### Physics 1022: Chapter 14 Waves

```Phys 1022: Introduction, Pg 1
Physics 1022: Chapter 14
Waves

Anatomy of a wave

Simple harmonic motion

Energy and simple harmonic motion
Phys 1022: Introduction, Pg 2
Page 1
1
Waves
New Topic
Phys 1022: Introduction, Pg 3
Simple Harmonic Motion: The restoring force is proportional
to the negative of the displacement (like F=-kx)
ma = F
m
d 2x
= −kx
dt 2
d 2x k
+ x=0
dt 2 m
d 2x
k
+ ω 2 x = 0, with ω =
2
dt
m
general solution:
x = A cos(ωt + φ )
A is amplitude, φ is phase angle. They are determined
by initial conditions (the value of x and v at t=0.)
Phys 1022: Introduction, Pg 4
Page 2
2
Phase angle
x = A cos(ωt + φ )
v=
dx
= − Aω sin(ωt + φ )
dt
A measure of different
starting positions (and
velocities)
Both A and φ can be
determined by initial
conditions:
At t = 0 :
x0 = A cosφ ,
v0 = −ωA sin φ
A=
2
x02 + (v0 / ω ) ,
tanφ = −
v0
x0ω
Phys 1022: Introduction, Pg 5
The graph shows a particle in SHM.
(a) What is the phase constant φ0?
(b) What is the phase of the particle at the numbered
points on the graph?
(c) Place dots (with labels) on the circle to indicate the
particle s position corresponding to the numbered points.
Phys 1022: Introduction, Pg 6
Page 3
3
What does the v vs. t graph look like?
A particle moves CCW around a circle at constant speed.
From the phase constants, show the particle s initial
position and sketch two cycles of the x vs. t graph.
Phys 1022: Introduction, Pg 7
Simple Harmonic ↔ Circular Motion
If we look at uniform circular motion from the side, the object appears to
move in simple harmonic motion
y
Top View:
v = v0
vx = v0
Consider the x-position of the object:
x = Acos θ
A
now θ = ω t
θ
x
v = v0
vx = 0
Side View:
x
x = Acos (ωt)
Click here for demo
y
x = +A
Phys 1022: Introduction, Pg 8
Page 4
4
ConcepTest 4(Post)Energy in SHM
This is the potential energy diagram
of a particle oscillating on a spring.
What is the equilibrium length of
the spring?
1.
12 cm
2.
16 cm
3.
20 cm
4.
24 cm
5.
28 cm
6.
cannot be determined
from the graph
Phys 1022: Introduction, Pg 9
ConcepTest 5(Post)Energy in SHM
If the particle s turning points are
14 cm and 26 cm, draw a line that
indicates the total energy and then
determine the particle s maximum
kinetic energy.
1.
2.5 J
2.
5 J
3.
7.5 J
4.
10 J
5.
12.5 J
6.
15 J
7.
other
Etot ~ 6.5 J
Phys 1022: Introduction, Pg 10
Page 5
5
Energy in Simple Harmonic Motion

Energy of the oscillating system at any time is constant:
Etotal = PEspring + KEmass
k
ENERGY IS CONSERVED
xmax = A
m

Etotal = PEspring = 1/2kA2
x Frestore
k
m

at any point in between …
Etotal = 1/2kx2 + 1/2mv2
Frestore
m
at end, x = A and v = 0 (KE = 0)

at x = 0 , PE = 0 and v = v0
Etotal = KEmass = 1/2mv02
equilibrium position
Phys 1022: Introduction, Pg 11
1
2
2
kA 2 = 12 mvmax
What will be the turning
points if the particle s
total energy is doubled?
Draw a graph
of the particle s
kinetic energy
as a function of
position.
Phys 1022: Introduction, Pg 12
Page 6
6
ConcepTest 6(Post)Simple Harmonic Motion
A block oscillating on a spring has
a period of T = 4 s. If the mass
of the block is halved, what is the
new period?
