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Agenda

•  Today: HW quiz, More simple harmonic motion and waves

•  Thursday: More waves •  Midterm scores will be posted by

Thursday.

Chapter 13 Problem 28

Calculate the buoyant force due to the surrounding air on a man weighing 800N. Assume his average density is 1,000 kg/m3, about the same as water.

Density of air: 1.2 kg/m3 ρ = m/V FB = mliquidg

Frequency and Period The frequency of oscillation depends on physical properties of the oscillator; it does not depend on the amplitude of the oscillation.

f = 12π

km

f = 12π

gL

Equations of Motion for SHM:

•  Acceleration is NOT constant! We can’t use our equations from kinematics.

x(t) = A cos (2π f t) v(t) = -vmax sin (2π f t) vmax = 2π f A a(t) = -amax cos (2π f t) amax = (2π f)2 A

What if we have friction or drag?

•  This is called a “damped” oscillation: the amplitude will steadily decrease over time

•  Spring: eventually will resettle back at equilibrium (friction with sliding surface, air resistance, friction at attachments)

•  Pendulum: eventually will stop swinging (air resistance, friction at attachments)

Driven Oscillations; Resonance

•  Imagine pushing a child on a swingset, applying force on every swing. – What if you push at exactly the right time, with

exactly the right force one every swing? – This “exact right time” is the resonant

frequency •  Example: Conservation of energy problem

from the third midterm.

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Energy Changes

•  Mass-on-a-spring: Exchange between kinetic energy (½ mv2) and potential spring energy (½ kx2)

•  Pendulum: Exchange between kinetic energy (½ mv2) and gravitational potential energy (mgh)

•  This provides a faster way to find the position or velocity of the oscillator at a particular time!

Example: A 0.5-kg, 0.75-m pendulum is pulled back at an

angle of 10º. •  What are the frequency and period of the

pendulum? •  If it just skims the ground at the bottom of

the swing, how much energy does the pendulum have?

•  How fast is it going at the bottom of its swing?

Properties of Waves

•  Wavelength is the distance between two wave peaks

•  Frequency is the number of times per second that a wave vibrates up and down

wave speed = wavelength x frequency

More Wave Properties

•  Amplitude: Distance from midpoint to crest of wave

•  Period: Time to complete one vibration

Speed of a Wave

•  v = λƒ = λ/T –  Is derived from the basic speed equation of

distance/time •  This is a general equation that can be

applied to many types of waves •  For sound in air, v = 343 m/s •  For light in a vacuum, v = c = 3.0 x 108 m/s

Types of Waves

•  Transverse Waves: like ocean waves or waves created by shaking a string

•  Longitudinal Waves: compression waves, such as sound

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Types of Waves – Transverse

•  In a transverse wave, each element that is disturbed moves in a direction perpendicular to the wave motion

Types of Waves – Longitudinal

•  In a longitudinal wave, the elements of the medium undergo displacements parallel to the motion of the wave

•  A longitudinal wave is also called a compression wave

What exactly is doing the waving?

•  Many waves require a medium: substance or object to move around – Sound waves – Water waves – Waves in a string – Waves in a spring/slinky

•  Waves transmit energy without moving the entire medium through space

Agenda

•  Today: Wave speed, sound vs. light, interference and standing waves

•  Tuesday: HW #12 quiz, more superposition and interference, doppler effect

Waveform – A Picture of a Wave

•  The brown curve is a “snapshot” of the wave at some instant in time

•  The blue curve is later in time

•  The high points are crests of the wave

•  The low points are troughs of the wave

Sinusoidal Waves

•  A wave can take on a variety of shapes, we’ll mostly look at sinusoidal waves

•  These waveforms can be described by a sine function that changes in both space and time:

y(x, t) = Acos 2π x

λ+tT

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Longitudinal Wave Represented as a Sine Curve

•  A longitudinal wave can also be represented as a sine curve

•  Compressions correspond to crests and stretches correspond to troughs

•  Also called density waves or pressure waves

Waves Lecture-Tutorial

•  Work with a partner or two •  Read directions and answer all questions

carefully. Take time to understand it now! •  Come to a consensus answer you all agree

on before moving on to the next question. •  If you get stuck, ask another group for help. •  If you get really stuck, raise your hand and I

will come around.

Speed of a Wave on a String

•  The speed on a wave stretched under some tension, F

– µ is called the linear density – F is the tension force in the string

•  The speed depends only upon the properties of the medium through which the disturbance travels

F mv where

Lµ

µ= =

Speed of sound waves •  “Room temperature” air allows sound waves

to travel at a speed of about 343 m/s. •  Temperature of air can change the speed of

sound! – General rule of thumb: sound travels more slowly

in cold air, faster in hot air – Determine distance to a lightning strike by

counting seconds until you hear the thunder (every 5 s ~ 1 mile)

– Unless otherwise specified, assume room temperature air.

Sound waves:

•  Human range of hearing is from about 20-20,000 Hz

•  Frequency pitch

Example: A violin string playing an “A” note vibrates at a frequency of

440 Hz. It has a linear density of µ = 7 x 10-4 kg/m.

•  What is the wave speed if the string is under a tension of 10 N?

•  What is the wavelength in the string? •  What is the wavelength of the 440 Hz

sound wave travelling through the air?

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Sound vs. Light

•  Sound waves and light waves are not the same thing!

•  You don’t actually hear radio waves, you hear sound waves that were picked up by the radio

•  Ears are sensitive to vibrations/compressions in the air

•  Eyes are sensitive to light waves (but only a very small portion of them…)

Wave Model of Light

•  Can characterize light as either a wave or a particle (modern physics says both!)

•  Light is electromagnetic radiation: waves that arise from the

•  Light needs no medium to travel! (Michelson & Morley showed this in 1887, effectively disproving the existence of the “luminiferous ether”)

The Electromagnetic Spectrum

•  Gamma Rays •  X-rays •  Ultraviolet Light •  Visible Light (ROY G BIV)

– λ = 400-700nm •  Infrared Light •  Microwaves & Radio Waves

Interference & Superposition •  Two traveling waves can meet and pass through

each other without being destroyed or even altered

•  Waves obey the Superposition Principle –  If two or more traveling waves are moving through a

medium, the resulting wave is found by adding together the displacements of the individual waves point by point

–  Actually only true for waves with small amplitudes

Constructive Interference

•  Two waves, a and b, have the same frequency and amplitude –  Are in phase

•  The combined wave, c, has the same frequency and a greater amplitude

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Constructive Interference in a String

•  Two pulses are traveling in opposite directions •  The net displacement when they overlap is the

sum of the displacements of the pulses •  Note that the pulses are unchanged after the

interference

Destructive Interference

•  Two waves, a and b, have the same amplitude and frequency

•  They are 180° out of phase

•  When they combine, the waveforms cancel

Destructive Interference in a String

•  Two pulses are traveling in opposite directions •  The net displacement when they overlap is

decreased since the displacements of the pulses subtract

•  Note that the pulses are unchanged after the interference

Reflection of Waves – Fixed End

•  Whenever a traveling wave reaches a boundary, some or all of the wave is reflected

•  When it is reflected from a fixed end, the wave is inverted

•  The shape remains the same

Reflected Wave – Free End

•  When a traveling wave reaches a boundary, all or part of it is reflected

•  When reflected from a free end, the pulse is not inverted

Standing Waves on a String

•  Nodes must occur at the ends of the string because these points are fixed

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Standing Waves on a String

•  The lowest frequency of vibration (b) is called the fundamental frequency

•  ƒ1, ƒ2, ƒ3 form a harmonic series

1ƒ ƒ2n

n Fn

L µ= =