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McMaster University
Physics
PHYSICS 1B03
Reza Nejat
Winter
Description
Chapter 15 Oscillatory Motion
• Simple Harmonic Motion: an object moves with SHM when its acceleration is
proportional to its position and is oppositely directed to the displacement from
equilibrium
o A x (k/m)(x)
o Speed is maximum whenA=0 since acceleration changes signs
o Acceleration is maximum whenA= positive or negative X and speed is zero
max
o Block oscillates between its turning point x= positive and negative x
• Particle in SHM
o ω = k/m
o position of particle as a function of time
x (t) =Acos (ωt+ϕ)
• A, ω, ϕ are constant
• A= amplitude of the motion; max value of particle in the positive
or negative direction
• ω = angular frequency with units radians per second; measure of
how frequently the oscillation are occurring; the greater the
oscillations per unit time, the greater value of ω
k/m
o ω = √
• ϕ = phase constant (or initial phase angle); determined by the
position and velocity of particle at t=0
o to get phase constant, set t=0 and solve for v(0) and x(0)
and then divide the eqn from each other; find phase
constant, then amplitude from the x(0) eqn and then v and
max and then the 3 eqn for position, velocity and
acceleration
o if particle is at max position x=Aat t=0, then ϕ= 0
• (ωt+ϕ)= phase of motion Period T= time interval required for the particle to go through one full
cycle of its motion
• T= 2π/ ω = 2π/ √ k/m or 2π* √ m/k ; measures seconds per
cycle
• f= 1/T=1/2π/ ω= ω/2π = [( √ k/m )/2π] ;units are Hz and
measures cycles per second
• period and frequency only depend on the mass of the object and
the force constant of the spring and notAor phase constant
v= ωA sin (ωt+ϕ) and v max/minv (plus or minus) ωA
a= ω Acos (ωt+ϕ) and a max/mina (plus or minus) ω A
x and v differ in phase by 90 degrees and when x is max or min, v=0 and
when x=0 v= max or min
x and a differ in phase by 180 degrees so when x is max, a is max in
opposite direction
when block moves to right, the ϕ is negative and when block moves to
left, ϕ is positive
period is independent of how the oscillator is set into motion (i.e initial
velocity)
• Comparing SHM with Uniform Circular Motion
o Angular speed of the particle in uniform circular motion (UCM) is same as
angular frequency ω
o The initial angle particle in UCM makes with x axis is same as phase constant ϕ
o The radius of particle in UCM is the same as amplitude
o Speed of particle in UCM is v=rω which is same as speed of particle ωA
2
o Acceleration of particle in UCM is same as acceleration of particle ω A
• The Pendulum o Exhibits periodic motion; consists of particlelike bob of mass, m and is
suspended by a light spring of length, L; motion occurs in vertical plane and is
driven by gravitational force
o Provided that angle is small (less than 10 degrees) between the string and the
vertical plane, the motion is very close to SHM
o mgsinϴ is the restoring force heading towards the equilibrium of 0 degrees and
mgsinϴ means that the restoring force is opposite the displacement
o if we assume that angle is small, then we can use small angle approximation
where sinϴ=ϴ and assume than pendulum follows SHM
velocity= (g/L)ϴ (whereϴ is smaller than 10 degrees)
ω= √g/L angular frequency for simple pendulum
T = 2π/ω = 2π* √ L/g period of simple pendulum
• Period and frequency of simple pendulum depend only on length
of the string and acceleration on gravity; period is independent of
mass so all simple pendula of equal length and at same location
(for constant g) on x axis have the same period
• Energy of Simple Harmonic Motion
2 2 2 2
o K=0.5mv = 0.5 m (ω A sin (ωt+ϕ)) Kinetic energy of a simple harmonic
oscillator (the block)
2 2 2
o U= 0.5 k x = 0.5 k (A cos (ωt+ϕ)) Potential energy of simple harmonic
oscillator
2 2 2 2
o E= K+U = 0.5 kA (cos (ωt+ϕ)) + sin (ωt+ϕ)) note: ω =k/m so that is taken
out as a factor and since cos (ωt+ϕ)) + sin (ωt+ϕ)=1 so E=0.5 kA 2
this is the total energy of a simple harmonic oscillator and is equal to
maximum potential energy of a spring at x=positive or negative x since
velocity=0 at that point
o Velocity of block at an arbitrary position when both K and U exist A −x 2
2 2
v= ± k/m(¿)=±ω A −√
√¿


Chapter 16Waves
• Propagation of Disturbance
o Wave motion is the transfer of energy through space without the accompanying
transfer of matter
o Mechanical waves and electromagnetic radiations depend on waves
o All mechanical waves require: (e.g. a string under tension with hand flicking the
