123
9.1. THE IMPORTANT STUFF
F1
f1
F2
f2
Figure 9.7: Given that our positive rotation direction is counter-clockwise, force F1 gives a positive torque.
Force F2 gives a negative torque.
we might call the z direction). For our present purposes, the rotatio
150
CHAPTER 10. OSCILLATORY MOTION
10.2.3
Simple Pendulum
3. What is the period of a simple pendulum which has a length of 3.00 m?
Use Eq. 10.16 with L = 3.00 m. Get:
L
= 2
g
T = 2
(3.00 m)
m = 3.48 s
9.80 s2
The period of the pendulum is 3.48 s.
4. What
148
CHAPTER 10. OSCILLATORY MOTION
q L
CM
M
Mom. of Inertia =
Figure 10.10: A physical pendulum. (What pendulum is not physical?) Object has moment of inertia
I about the pivot; center of mass is a distance L from the pivot.
10.1.8
Physical Pendulum
For c
149
10.2. WORKED EXAMPLES
10.2
Worked Examples
10.2.1
Harmonic Motion
1. Atoms in a solid are not stationary, but vibrate about their equilibrium
positions. Typically, the frequency of vibration is about f = 2.0 1012 Hz, and the
amplitude is about 1.1 101
Chapter 11
Waves I
11.1
The Important Stu
11.1.1
Introduction
A wave (as well use the term in this chapter) is a disturbance in some continuous, deformable
and otherwise uniform medium which travels over long distances while the little bits of
the medium
146
CHAPTER 10. OSCILLATORY MOTION
q0
q
L
T
m
mg sinq
m
mg cosq
(a)
(b)
mg
Figure 10.9: (a) Simple pendulum of length L with small mass m attached to the end. Pendulum is pulled
back by 0 and released. (b) Forces acting on the mass when pendulum is at ang
147
10.1. THE IMPORTANT STUFF
From = I and I = mL2 we then get
= mgL sin = (mL2)
and then some algebra gives
g
sin
(10.11)
L
One more step is needed so that we can get f from this equation. It turns out that when
the angle is small (and measured in radi
144
CHAPTER 10. OSCILLATORY MOTION
k
k
x
m
m
(a)
(b)
Figure 10.8: (a) Mass is attached to the end of a vertical spring; the spring stretches from its original
length so as to support the mass. (b) Mass will now oscillate around the new equilibrium positio
145
10.1. THE IMPORTANT STUFF
(as shown in Fig. 10.8(b) with the same frequency as in the horizontal case, that is, T =
2 m/k. It might seem surprising that the frequency doesnt depend on the value of g, but
it doesnt. The eect of gravity is that the mass
152
CHAPTER 11. WAVES I
(a)
(b)
(c)
(d)
(e)
Figure 11.1: Two positive pulses are put on a string; they travel towards one another, add constructively
and then continue on as if nothing had happened!
the coils are slightly compressed. This type of wave is
11.1. THE IMPORTANT STUFF
153
(a)
(b)
(c)
(d)
(e)
Figure 11.2: Positive and negative pulses of similar shape are created on a string and travel toward
each other. When they overlap the pulses cancel such that the rope is nearly at at some point. Then the
160
CHAPTER 11. WAVES I
radius r centered on the source then the rate at which energy crosses this surface is also
P . The energy ux at this distance can then be found by taking the total power crossing
the surface (P ) and dividing by the total area of t
161
11.2. WORKED EXAMPLES
encounters the wave maxima at a greater rate because of his motion. In eect, the speed of
the waves is greater and so again the observer hears a larger frequency.
If the observer had been running away from the source he would hav
159
11.1. THE IMPORTANT STUFF
v
v
v
v
Figure 11.9: Spherical wave.
It is more useful to describe the loudness of a sound wave in terms of the rate at which it
transports energy. At a large distance from a source of sound, the wave maxima are close to
a se
158
CHAPTER 11. WAVES I
v
Figure 11.8: Sound wave travels down air air-lled pipe with speed v.
Damn.
