Vibrations and Waves
Simple harmonic is the simplest modelpossible of oscillatory motion, yet it isextremely important.
Examples:
›a grandfather clock
›the vibrations of atom inside a crystal
›the oscillations of a buoy due to wavemotion in a lake
SHM forms the basis for understandingwave motion itself.
Fs = - k x
›Fs is the spring force
›k is the spring constant
It is a measure of the stiffness of the spring
A large k indicates a stiff spring and a small k indicates asoft spring
›x is the displacement of the object from itsequilibrium position
x = 0 at the equilibrium position
›The negative sign indicates that the force is alwaysdirected opposite to the displacement
The force always acts toward theequilibrium position
›It is called the restoring force
The direction of the restoring force issuch that the object is being eitherpushed or pulled toward the equilibriumposition
When x is positive (to theright), F is negative (tothe left)
When x = 0 (atequilibrium), F is 0
When x is negative (tothe left), F is positive (tothe right)
Assume the object is initially pulled to adistance A and released from rest
As the object moves toward the equilibriumposition, F and a decrease, but v increases
At x = 0, F and a are zero, but v is a maximum
The object’s momentum causes it to overshootthe equilibrium position
The force and acceleration start toincrease in the opposite direction andvelocity decreases
The motion momentarily comes to a stopat x = - A
It then accelerates back toward theequilibrium position
The motion continues indefinitely
Motion that occurs when the net forcealong the direction of motion obeysHooke’s Law
›The force is proportional to the displacementand always directed toward the equilibriumposition
The motion of a spring mass system is anexample of Simple Harmonic Motion
Not all periodic motion over the samepath can be considered SimpleHarmonic motion
To be Simple Harmonic motion, the forceneeds to obey Hooke’s Law
Amplitude, A
›The amplitude is the maximum position ofthe object relative to the equilibrium position
›In the absence of friction, an object in simpleharmonic motion will oscillate between thepositions x = ±A
X = A
X = -A
X = 0
The period, T, is the time that it takes for theobject to complete one complete cycle ofmotion
›From x = A to x = - A and back to x = A
The frequency, ƒ, is the number of completecycles or vibrations per unit time
›ƒ = 1 / T
›Frequency is the reciprocal of the period
X = A
X = -A
X = 0
Newton’s second law will relate force andacceleration
The force is given by Hooke’s Law
F = - k x = m a
›a = -kx / m
The acceleration is a function of position
›Acceleration is not constant and therefore theuniformly accelerated motion equation cannot beapplied
A compressed spring has potentialenergy
›The compressed spring, when allowed toexpand, can apply a force to an object
›The potential energy of the spring can betransformed into kinetic energy of the object
The energy stored in a stretched orcompressed spring or other elastic material iscalled elastic potential energy
›PEs = ½kx2
The energy is stored only when the spring isstretched or compressed
Elastic potential energy can be added to thestatements of Conservation of Energy andWork-Energy
A block sliding on africtionless systemcollides with a lightspring
The block attachesto the spring
The systemoscillates in SimpleHarmonic Motion
The block is moving on a frictionless surface
The total mechanical energy of the system is the kineticenergy of the block
The spring is partially compressed
The energy is shared between kinetic energy andelastic potential energy
The total mechanical energy is the sum of the kineticenergy and the elastic potential energy
The spring is now fully compressed
The block momentarily stops
The total mechanical energy is stored as elasticpotential energy of the spring
When the block leaves the spring, the total mechanicalenergy is in the kinetic energy of the block
The spring force is conservative and the total energy ofthe system remains constant
Conservation of Energy allows a calculation ofthe velocity of the object at any position in itsmotion
›Speed is a maximum at x = 0
›Speed is zero at x = ±A
›The ± indicates the object can be traveling in eitherdirection
Example: An automobile having a massof 1000 kg is driven into a brick wall in asafety test. The bumper acts like a springwith a constant 5.00 x 106 N/m and iscompressed 3.16 cm as the car isbrought to rest. What was the speed ofthe car before impact, assuming that noenergy is lost in the collision.
A ball is attached to the rim ofa turntable of radius A
The focus is on the shadowthat the ball casts on thescreen
When the turntable rotateswith a constant angularspeed, the shadow moves insimple harmonic motion
The angular frequency is related to thefrequency
The frequency gives the number of cycles persecond
The angular frequency gives the number ofradians per second
Period
›This gives the time required for an object of mass mattached to a spring of constant k to complete onecycle of its motion
Frequency
›Units are cycles/second or Hertz, Hz
Example: An object-spring systemoscillates with an amplitude of 3.5 cm. Ifthe spring constant is 250 N/m and theobject has a mass of 0.50 kg, determinethe (a) mechanical energy of thesystem, (b) the maximum speed of theobject, and (c) the maximumacceleration of the object.