string)
Source of disturbance hand
Amedium containing elements that can be disturbedstring
Some physical mechanism through which elements of the medium can
influence each otherthe elements of the string are connected to one
another so influence each other
o Wave: periodic disturbance travelling through a medium
o Transverse wave a travelling wave or a pulse that causes the elements of the
disturbed medium to move perpendicular to the direction of propagation
a string under tension with hand flicking the string
o Longitudinal wave a travelling wave or a pulse that causes the elements of the
disturbed medium to move parallel to the direction of propagation
Stretching and compressing coil with one end attached to wall and parallel
to ground
o Vertical position of an element of the medium at any point, P is given by
Y= f(x+vt) (movement to left) and Y= f(xvt) (movement to right)
• This function depends on both x and t thus written y(x,t) Y(x,t) represents the y coordinatethe transverse position or any element
located at position x at any t; the element at position x moves up, gets to
max and then decline once the wave had passed (the vertical position of a
particle located at x at time t when the wave passes through it)
• The Speed of Waves on Strings
o speed of a transverse pulse travelling on a taut string
o acceleration of element increases with increasing tension and so wave speed
increases with increasing tension
o likewise, because it is more difficult to accelerate an element of a massive string
than that of a light string, the wave speed should decrease as the mass per unit
length increases
light string and high tension= greater wave speed
v= √T/μ Speed of a wave on a stretched string
The wave speed is determined by the medium, so it is unaffected by
changing the frequency
• Analysis Model: Travelling Wave
o motion of elements of the medium is in SHM, moving up and down
o motion of the wave is right or left
o analysis model of a traveling wave
used where wave moves through space without interacting with other
waves
o crest is the displacement of the element from its normal position and lowest point
is called the trough
o wavelength is minimum distance between any 2 identical points on adjacent
waves
o period is the time interval required for 2 identical points of adjacent waves to pass
by a point
o frequency of a periodic wave if the number of crest (or trough or any other point
on wave) that pass a given point in a unit time interval; units= Hz f=1/T
o amplitude is the maximum position of an element of a medium relative to its
equilibrium position
o speed of wave depends on the properties of the medium
o function describing the position of the element of the medium through which the
sinusoidal wave is travelling through is:
2π (x)¿
y(x,0)=Asin[ λ
at any given time t, y has the same value at the position x, x + λ, x+2λ
o if the wave move to the right with a speed, v, the wave function at some later
time, t is:
2π (x−vt)¿
y(x,t)=Asin[ λ , if wave travelling to left then replace –
with +
o angular wave number
2π
k= λ , in rad/m
o angular frequency
ω=2π/T=2πf , in rad/s
o wave function for sinusoidal wave
y=Asin (kxωt) , this is when y=0 when x=0 and t=0 (and so ϕ=0)
y=Asin (kx ωt+ ϕ), this is the general case when y=0 when x=0 and t=0
is not the case
• determine ϕ by setting t=0 and x=0 (initial conditions)
o Speed of a sinusoidal wave
ω
v = λ/T = fλ = k o transverse speed (speed of element) and transverse acceleration respectively
(when y=0 when x=0 and t=0) note: either x or t must be held constant since y
depends on both so this is the speed of the element when either the particle is at a
constant position x or at a constant time t
vy= ωA sin (kxωt) and v max/min v ±ωA
• reaches max value when y=0
ay=ω Acos (kxωt) and a max/mina ±ω A
• reach max value when y is ±A
• Rate of Energy Transfer by Sinusoidal Waves on String
o Waves transfer energy through a medium as they propagate
o The element moves in SHM in the y direction
o Power of a wave
P=1/2 µω A (λ/T)= 1/2 µω A v and units are watts
this is the rate of energy transfer by a sinusoidal wave on a string which is
proportional to the square of the frequency, the square of the amplitude
and the wave speed
o Each element can be considered to have a mass of dm
o This gives a total energy of
El= Kl+ Ul = ½μ w A λ 2
• Reflection and Transmission
o How a traveling wave is affected when it interacts with a change in medium
o When a wave traveling on a taut string reaches the fixedend (hits a wall), it
