oscillations and waves (4)

Oscillations and Waves (4)Many things go back and forth; they vibrate; they

oscillate. Examples include a swing, a leaf on a tree, a yo-yo, and the stock market.

Without waves there would be no light, no sound, no music, no Picasso, no communication, no

society, and therefore, no humanity as we know it.

Mr. KlapholzShaker Heights

High School

A Pendulum and a mass on a spring.

A Pendulum and a mass hanging from a spring.

• Why do the systems go back and forth?• When you take the system out of equilibrium, and

release it, it goes back to equilibrium. Why?• In each system, what causes the restoring force?• In each system, where is the mass when the force is

large?• Why doesn’t it just go back to equilibrium and stay

there?

A Pendulum and a mass hanging from a spring.

• In each case, the force is opposite in direction to the displacement. This helps define Simple Harmonic Motion.

Force = -kxma = -kx

a = -(k/m)x {This always gives SHM}

Both systems show:

• Equilibrium Position• Natural frequency (f)– How frequently the system cycles in one second. – Measured in Hertz (Hz = 1/s)– Example: the frequency of the wings of a housefly is

about 200 Hz.• Amplitude– How large does the displacement ever get– Measured in meters

Both systems show:• Displacement– How far the mass is from equilibrium at any moment

• Period (T)– How much time it takes for the motion to repeat– Measured in seconds– T = 1 / f

• Angular Frequency– w = 2pf– Used for describing motion when using trig functions;

one cycle is 2p radians.– Housefly Example: w = 2pf = 2p(200 Hz) ≈ 1260 Hz

Graph of Displacement vs. Time

• You might need a separate page of paper to draw 4 graphs, one below the other, like a tower.

Graph of Displacement vs. Time

Graph of Velocity vs. Time

Graph of Acceleration vs. Time

Graph of Force vs. Time

Graph of Force vs. Time

• Compare this graph to the defining idea: the force is opposite in direction to the displacement.

Review the Basic Math of SHM

• The force is opposite in direction to the displacement.

• Force = -kx {This always gives SHM}• ma= -kx {This always gives SHM}• a = -(k/m)x {This always gives SHM}

The Solution of SHM

• If you take a system out of equilibrium, by a distance A (for ‘amplitude’) then the system will oscillate.

• The sine and cosine functions are good descriptions of SHM.

• x = A cos(wt) (This is the solution.)• velocity = -Aw sin(wt)

(what is the fastest that the object will move?)• acceleration = -Aw2 cos(wt)

Check the Solution of SHM• Recall:• a = -(k/m)x {This always gives SHM} • x = A cos(wt) and a = -Aw2 cos(wt)• Put it all together:

-Aw2 cos(wt) = -(k/m) A cos(wt)

w2 = (k/m)• For large restoring forces, the frequency is large.• For large masses, the frequency is small.

a = -w2x

Connection between speed & position

v2 = w2 (A2 – x2)• This equation tell us that when the

displacement is equal to the amplitude, the speed is _______.


v2 = w(A2 – x2)• This equation tell us that when the

displacement is equal to the amplitude, the speed is zero.




• And, when the displacement is zero, the speed is




• And, when the displacement is zero, the speed is

€

v = ωA

Kinetic Energy One

• KE = ½mv2

• Speed depends on time: v = -Aw sin(wt) • So, KE = ½m(-Aw sin(wt))2

• KE = ½mA2w2 sin2wt• The Kinetic energy is a varying function.

What is its greatest value?• KEmax = ½mw2A2

Kinetic Energy Two

• KE = ½mv2

• Also, speed depends on position:

v2 = w2 (A2 –x2)• So KE = ½mw2(A2 – x2)• What is the greatest value that the Kinetic

Energy can have?• KEmax = ½mw2A2

Total Energy = KE + PE

• The total energy stays the same for a closed system.

• As the system passes through equilibrium, the Potential Energy is zero; all of the energy is kinetic. This ‘fixes’ the value of the total energy to be the same as the maximum of the kinetic energy.

• Total Energy = ½mwA2

Total Energy = KE + PE• PE = Total Energy - ?• PE = Total Energy – KE• We can write it as a function of position or of time.• Time View: • PE = ½mwA2 - ½mA2w2 sin2wt • PE = ½mwA2 {1-sin2wt} = ½mwA2 {cos2wt} • Position View:• PE = ½mwA2 - ½mw(A2 – x2)• PE = ½mwx2

Example: How much does this pendulum have as it passes through equilibrium?

• The mass of the ‘bob’ is 200 g = 0.2 kg• The amplitude of the motion is 3 cm = 0.03 m.• The frequency of the motion is 0.5 Hz.

Example: How much energy does this pendulum have as it passes through equilibrium?

• The KE at the equilibrium position is the greatest KE the bob will ever have.

