Elements of Waves and Thermal Physics

Elements of Waves and Thermal Physics

Wed. 14:50 – 16:20Place: Room 2214

Assoc. Prof. H. SAIBI, West building 2, 431, Ito Campus

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Textbooks

Oscillations and Wave

H. SAIBIOctober 8th , 2014

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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 Force constant of the spring or spring stiffness or spring force

Newtons per metre

Angular frequency Radians/second

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) Radians/second2

k Torsional constant

Torque (moment of force) (Tau) Newton-meter

g gravity m/s2

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Outline

Simple Harmonic Motion

Energy in Simple Harmonic Motion

Some Oscillating Systems

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

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• Very basic kind of oscillatory motion is: Simple Harmonic Motion.

• Hooke’s Law: … (1) (restoring force)

• Newton’s Second Law: … (2)

• Combination of (1) and (2):

Simple Harmonic Motion

Force constant

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.

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.

…(3)

…(4)

Amplitude

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• Frequency (reciprocal of the period T): … (5) (cycle/second or Hz)

• T is the time it takes for a displaced object to execute a complete cycle of oscillatory motion.

• Position in simple harmonic motion: … (6)

• For info: … (7)

• 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.

… (8)

… (9)

Simple Harmonic Motion

Phase of motion

Phase constant

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Simple Harmonic Motion• Velocity in simple harmonic motion: … (10)

• Acceleration is: … (11)

… (12)

• 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 .

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Acceleration in simple harmonic motion

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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: … (21)

• Because of … (22) “the frequency increases with increasing k and decreases with increasing mass”

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Simple Harmonic Motion

Figure: Two identical mass-spring systems.

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Simple Harmonic Motion

Figure: Plots of x versus t for the systems in Fig. Both reach their equilibrium positions at the same time.

The frequency (and thus the period) of simple harmonic motions is independent of the amplitude

The frequency (and thus the period) of simple harmonic motions is independent of the amplitude

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Simple Harmonic Motion• If the phase constant is zero, Equations6, 10, and 11 then become:

… (23)

… (24)

… (25)

These functions are plotted in this Figure

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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: … (26)

• Angular speed of the particle is: … (27)

• The x component of the particle’s position is:

… (28) “simple harmonic motion”

… (29)

Angular displacement at t=0

When a particle moves with constant speed in a circle, its projection onto a diameter of the circle moves with

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.

Same as Eq. 10 (simple harmonic motion).

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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

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Energy in Simple Harmonic Motion

Fig. Plots of x, U, and K versus t.

Fig. The potential energy function.

• 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

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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)• The force is related to the potential energy curve by:

… (36)

Figure: Plot of U vs. x.Figure: Plot of U vs. x.

A and B are constants.

Unstable equilibrium

Stable equilibrium

A

Stable equilibrium at x=0

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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:• …

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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)

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The Simple Pendulum

bob

string

String tension

weight

Arc length

The mass m does not appear in Eq.(43), the motion of a pendulum does not depend on its mass. For small

and … (44)

• Period:• Using tangential components, Newton’s Second Law( ) gives: …(42) (s: arc length)

• Differentiating both sides of: gives:

Rearranging gives: … (43)

• Eq.(44) is the same form with Eq.(4). Thus, the motionof a pendulum approximates simple harmonic motion for small angular displacements. Eq.(44) can be written:

…(45) and …(46) (period of a simple pendulum for small oscillations)

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The Simple Pendulum

• The solution of the Eq.(45) is: where 0 is the maximum angular displacement.

• According to Eq.(46) : the greater the length of a pendulum, the greater the period.

– 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

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Pendulum in an accelerated reference frame

The clock keeps time by using a torsional oscillator

(a) Simple pendulum in apparent equilibrium. (b) in accelerated frame.

a

0 is the equilibrium angle and

• Applying Newton’s second law to the bob gives:

• If the bob remains at rest relative to the boxcar, then: 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:

… (47)

Some rearrangement

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Large-Amplitude Oscillations

Fig. T/To vs. Amplitude.

Period for 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:

where:

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The Torsional Oscillator

• A system that undergoes rotational oscillations in a variation of simple-harmonic motion is called a torsional oscillator.

and

and

Eq.(49) is similar to Eq.(4) and its solution is:

where is the angular frequency (NOT the angular speed) of the motion. The period is therefore:

Fig. The torsional pendulum consists of a solid disk suspended by a steel wire.

Fig. Mechanical clock (period of oscillation is constant = keep time).

Torsional constant of the wire.

Newton’s second law for rotational motion

Angular acceleration

…(48)

…(49)

…(50)

Period of a torsional oscillator

…(51)

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The Physical Pendulum

Fig. A 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.

• 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:

• Thus the motion of the pendulum is described by Eq.(50) with . The period is therefore:

… (52)

Period of a physical pendulum depends on distribution of the mass.

… (53)Moment of Inertia