Elements of Waves and Thermal
Physics
Wed. 14:50 – 16:20
Place: Room 2214
Assoc. Prof. H. SAIBI, West building 2, 431, Ito Campus
Textbooks
2
Oscillations and Wave
H. SAIBI
October 8th , 2014
Name
Unit

Phase constant (delta)
radians

Theta
Angle in degrees

Phi
Angle in degrees
T
Period (T=2π/)
second
t
time
second
f
Frequency (f=1/T)
Cycle per second (cy/s) or Hertz (Hz)
m
mass
kg
(t+)
Phase of motion
Fx
Linear restoring force
Dynes in m (grams) and ax in cm/s2.
k
Newtons per metre

Force constant of the spring or spring
stiffness or spring force
Angular frequency

Angular speed or angle of velocity
Radians/second
vx
velocity
cm/s
ax
acceleration
cm/s2
A
amplitude
cm
E
Mechanical energy
Joul if A in m., T in sec., m in kg.
I
Moment of inertia

Angular acceleration (alpha)
k
Torsional constant

Torque (moment of force) (Tau)
Newton-meter
g
gravity
m/s2
Radians/second
Radians/second2
4
Outline
Simple Harmonic Motion
Energy in Simple Harmonic Motion
Some Oscillating Systems
5
Simple Harmonic Motion
• Position x vs. time t
• Definition of period T
• Definition of amplitude A
Phase
Phase (Time)
Phase difference
Frequency and Period
• Period (T) in seconds (s) and Frequency (f) in Hertz (Hz).
• Frequency range of human hearing (20 to 20,000 Hz):
f = 20 Hz
T = 0.05 s = 50 ms
f = 20,000 Hz = 20 kHz
T = 0.000,05 s = 0.05 ms = 50 ms
Simple Harmonic Motion
• Very basic kind of oscillatory motion is: Simple
Harmonic Motion.
Amplitude
Force constant
• Hooke’s Law:
(restoring force)
… (1)
• Newton’s Second Law:
… (2)
• Combination of (1) and (2):
…(3)
…(4)
In simple harmonic motion, the acceleration, and thus the net force, are both proportional to, and
oppositely directed from, the displacement from the equilibrium position.
9
Simple Harmonic Motion
•
Frequency (reciprocal of the period T):
•
T is the time it takes for a displaced object to execute a complete cycle of oscillatory
motion.
… (5)
(cycle/second or Hz)
Phase constant
•
Position in simple harmonic motion:
… (6)
Phase of motion
•
For info:
•
System in phase: phase difference is ZERO or integer times 2π.
•
System 180°out of phase: phase difference is π or an odd integer times of π.
… (7)
… (8)
… (9)
10
Simple Harmonic Motion
• Velocity in simple harmonic motion:
• Acceleration is:
… (10)
2
… (11)
… (12)
Acceleration in simple harmonic motion
• Comparing Eq. 12 with Eq. 4, we see that Eq. 6 is a solution of Eq. 4. if:
• The amplitude A and the phase constant can be determined from the initial
position x0 and the initial velocity v0x of the system. Setting t=0 in
… (13) ; Similarly, setting:
… (14) gives:
… (15)
By these equations we can determine A and  in terms of x0, vox, and .
11
Simple Harmonic Motion
• The period T is the shortest time interval satisfying the relation:
… (16)
For all t. Substituting into this relation using:
… (17)
… (18)
• The cosine (and sine) function repeats in value when the phase increases by 2π,
so:
… (19)
• The constant  is called the angular frequency. It has the unit of radians per
second and dimensions of inverse time, the same as angular speed. Substituting
2π/T for  in
gives:
… (20)
• The frequency is related to the angular frequency by:
• Because of
… (21)
… (22) “the frequency increases with increasing k and decreases with
increasing mass”
12
Simple Harmonic Motion
Figure: Two identical mass-spring systems.
13
Simple Harmonic Motion
The frequency (and thus the period) of simple harmonic
motions is independent of the amplitude
Figure: Plots of x versus t for the systems in Fig. Both reach
their equilibrium positions at the same time.
14
Simple Harmonic Motion
• If the phase constant  is zero, Equations
6, 10, and 11 then become:
… (23)
… (24)
… (25)
These functions are plotted in this Figure
15
Simple Harmonic Motion and Circular Motion
•
A relation exists between simple harmonic motion and circular motion with constant speed.
•
The angular displacement relative to the +x direction is given by:
•
Angular speed of the particle is:
•
The x component of the particle’s position is:
… (27)
… (26)
Angular displacement at t=0
… (28)
“simple harmonic motion”
When a particle moves with constant speed in a circle,
its projection onto a diameter of the circle moves with
simple harmonic motion.
… (29)
Same as Eq. 10 (simple harmonic motion).
16
Energy in Simple Harmonic Motion
• When an object on a spring undergoes simple harmonic motion, the system’s
potential energy and kinetic energy vary with time. Their sum, the total
mechanical energy is constant:
… (30)
• The potential energy is:
… (31)
• The kinetic energy of the system is:
… (32)
• The total mechanical energy E is:
… (33)
