### (−x 2 )+ - tenaya.physics.lsa.umich.edu

```Physics 140 – Winter 2014
April 15
Mechanical
Waves
Prof. Andrew Davidhazy - Rochester Institute of Technology
1
Questions concerning
3
The Jacob’s Ladder demo illustrated
torsional waves. What kind of waves
are shown?
A) Transverse
B) Longitudinal
4
The table slinky demo can be used to
illustrate what kind of waves?
A) Transverse
B) Longitudinal
C) Both
5
When I hang this slinky by one end and then drop it,
what will happen?
1) The bottom end of the slinky will immediately
start falling at the same rate as the top.
2) The bottom end of the slinky will rise up, and
the two ends will meet in the middle. Then the
whole thing will fall to the floor.
3) The bottom end of the slinky will hang
suspended momentarily, then start falling.
Information that top is
no longer supported
travels down as a wave!
Mathematics of a wave
• Below is a graph of function f(x) = exp(−x2). Which
mathematical operation needs to be applied to this
function in order to have the pulse in this graph shifted
by 2 units to the right?
A.
B.
C.
D.
E.
exp(−(x+2)2)
exp(−(x−2)2)
exp(−x2)+2
exp(−x2)−2
Some other
operation
Consider a wave pulse moving to the right with speed 2 units/second
• Red = f(x)
exp(−x2)
t=0
Consider a wave pulse moving to the right with speed 2 units/second
• Red = f(x)
• Blue = f(x−v(1 s))
exp(−x2)
exp(−(x−2)2)
t=0
t=1s
Consider a wave pulse moving to the right with speed 2 units/second
• Red = f(x)
• Blue = f(x−v(1 s))
• Green = f(x−v(2 s))
exp(−x2)
exp(−(x−2)2)
exp(−(x−4)2)
t=0
t=1s
t=2s
Consider a wave pulse moving to the right with speed 2 units/second
•
•
•
•
Red = f(x)
Blue = f(x−v(1 s))
Green = f(x−v(2 s))
Yellow = f(x−v(3 s))
exp(−x2)
exp(−(x−2)2)
exp(−(x−4)2)
exp(−(x−6)2)
t=0
t=1s
t=2s
t=3s
Right vs left motion
• Thus, function f(x-vt) represents a pulse of shape f(x)
travelling to the right with velocity v
• Similarly, function f(x+vt) represents a pulse of shape
f(x) travelling to the left with velocity v.
f(x+vt)=f(x−(−vt))
Jerking a rope
A man jerks a taut rope to produce a
pulse that travels away from him at a
speed v1. If he pulls the rope tauter and
generates another pulse which moves
away at speed v2, how do the speeds
compare?
A) v1 < v2
B) v1 = v2
C) v1 > v2
v
F

Hanging rope (non-negligible mass)
A long rope with mass m is suspended from the ceiling and hangs
vertically. A wave pulse is produced at the lower end of the rope,
and the pulse travels up the rope. How does the speed of the pulse
change as it moves up the rope?
A. Increases
B. Decreases
C. Stays the same
v
F

and F increases
Tension F
higher here
(supporting
more weight)
String Composed of Two Parts
A weight is hung over a pulley and attached to a string
composed of two parts, each made of the same material
but one having 4 times the diameter of the other. The
string is plucked so that a pulse moves along it, moving at
speed v1 in the thick part and at speed v2 in the thin part.
What is v1/ v2?
A.
B.
C.
D.
E.
1/4
1/2
1
2
4
v
Ft

1  16 2
String Composed of Three Parts
Three pieces of string, each of length L, are joined together end
to end, to make a combined string of length 3L. The first piece of
string has mass per unit length μ1, the second piece has mass per
unit length μ2 = 4μ1, and the third piece has mass per unit length
μ3 = μ1/4. If the combined string is under tension F, how much
time does it take a transverse wave to travel the entire length 3L?
3
1
A. L
4
F
D. 3L
3
1
B. L
2
F
1
F
18
1
C.
L
7
F
7
1
E. L
2
F
L
t1 

v1
L
1
L
F
F / 1
L
t2 

v2
L
2
4 1
L
L
 2t1
F
F
F / 2
L
t3 

v3
L
 3  L 1  1 t
L
1
4F 2
F
F / 3
t total
1
7
1

 t1  t 2  t 3   1  2   t1  L

2
2
F
L
μ1
L
L
μ2 = 4μ1
μ3 = μ1 / 4
17
Three waves on three strings
• Three waves are traveling along identical strings under
the same tension.
• Wave B has twice the amplitude of the other two.
• Wave C has half the wavelength of A or B.
• Which wave moves
v
y
A
fastest?
wave
x
A.
B.
C.
D.
E.
Wave A
Wave B
Wave C
Waves A and B (tie)
All have same v
y
B
x
y
C
x
Three waves on three strings
• Three waves are traveling along identical strings under
the same tension.
• Wave B has twice the amplitude of the other two.
• Wave C has half the wavelength of A or B.
• Which wave has the
v
y
A
highest frequency?
wave
x
A.
B.
C.
D.
