Chap. 16: Intro to Waves

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Mass on a spring We want a solution to the equation

We now know that for

This is a solution with

Constants A and are not determined by the equations of motion. They are determined by the initial conditions of the problem.

We found the force on a pendulum along the θ direction is . Using the small angle approximation gives us .

Pendulum L

Note also that so

Which is similar to a spring force: Identifying k with mg/L we can simply use all of our results for a mass on a spring.

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ω =km

=mgmL

=gLThe angular frequency is

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d2sdt 2

= −gLs

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s = Acos ωt + φ( )

Torsion Pendulum •  Consider a disk suspended from a torsion wire attached to its centre.

Called a torsion pendulum. A torsion wire is essentially inextensible, but is free to twist about its axis. As the wire twists it also causes the disk attached to it to rotate in the horizontal plane. Let be the angle of rotation of the disk, and equilibrium position is when the wire is untwisted.

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T = 2π Iκ

Note: I thru center of wire

k is the torsion constant > 0

Clicker question 1 Set frequency to BA

A kid is swinging on a swing with a period T. A second kid climbs on with the first, doubling the weight on the swing. The period of the swing is now...

A: the same, T B: 2T C: T Sqrt[2] D: T/Sqrt[2] E: None of these

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T = 2π lg

Clicker question 1 Set frequency to BA

A kid is swinging on a swing with a period T. A second kid climbs on with the first, doubling the weight on the swing. The period of the swing is now...

A: the same, T B: 2T C: T Sqrt[2] D: T/Sqrt[2] E: None of these

For a pendulum, Adding weight doesn't impact the period.

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T = 2π lg

Clicker question 2 Set frequency to BA

The period of a physical pendulum is . Compare the periods of two physical pendula. One is a solid disk of mass m, radius R, supported at the edge. The other is a hoop also of mass m, radius R, supported at the edge.

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T = 2π Imgl

pivot

C.M.

pivot

C.M.

Disk Hoop

Which has the longer period?

A: Disk B: Hoop C: The periods are the same.

Clicker question 2 Set frequency to BA

The period of a physical pendulum is . Compare the periods of two physical pendula. One is a solid disk of mass m, radius R, supported at the edge. The other is a hoop also of mass m, radius R, supported at the edge.

€

T = 2π Imgl

pivot

C.M.

pivot

C.M.

Disk Hoop

Which has the longer period?

A: Disk B: Hoop C: The periods are the same.

Hoop has I=MR2 about it's CM, Disk has I=(1/2) MR2. Then you have to use the parallel axis theorem to add MR2 to each of those to get I about the pivot point. The I of the hoop is *larger*, and therefore T is longer.  

Clicker question 3 Set frequency to BA

A person swings on a swing. When the person sits still, the swing oscillates back and forth at its natural frequency. If, instead, the person stands on the swing, the natural frequency of the swing is..

A: greater. B: the same. C: smaller.

Clicker question 3 Set frequency to BA

A person swings on a swing. When the person sits still, the swing oscillates back and forth at its natural frequency. If, instead, the person stands on the swing, the natural frequency of the swing is..

A: greater. B: the same. C: smaller.

Also can use T = 2 pi Sqrt[ I/ (m L g)]. I goes from mL2 (When the person sits, and all the mass is a distance L away from the pivot) to I = (fraction)*mL2, because the standing person has shifted some mass to smaller radii. So the period gets shorter or frequency gets larger. 

Intuitively, if you stand, you have effectively a shorter pendulum, which has a shorter period, or HIGHER frequency. 

What is a Wave?   Nature of waves:

 A wave is a traveling disturbance that transports energy from place to place.

 There are two basic types of waves: transverse and longitudinal.   Transverse: the disturbance travels perpendicular to the direction of

travel of the wave.   Longitudinal: the disturbance occurs parallel to the line of travel of the

wave.   Examples:

 Longitudinal: Sound waves (e.g. air moves back & forth)  Transverse: Light waves (electromagnetic waves, i.e. electric and

magnetic disturbances)

The source of the wave, i.e. the disturbance, moves continuously in simple harmonic motion, generating an entire wave, where each part of the wave also performs a simple harmonic motion.

Types of Waves   Transverse: The medium oscillates perpendicular to the

direction the wave is moving.   Water waves (also have a longitudinal component)

  Longitudinal: The medium oscillates in the same direction as the wave is moving

 Sound

Wave Properties   Period: The time T for a point on the wave to undergo one

complete oscillation.

  Speed: The wave moves one wavelength λ in one period T so its speed is v = λ / T.

Clicker question 4 Set frequency to BA Consider the following waves:

i) The "wave" made by fans at sports events

ii) waves on the surface of a lake

iii) Music in the auditorium

iv) TV signals transmitting pictures

How many of the above four are transverse waves?

A) All four B) 3 of them C) 2 of them D) just one of them E) None are transverse

Clicker question 4 Set frequency to BA Consider the following waves:

i) The "wave" made by fans at sports events

ii) waves on the surface of a lake

iii) Music in the auditorium

iv) TV signals transmitting pictures

How many of the above four are transverse waves?

A) All four B) 3 of them C) 2 of them D) just one of them E) None are transverse

All are transverse except "music", which is a longitudinal sound wave. (Although depending on the instrument, it could be CREATED by a transverse wave, like a string on a guitar!)

Transverse Sinusoidal Wave

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Tranverse Displacement = y(x, t)

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y(x, t) = ym sin(kx −ωt + φ)

Amplitude Phase

Angular wave number

Position Angular Frequency

Time

Wave traveling in a positive x direction is given by:

Phase Constant

Wavelength and Angular Wave Number

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y(x, t) = ym sin(kx −ωt)

At t=0;

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y(x,0) = ym sinkx

By definition, the displacement is the same at both ends of the wavelength or :

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ym sinkx = ym sink(x + λ) = sin(kx + kλ)

The sine function repeats itself when the angle is increased by , so ; Hence,

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2π

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kλ = 2π

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k =2πλ

(angular wave number)

Period, Angular Frequency, Frequency

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y(x, t) = ym sin(kx −ωt)

At x=0;

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y(0,t) = ym sin(−ωt) = −ym sinωt

Period is defined as the time T for any element to move through a full oscillation.

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−ym sinωt = −ym sin(ωt +ωT)

The sine function repeats itself when the angle is increased by , so ; Hence,

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2π

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ωT = 2π

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ω =2πT

(angular frequency); f =1T

=ω2π

(frequency)

Speed of a Traveling Wave

x

y

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Δx

Wave at t=0

Wave at t=Δt

Wave is moving to the right and the wave speed is

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v =ΔxΔt

If the blue dot retains it’s displacement y, then

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y(x, t) = ym sin(kx −ωt + φ)

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kx −ωt = constant

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kΔx −ωΔt = 0

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k ΔxΔt

−ω = 0; ΔxΔt

= v =ωk

=2πf

2π /λ= λf (wave speed)

Wave moving to the left

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y(x, t) = ym sin(kx −ωt + φ) Wave moving to positive x in time.

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y(x, t) = ym sin(kx +ωt + φ) Wave moving to negative x in time.

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kx +ωt = constant

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ΔxΔt

= v = −ωk

=2πf

2π /λ= −λf (wave speed)