The total mechanical energy in simple harmonic motion is proportional to the
square of the amplitude
17
Energy in Simple Harmonic Motion
• The average value of the potential
energy and the kinetic energy over
one or more cycles are equal, their
average values are given by:
… (34)
Turning points
Fig. The potential energy function.
Fig. Plots of x, U, and K versus t.
18
General Motion Near Equilibrium
• Simple harmonic motion occurs when a particle is displaced slightly from a
position of stable equilibrium.
• The equation for a parabola that has a minimum at point x1 is:
… (35) A and B are constants.
• The force is related to the potential energy curve by:
… (36)
Unstable
equilibrium
A
Stable
equilibrium
Stable equilibrium at
x=0
Figure: Plot of U vs. x.
Figure: Plot of U vs. x.
19
Some Oscillating Systems – Object on a vertical spring • The net force of the spring is:
… (37)
• If we set y’=y-y0 in Eq. (37) we get:
… (38)
… (39)
… (40)
• Newton’s Second Law:
•
…
20
Some Oscillating Systems – Object on a vertical spring is the same Eq. (4) with y’ replacing x. It has now the familiar
solution:
where:
(angular frequency)
 A force is conservative if the work done by it is independent of the path.
Both the force of spring and the force of gravity are conservative, and the
sum of these forces (Eqs. 37 & 40) also is conservative.
 The potential energy function U associated with the sum of these forces is
the negative of the work done plus an arbitrary integration constant, that is:
… (41)
21
The Simple Pendulum
• Period:
• Using tangential components, Newton’s Second Law
(
) gives:
…(42) (s: arc length)
• Differentiating both sides of:
 Rearranging gives:
gives:
… (43)
string
The mass m does not appear in Eq.(43), the motion of a
pendulum does not depend on its mass. For small 
and
String
tension
… (44)
• Eq.(44) is the same form with Eq.(4). Thus, the motion
of a pendulum approximates simple harmonic motion for
small angular displacements. Eq.(44) can be written:
…(45) and
…(46)
bob
Arc
length
weight
22
(period of a simple pendulum for small oscillations)
The Simple Pendulum
• The solution of the Eq.(45) is:
maximum angular displacement.
• According to Eq.(46)
the greater the period.
where 0 is the
: the greater the length of a pendulum,
– The period and therefore the frequency are independent of the amplitude of oscillation
(as long as the amplitude is small). This statement is a general feature of simple
harmonic motion.
We can measure gravity “g” using a simple pendulum
undergoing small oscillations. We need only to measure the
length l and period T of the pendulum, and using Eq.(46), solve
for “g”.
Lab. Experiment on Oct. 25th, 2013
23
Pendulum in an accelerated reference frame
•
Applying Newton’s second law to the bob gives:
… (47)
•
If the bob remains at rest relative to the boxcar, then:
and
The clock keeps time by using a torsional oscillator
0 is the equilibrium angle and
•
If the bob is moving relative to the boxcar, then:
(a’ is the acceleration of the bob relative to the boxcar).
Substituting for a in Eq.(47) gives:
a
Some rearrangement
(a) Simple pendulum in apparent equilibrium. (b) in
accelerated frame.
24
Large-Amplitude Oscillations
•
When the amplitude of a pendulum’s oscillation
becomes large, its motion continues to be
periodic, but it is no longer a simple harmonic.
For an angular amplitude 0 , the period can be
shown to be given by:
Period for large-amplitude oscillations
where:
Fig. T/To vs. Amplitude.
25
The Torsional Oscillator
•
A system that undergoes rotational oscillations in a
variation of simple-harmonic motion is called a
torsional oscillator.
Torsional constant of the wire.
…(48) and
Newton’s second law for rotational motion
and
…(49)
Fig. The torsional pendulum consists of
a solid disk suspended by a steel wire.
Angular acceleration
Eq.(49) is similar to Eq.(4) and its solution is:
…(50)
where
is the angular frequency (NOT the
angular speed) of the motion. The period is
therefore:
…(51)
Period of a torsional oscillator
Fig. Mechanical clock (period of oscillation is
constant = keep time). 26
The Physical Pendulum
• A rigid object free to rotate about a horizontal
axis that is not through its center of mass will
oscillate when displaced from equilibrium. Such
system is called a physical pendulum.
… (52)
• Comparing Eq.(52) with Eq.(48), we can see that
for small angular displacements the physical
pendulum is a torsional oscillator with a torsional
constant given by:
Fig. A physical Pendulum.
• Thus the motion of the pendulum is described by
Eq.(50) with
. The period is
therefore:
Moment of Inertia
… (53)
Period of a physical pendulum depends on distribution of the mass.
27