Use of a referencecircle allows adescription of themotion
x = A cos (2ƒt)
›x is the position attime t
›x varies between +Aand -A
When x is a maximumor minimum, velocity iszero
When x is zero, thevelocity is a maximum
When x is a maximum inthe positive direction, ais a maximum in thenegative direction
Remember, the uniformly acceleratedmotion equations cannot be used
x = A cos (2ƒt) = A cos t
v = -2ƒA sin (2ƒt) = -A sin t
a = -42ƒ2A cos (2ƒt) =
-A2 cos t
This experiment shows thesinusoidal nature of simpleharmonic motion
The spring mass systemoscillates in simpleharmonic motion
The attached pen tracesout the sinusoidal motion
Example: The motion of an object is describedby the following equation:
x = (0.30 m) cos ((πt)/ 3)
Find (a) the position of the object at t=0 andt=0.60 s,
(b) the amplitude of the motion,
(c) the frequency of the motion, and
(d) the period of the motion.
The simple pendulumis another example ofsimple harmonicmotion
The force is thecomponent of theweight tangent to thepath of motion
›Ft = - m g sin θ
In general, the motion of a pendulum is notsimple harmonic
However, for small angles, it becomes simpleharmonic
›In general, angles < 15° are small enough
›sin θ = θ
›Ft = - m g θ
This force obeys Hooke’s Law
This shows that the period is independentof the amplitude
The period depends on the length of thependulum and the acceleration ofgravity at the location of the pendulum
A physical pendulumcan be made froman object of anyshape
The center of massoscillates along acircular arc
The period of a physical pendulum is given by
›I is the object’s moment of inertia
›m is the object’s mass
For a simple pendulum, I = mL2 and theequation becomes that of the simplependulum as seen before
Only ideal systems oscillate indefinitely
In real systems, friction retards the motion
Friction reduces the total energy of thesystem and the oscillation is said to bedamped
Damped motion variesdepending on thefluid used
›With a low viscosityfluid, the vibratingmotion is preserved,but the amplitude ofvibration decreases intime and the motionultimately ceases
This is known asunderdampedoscillation
With a higher viscosity, the object returnsrapidly to equilibrium after it is released anddoes not oscillate
›The system is said to be critically damped
With an even higher viscosity, the piston returnsto equilibrium without passing through theequilibrium position, but the time required islonger
›This is said to be over damped
Plot a shows anunderdampedoscillator
Plot b shows acritically dampedoscillator
Plot c shows anoverdampedoscillator
Oscillations cause waves
A wave is the motion of adisturbance
Mechanical waves require
›Some source of disturbance
›A medium that can be disturbed
›Some physical connection betweenor mechanism though whichadjacent portions of the mediuminfluence each other
All waves carry energy andmomentum
Flip one end of along rope that isunder tension andfixed at one end
The pulse travels tothe right with adefinite speed
A disturbance of thistype is called atraveling wave
In a transverse wave, each element that isdisturbed moves in a direction perpendicularto the wave motion
In a longitudinal wave, the elements of themedium undergo displacements parallel to themotion of the wave
A longitudinal wave is also called acompression wave
The brown curve is a“snapshot” of the wave atsome instant in time
The blue curve is later intime
The high points are crestsof the wave
The low points are troughsof the wave
A longitudinal wave can also be represented as a sinecurve
Compressions correspond to crests and stretchescorrespond to troughs
Also called density waves or pressure waves
A steady stream ofpulses on a very longstring produces acontinuous wave
The blade oscillates insimple harmonic motion
Each small segment ofthe string, such as P,oscillates with simpleharmonic motion
Amplitude is themaximum displacementof string above theequilibrium position
Wavelength, λ, is thedistance between twosuccessive points thatbehave identically
v = ƒ λ
›Is derived from the basic speed equation ofdistance/time
This is a general equation that can beapplied to many types of waves
Example: A wavetraveling in thepositive x-directionhas a frequency of 25Hz as in the figure.Find the (a)amplitude,
(b) wavelength,
(c) period, and
(d) speed of thewave.
The speed on a wave stretched undersome tension, F
› is called the linear density
The speed depends only upon theproperties of the medium through whichthe disturbance travels
Two traveling waves can meet and passthrough each other without being destroyed oreven altered
Waves obey the Superposition Principle
›If two or more traveling waves are moving through amedium, the resulting wave is found by addingtogether the displacements of the individual wavespoint by point
›Actually only true for waves with small amplitudes
Two waves, a and b,have the samefrequency andamplitude
›Are in phase
The combinedwave, c, has thesame frequency anda greater amplitude
Two pulses are traveling in opposite directions
The net displacement when they overlap is the sum ofthe displacements of the pulses
Note that the pulses are unchanged after theinterference
Two waves, a and b,have the sameamplitude andfrequency
They are 180° out ofphase
When they combine,the waveforms cancel
Two pulses are traveling in opposite directions
The net displacement when they overlap is decreasedsince the displacements of the pulses subtract
Note that the pulses are unchanged after theinterference
Whenever a travelingwave reaches aboundary, some or all ofthe wave is reflected
When it is reflected from afixed end, the wave isinverted
The shape remains thesame
When a traveling wave reaches a boundary,all or part of it is reflected
When reflected from a free end, the pulse isnot inverted