• KEmax = ½mw2A2

– w = ?– w = 2pf = 2p(0.5) = 3.14

• KEmax = ½mw2A2

• KEmax = ½ (0.2)(3.14)2(0.03)2

• KEmax = 9 x 10-4 J

In the previous example…

• How much Gravitational Potential Energy does the bob have when it is at equilibrium?

• 0 J• How much Gravitational Potential Energy does

the bob have when it is at the extreme part of its motion?

• 9 x 10-4 J

A story of damping:

• Use a tea bag to make some tea. Pull the the little label of the tea bag so that the bag is suspended above the hot liquid.

• The bag will begin to spin. Why?• Then the bag will spin faster. Why?• Then the bag will slow to a stop. Why?• Then the bag will spin the other way. Why?• Now change one thing…

Do the same experiment, but…

• Suspend the bag so that it can spin, but keep the bag in the liquid, but.

• What will be different?• The liquid ‘damps’ the motion. Oscillations

might still occur, but with less and less amplitude.

• Imagine doing the same experiment in honey.• See the figure…

Find the bag in water and in honey.

http://www.splung.com/content/sid/2/page/damped_oscillations

Natural Frequency

• If you disturb a system, and let it go, it will often vibrate. That vibration will be at its natural frequency.

• Examples: – Pendulum– mass on spring– guitar string

Forced Vibration

• If you push and pull on a system, then that has a big effect on what the system does.

• Examples: – Pendulum– mass on spring– guitar string

Resonance

• If you push and pull on a system, at the natural frequency of the system, then the amplitude can get really big.

• Examples: – kid on a swing.– wine glass– Tacoma Narrows Bridge Disaster– St. Louis Bridge in a hotel

• See the resonance curve:

Amplitude vs. Frequency

Phase

• Imagine that you are keeping a nice rhythm, by clapping once per second.

• Another person could clap with the same frequency, but be out of step with you, by 0.5 seconds, or 0.25 seconds, whatever.

• This difference is called “phase”.

Waves

• A wave is a disturbance that moves. • If you make disturb the surface of a swimming

pool in a simple harmonic way, then at any one point on the surface the water will oscillate with SHM.

• Also, the disturbances move across the pool, and that’s what we call waves.

• Examples: sound, light, garden hose• What follows are properties of waves…

Transverse vs. Longitudinal

• If the disturbance is perpendicular to the motion of the disturbance, we call the wave “Transverse”.

• Examples: Classic Slinky wave, waves on the surface of a swimming pool, light.

• If the disturbance is in the same direction as the motion of the disturbance, we call the wave “Longitudinal”.

• Examples: Sound, waves on a slinky made by pulling and pushing.

Wavelength (l)

http://electron9.phys.utk.edu/phys135d/modules/m10/waves.htm

Amplitude (A, x0)

Wave Speed (v)

• v = f l• Practice: If the frequency is 4 Hz, and the

wavelength is 0.5 m, then what is the speed of the wave?

• v = f l = (4 s-1) (0.5 m) = 2 m s-1

Guitar Basics

• If the string is thicker, then the speed of the wave is slower. What does that do to frequency?

• v = f l. So f = v / l. We see that thick strings cause a low values for frequency. Thick strings make low sounds.

• Tightening the tension in the string makes the speed greater. What does that do to pitch?

• Putting your finger on the string makes it shorter, what does that do to frequency? Hint: f = v / l

Reflection

• If you are standing in a pool, and you make disturbances on the surface, then when the pulses reach the wall of the pool, they will bounce back.

• Waves on a slinky show this.• Mirrors reflect light.• Walls reflect sound.

Reflection


Interference (Superposition)

• One of the craziest things about waves is that they can can go through each other.

• Examples: Slinky, Light and Sound• But while two waves are in the same spot, the

result is the addition (or subtraction of the waves).

Constructive Interference


Destructive Interference


Diffraction

http://learn.uci.edu/oo/getOCWPage.php?course=OC0811004&lesson=005&topic=006&page=10

Diffraction

• Sound goes around corners.• Water waves bend around edges.

Diffraction depends on the size of the opening.

Diffraction caused by an obstacle

http://mcat-review.org/waves-periodic-motion.php

Refraction

http://hyperphysics.phy-astr.gsu.edu/hbase/geoopt/refr2.html

Refraction(this could be water waves or light waves)

http://www.school-for-champions.com/SCIENCE/waves_obstacle.htm

Refraction

• Waves change direction (they bend) when they change to a medium in which they must travel at a different speed.

Air GlassThe angle in glass is less than the angle in air.

A

Normal lineNormal lineqq

The Math of Refraction: Snell’s Law

v2 sin(q1) = v1 sin(q2)

Math of Refraction of Light

The “index of refraction” of a medium depends on the speed of light in that

medium:n = c / v

Where c = speed of light in a vacuumAnd v = speed of light in the medium

Snell’s Law: n1sin(q1) = n2sin(q2)

oscillations and waves (4)

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