1.
1 s
2.
2 s
3.
2.8 s
4.
4 s
5.
5.6 s
6.
8 s
7.
16 s
Phys 1022: Introduction, Pg 13
ConcepTest 7(Post)Simple Harmonic Motion
A block oscillating on a spring has
a period of T = 4 s. If the spring
constant is quadrupled, what is the
new period?
1.
1 s
2.
2 s
3.
2.8 s
4.
4 s
5.
5.6 s
6.
8 s
7.
16 s
Phys 1022: Introduction, Pg 14
Page 7
7
ConcepTest 8(Post)Simple Harmonic Motion
A block oscillating on a spring
has a period of T = 4 s. If the
oscillation amplitude is doubled,
what is the new period?
1.
1 s
2.
2 s
3.
2.8 s
4.
4 s
5.
5.6 s
6.
8 s
7.
16 s
Phys 1022: Introduction, Pg 15
The graph shows x vs. t for a particle in SHM.
(a) Draw the v vs. t and the
a vs. t graphs.
(b) When x is greater than
zero, is a ever greater than
zero? When?
(c) When x is greater than
zero, is v ever greater than
zero? When?
Can we describe the entire motion of this oscillating system?
Phys 1022: Introduction, Pg 16
Page 8
8
Based on this reference graph
of x vs. t, draw the new graphs
that represent the following
conditions:
(a) amplitude and frequency are doubled
(b) amplitude is halved and mass is quadrupled
(c) phase constant is increased by π/2 rad
(d) max. speed is doubled while amplitude stays constant
Phys 1022: Introduction, Pg 17
Simple Pendulum
New Topic
Phys 1022: Introduction, Pg 18
Page 9
9
ConcepTest 9(Post)Simple Pendulum
A pendulum on Planet X, where the
value of g is unknown, oscillates
with a period of T = 4 s. If the
mass is quadrupled, what is the
new period?
1.
1 s
2.
2 s
3.
2.8 s
4.
4 s
5.
5.6 s
6.
8 s
7.
16 s
Phys 1022: Introduction, Pg 19
The Simple Pendulum
Consider only small oscillations ⇒ sinθ ≈ θ
(Try this on your calculator, but θ must be in radians!)
Restoring force:
Spring Oscillator
F = –kx
T = 2π √ m / k
F = –mg sin θ = - mg θ
= –mg x/L
= –(mg/L) x
Pendulum
F = –mg/L x
T = 2π √ m / (mg/L)
T = 2π √ L / g
Period of a pendulum does NOT depend on
ý  mass
ý  amplitude
Phys 1022: Introduction, Pg 20
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10
ConcepTest 12(Post)Simple Pendulum
If a pendulum with period T
on Earth is taken to the Moon,
how will the period change?
1.  increases
2.  decreases
3.  stays the same
4.  no way to tell
Phys 1022: Introduction, Pg 21
ConcepTest 13(Post)Simple Pendulum
If a mass-spring system with
period T on Earth is taken to
the Moon, how will the period
change?
1.  increases
2.  decreases
3.  stays the same
4.  no way to tell
Phys 1022: Introduction, Pg 22
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11
ConcepTest 14(Post)Simple Pendulum
A pendulum in an elevator has
period T when the elevator is
at rest. If the elevator is
accelerating upward, how will
the period change?
1.  increases
2.  decreases
3.  stays the same
4.  no way to tell
What happens to the period
of the pendulum if the
elevator is in free fall?
Phys 1022: Introduction, Pg 23
ConcepTest 15(Post)To the Center of the Earth
A hole is drilled through the
center of the Earth and
emerges on the other side.
You jump into the hole.
What happens to you?
1.
2.
3.
4.
5.
you fall to the center and stop
you go all the way through and
continue off into space
you fall to the other side of
the Earth and stay there
you fall to the other side of
the Earth and then return
you won t fall at all
Phys 1022: Introduction, Pg 24
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