E.
v = λ/T = λf
Wave A
Wave B
Wave C
Waves A and B (tie)
All have same v
y
B
x
y
C
x
y
k  2 m 1 and
  4 s1

 vx    2 m/s
k
x
Leftward
moving
A wave traveling along a string (shown above) takes the form
y(x,t) = A cos (2x + 4t)
with x in meters and t in seconds. Imagine following the motion
of the point of constant phase, P, as the wave evolves. After 2s
elapse, where will point P be located?
1.
2.
3.
4.
5.
(2 m/s)(2 s)  4 m
at the same x position
2m to the right of its current position
2m to the left of its current position
4m to the right of its current position
4m to the left of its current position
Wave vs particle speed A
k
ω
• A wave travelling along a string takes the form
y( x, t )  5 cos(2 x  3t )
where x and y are in meters and t is in seconds. What is
the ratio of the wave velocity to the maximal velocity of
an oscillating particle on that string?
A.
B.
C.
D.
E.
1/15
1/10
1/9
1/6
9/10
1
1
1
vwave  / k




kA 2  5 10
v p,max
A
L
Hanging weight – part 1
W
A 1.50-m string of weight 0.0125 N is tied to the ceiling at its upper
end, and the lower end supports a weight W. When you pluck the
string slightly, the waves traveling up the string obey the equation:
k
ω

1
1
v
y(x,t)  (8.50 mm)cos(172m x  4830s t)
k
How much time does it take a pulse to travel the full length of the
string?
Use
A)
B)
C)
D)
E)
0.0072 s
0.0137 s
0.0534 s
0.0661 s
0.0945 s
y(x,t)  Acos(kx  t)

4830 1 s1
v 
 28.1 m/s
1
k
172m
L 1.50 m
t 
 0.0534 s
v 28.1 m/s
22
L
Hanging weight – part 2
W
A 1.50-m string of weight 0.0125 N is tied to the ceiling at its upper
end, and the lower end supports a weight W. When you pluck the
string slightly, the waves traveling up the string obey the equation:
v
y(x,t)  (8.50 mm)cos(172m x  4830s t)
What is the weight W? (The wavespeed we found is 28.1 m/s.)
1
1)
2)
3)
4)
5)
1
F

0.47 N v  F  F  W  v2  mstring v2  wstring / g v2
0.67 N

L
L
0.87 N
2
(0.0125
N)
/
(9.8
m/s
)
1.07 N
2

F

(28.1
m/s)
1.27 N
1.50 m
= 0.671 N
23
Hanging weight – part 3
L
W
A 1.50-m string of weight 0.0125 N is tied to the ceiling at its upper
end, and the lower end supports a weight W. When you pluck the
string slightly, the waves traveling up the string obey the equation:
k
y(x,t)  (8.50 mm)cos(172m1x  4830s1t)
How many wavelengths are on the string at any time? (
A)
B)
C)
D)
E)
11
21
31
41
51
1 2

L
 2 
L
1

kL
2
1

(172 m 1 )(1.50 m)  41.
2
24
Dependence on space and time
y(x,t)  Acos(kx   t)
A transverse sine wave with an amplitude of 2.50 mm and a
wavelength of 1.80 m travels from left to right along a long, horizontal,
stretched string with a speed of 36.0 m/s. Take the origin at the left end
of the undisturbed string. At time t=0 the left end of the string has its
maximum upward displacement.
What is the velocity of a particle 1.35 m to the right of the origin at
time t = 0.0625 s?
2
2
Hint:
1
k



3.49
m
A) −0.20 m/s
 1.80 m
y(x,t)
v
(x,t)

y
B) 0.00 m/s

t
v
   kv
C) +0.20 m/s
k
D) +0.40 m/s
E) +0.60 m/s
  (3.49 m1 )(36.0 m/s)  126. s1
 y(x,t)  (2.50 mm)cos[(3.49m1 )x  (126.25s1 )t]
Dependence on space and time
A transverse sine wave with an amplitude of 2.50 mm and a
wavelength of 1.80 m travels from left to right along a long, horizontal,
stretched string with a speed of 36.0 m/s. Take the origin at the left end
of the undisturbed string. At time t=0 the left end of the string has its
maximum upward displacement.
What is the velocity of a particle 1.35 m to the right of the origin at
time t = 0.0625 s?
y(x,t)  Acos(kx   t)
y(x,t)
vy (x,t) 
 ( )A[sin(kx   t)]   Asin(kx   t)
t
 vy  (0.314 m/s)sin[(3.49m1 )(1.35m)  (126. s1 )(0.0625s)]
= (0.314 m/s)sin(  3.14) = 0.
26
Tossing an ant – exercise for home
An ant with mass m is standing peacefully on top of a horizontal,
stretched rope. The rope has mass per unit length μ and is under
tension F. Without warning, a boy starts a sinusoidal transverse
wave of wavelength λ propagating along the rope. The motion of
the rope is in a vertical plane. What minimum wave amplitude will
make the ant become momentarily weightless?
(Assume that m is so small that the presence of the ant has no effect
on the propagation of the wave.)
AMin
 2 g

4 2 F
27
```