy the same
space at the same time - When two waves occupy the same space at the same
time, the resultant wave is the sum of the two
waves - Constructive interference displacements of the
two waves are on the same side of equilibrium,
producing greater amplitude - Destructive interference displacements of the
two waves are on opposite sides of equilibrium,
reducing amplitude
29Reflection
- Waves on a String (http//phet.colorado.edu/sims/w
ave-on-a-string/wave-on-a-string_en.html - Waves are reflected when they encounter a
boundary - At a free boundary, incident (incoming) and
reflected pulses occur on the same side of
equilibrium - At a fixed boundary, incident and reflected
pulses occur on opposite sides of equilibrium
30Standing Waves
- Standing wave a wave pattern that results when
two waves of the same frequency, wavelength, and
amplitude travel in opposite directions and
interfere - Node represents complete destructive
interference and is therefore no displacement (is
stationary) - Antinode mid-way between two nodes, maximum
displacement occurs - Fig 12-22, p. 463
31Nodes and Antinodes
Node Complete destructive interference
stationary zero displacement Antinode Midway
between nodes, maximum amplitude maximum
displacement
Standing wave - Wikipedia, the free encyclopedia
32A Standing Wave is a Stationary Wave
- A standing wave is a wave that remains in a
constant position. - The wave is constrained by two fixed boundaries,
forming nodes - Results from interference of two waves traveling
in opposite directions. - There is no net propagation of energy.
33Wavelengths of standing waves
- Because each end of the wave must be a node, only
certain frequencies produce standing waves - Consider a string of length L
- Describe wavelength in terms of L