Lecture 5
Summary of Fluids: applications of ...
• Newton’s laws
(i)
(ii) Archimedes principle
F B = ρf V f g
Summary of Fluids: applications of ... (b) Bernoulli’s principle
work and energy conservation
•
Work
KE
PE
2
(a) Hydraulic Lift: force multiplication ( F2 = A
A1 F1 (neglect h) ),
d1
d
=
distance division ( 2 A2 /A1 )
input work (F1 d1) = output..(F2 d2)
Waves (chapters 20, 21)
• particles (localized, individual, discrete) and
wave (collective, continuous): two
fundamental models of physics
• This week: (single) traveling waves (go
outward from source thru’ medium), e.g.
ripples on water, wave on a string, sound,
light...(theory applicable to all waves)
• Next week: standing waves from combining
traveling waves (interference)
• Next course (Phys 270): light
Outline:
• Types of waves
• Graphs
• Displacement function
• Sinusoidal waves
• Waves on a string
Wave model
• describes behavior common to all waves
• Traveling wave: organized (collective)
disturbance traveling at a well-defined
speed, v
• 3 types:
(i) Mechanical (within a material medium) e.g.
sound in air, ripples on water
(ii) Electromagnetic (light): oscillation of field,
can travel in vacuum
(iii) Matter waves: electron has wave-like
characteristics (quantum physics in Phys 270)
Wave model (II)
•
medium: substance wave moves thru’, elastic,
restoring force brings back to equilibrium e.g.
tension in string, gravity for waves in water
•
disturbance: displacement from equilibrium as
wave passes (organized motion cf. random motion
of thermal energy)
•
wave speed (v): disturbance travels outward from
source, energy (but not material) transferred
•
Transverse waves (particles move perpendicular to
direction of wave: e.g. string) vs.
Longitudinal waves (...parallel...e.g. chain of
masses connected by springs)
•
Apply Newton’s laws to particles of
medium e.g. forces on segment of string:
wave speed depends only on material of medium
One-Dimensional Waves
•
Waves on a string:
•
function of two variables: t (when)
and x (which point of wave)
“Snapshot” graph
sequence
•
History graph
Longitudinal Waves:
∆x vs. x
Displacement function: D(x, t)
• “particles” (segment of string, small volume
of fluid) of medium displaced from
equilibrium as wave travels
snapshot
•
D(x, t = t1 ) (function of x) = D(x − vt1 , 0), i.e., D(x, 0) shifted by vt1
⇒ D is function of (x − vt) e.g. sin(x − vt)
• Wave moving to left:
D is function of (x + vt)
Sinusoidal waves (graphical)
•
•
generated by source in SHM
•
Wavelength (λ): spatial analog of T,
distance disturbance repeats
•
In time T: (one oscillation for
point) wave (crest) moves λ
snapshot and history graphs
sinusoidal/periodic in space, time
v = distance
=
time
λ
T
= λf
Sinusoidal waves:mathematical
•
Set wave in motion by x → (x − vt)
!
D(x, t) = A sin 2π
wave number, k =
"x
2π
λ ;
λ
−
#
t
T
+ φ0
ω = νk
φ0 sets initial condition:
D(x = 0, t = 0) = A sin φ0
$
Example
• A cork bobs on the surface of the water making an
oscillation every 3.0 s, with an amplitude of 1.0 m.
Another cork 45.0 m away is observed to bob exactly
180 degrees out of phase with the first cork. What is
the velocity of the water waves?
•
Waves on a string
y(x, t)
vy
ay y
Newton’s laws applied to
string
(Fnet )y
= may
(Fnet )y
= 2Ts sin θ
= A sin (kx − ωt + φ0 )
= −ωA cos (kx − ωt + φ0 )
= −ω A sin (kx − ωt + φ0 )
2
segment of
= (µ∆x) ay
≈ −k 2 ATs ∆x
(evaluate slope of y = A cos(kx))
⇒v=
!
Ts
µ
(independent of A/shape)