Anyway, the speed of sound is strongly dependent on the type of medium in which the
wave travels. Even for sound waves in air at normal pressure the speed
155
11.1. THE IMPORTANT STUFF
Figure 11.4: The individual points on the harmonic wave act like harmonic oscillators, moving up and
down.
y
A
y
A
x
-A
t
-A
(a)
(b)
Figure 11.5: (a) At a xed time t the displacement y has a sinusoidal dependence on the coord
156
CHAPTER 11. WAVES I
y
A
x
-A
Figure 11.6: When we freeze the time t, the wavelength is the length along the string for one full cycle
of the displacement.
y
A
T
t
-A
T
Figure 11.7: When we choose a bit of the string at a particular x, the period T is
157
11.1. THE IMPORTANT STUFF
oscillators in the last chapter. This is what we mean by the frequency of the wave. The
frequency (f) is measured in Hz= cycle .
s
And now we want to think about the space and time dependence of the wave simultaneously. If yo
154
CHAPTER 11. WAVES I
-3
-2
-1
0
1
2
3
Figure 11.3: A harmonic wave is a sinusoidal pattern that travels to the left or right.
pulses emerge and continue on their separate ways.
How can this be? If the string is momentarily at, havent the two waves been
143
10.1. THE IMPORTANT STUFF
v
a
v
w
w
a
A
(a)
(b)
Figure 10.7: Man with big nose and bad haircut can only see the sideways components of the velocity
and acceleration vectors. (a) When the peg is at the = 90 position man sees the peg with maximum
veloci
142
CHAPTER 10. OSCILLATORY MOTION
w
A
q
x
x = Acosq
q = wt
Figure 10.6:
Thinking of the reference circle we can associate an angular velocity with the motion
of an oscillator. This makes sense even though the mass in question really moves back and
forth
131
9.1. THE IMPORTANT STUFF
R
M
I
q
Figure 9.12: Example: Object with mass M radius R and moment of inertia I rolls (without slipping)
down a slope inclined at above the horizontal.
FN
Mg sinq
fs
Mg cosq
Mg
q
Figure 9.13: Forces acting on the round objec
130
CHAPTER 9. ROTATIONAL DYNAMICS
in contact with the surface has an instantaneous velocity of zero which the top point has a
speed of 2vc in the forward direction.
Because the rolling object is in contact with the surface there may be a force of frictio
126
CHAPTER 9. ROTATIONAL DYNAMICS
T2
T1
T
(a)
(b)
Figure 9.9: (a) String wrapped around a pulley; tension T gives a torque on the pulley. (b) String is in
contact with pulley; tensions on the dierent sides are not the same.
Another one is shown in (b). H
129
9.1. THE IMPORTANT STUFF
Now, there many situations in the world when we wed like some object to be absolutely
motionless. For example, since gravity acts on all objects on the earth we might need to hold
them up and make sure they dont fall; we need
128
CHAPTER 9. ROTATIONAL DYNAMICS
Now look at the forces acting on the pulley, shown in Fig. 9.11(b). The string tension
acts to give a tangential force T at the edge of the wheel, and hence a torque of RT . Thats
the only force which gives a torque arou
125
9.1. THE IMPORTANT STUFF
Then Newtons 2nd law gives F = ma, a being the acceleration in the tangential direction.
Multiply both sides by r and get rF = mar.
Now make some substitutions in this last equation. Since the force is applied at 90 to
the rad
124
CHAPTER 9. ROTATIONAL DYNAMICS
F
r
F
r
f
f
f
l
F
(b)
(a)
Figure 9.8: (a) Force F pulls at angle from the radial line. (b) We drop a line of action along the
direction of the force and make a perpendicular lever arm from the axis. Now the (same) force
127
9.1. THE IMPORTANT STUFF
R
M
m
Figure 9.10: Mass hangs from a string which is wrapped around a wheel.
R
T
+
(a)
mg
T
(b)
Figure 9.11: (a) Forces on the block. Positive direction of motion will be taken as downward for simplicity.
(b) Force on the whee
132
CHAPTER 9. ROTATIONAL DYNAMICS
the surface FN is applied at the point of contact, as is the force of friction fs which acts
along the surface in the direction shown. (Actually, it may not be clear that friction acts in
the backward direction, but that