reflects
Reflection the pulse moves back along the spring in the opposite direction
Reflected pulse is inverted
o When a wave traveling on a taut string reaches the freeend , it reflects but is NOT
inverted
o When a wave traveling on a taut string reaches a boundary intermediate between
the freeend and fixedend, part of the energy in the incident pulse is reflected and part undergoes transmission, that is, some of the energy passes through the
boundary
E.g. light string is attached to a heavier string
Light string pulse, when reaches heavier string, part is reflected while part
energy goes into the heavier string
• speed of wave is faster and then becomes slower
• the reflected pulse is inverted since the heavier string is analogous
to the fixed end
• the reflected pulse has a smaller amplitude as compared with the
incident pulse (energy carried by wave is related to its amplitude)
• according to principal of conservation of energy, when the pulse
breaks into reflected and transmitted wave, the sum of the energies
must equal to the energy of the incident pulse
heavier string pulse, when reaches lighter string, part is reflected while
part energy goes into the lighter string
• speed of wave is slower and then becomes faster
• the reflected pulse is not inverted since the lighter string is
analogous to the free end
the relative heights of the reflected and transmitted pulse depends on the
relative densities of the 2 medium
• if same densities then no reflection
lighter string means faster wave speed if tension is remains constant


Chapter 18 Superposition
• Analysis Model: Waves in Interference
o this is combination of traveling waves
o superposition principle If two or more waves are travelling through a medium, the resultant value
of the wave function at any point is the algebraic sum of the values of the
wave function of the individual waves.
• Two travelling waves can pass through each other without being
destroyed or altered
waves that obey this principle are called linear waves
• have amplitudes much smaller than their wavelength
waves that do not obey this principle are called nonlinear waves
• have amplitudes that are much larger
o interference: combination of separate waves in the same region of space to
produce a resultant wave
o constructive interference: displacement caused by 2 pulses are in the same
direction (2 waves heading towards each other but have their pulses with same y
direction element displacement)
o destructive interference: displacement caused by 2 pulses are in the opposite
direction (2 waves heading towards each other but have their pulses with different
y direction element displacement)
o superposition principle in 2 sinusoidal waves travelling in the same direction in a
linear medium (have same frequency, wavelength, amplitude but differ in
phase)
y 1Asin (kxωt) and y =A2in (kxωt+ϕ)
resultant wave y = y 1+=2[sin (kxωt) + sin (kxωt+ϕ)]
• finally> y = 2Acos(ϕ/2) sin (kxωt+(ϕ/2))
• resultant wave has the same frequency, wavelength
• amplitude of resultant wave is 2Acos(ϕ/2) and if the phase
constant =0 then the resultant wave has an amplitude of 2A
waves interfere constructively when cos(ϕ/2)= ±1 and when ϕ = 0, 2 π, 4
π, 6π (even multiple of π)
waves interfere destructively when cos(ϕ/2)= cos(π/2)=0 where ϕ= odd
multiple of π • 180 degrees out of phase (crest of one wave meets trough of
another)
When ϕ= not an integer multiple of π then the amplitude is between 0 and
2A(not constructive nor destructive
o In general: waves in phase> wave amplitudes add
Waves out of phase> wave amplitudes subtract
o Interference of sound waves
The distance along any path from speaker to receiver is called the path
length, r
∣r2−r1 ∣
Difference in path lengths Δr=
• If the difference is equal toΔr=nλ (some integer of λ), 2 waves are
in a phase and interfere constructively= maximum sound intensity
• If the difference is equal toΔr=nλ/2 (n is odd), 2 waves are 180
degrees out of phase and interfere destructively= no sound
intensity
• Beats: Interference in Time
o Beating: periodic variation in amplitude at a given point due to the superposition
of two waves having slightly different frequencies (superposition of 2 waves
having slightly different frequencies where there is temporal alternations between
constructive and destructive interferences)
o Beat frequency: number of amplitude maxima one hears per second equals the
difference in frequency between 2 sources
Humans can only hear a maximum
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