…on to Chapter 16

• We have studied linear motion—objects moving in 1D or 2D at either constant velocity or constant acceleration.
• We have also studied objects rotating or moving at constant speed in a circle.
• In this chapter we encounter a new type of motion, in which objects oscillate.

Chapter 16: Oscillations

Some Examples:

• Sloshing of water in the bath-tub.
• Swaying of a tall building.
• The pendulum of a grandfather clock.
• The vocal cords in your larynx.

Grandfather Clock by Jason Jenkinson https://dribbble.com/shots/6464673-Grandfather-Clock

An oscillation is a periodic motion of some part of a system that occurs when that part is displaced from its equilibrium and allowed to respond to forces that tend to restore equilibrium.

Period and Frequency

• The oscillation frequency f is measured in cycles per second, or Hertz.

• The time to complete one full cycle, or one oscillation, is called the period, T.
• The frequency, f, is the number of cycles per second. Frequency and period are related by

Frequency and Period are inversely proportional to each other.

From today’s Preclass Survey

1. The plastic ruler has been released, and the restoring force is returning the ruler to its equilibrium position.
2. The net force is zero at the equilibrium position, but the ruler has momentum and continues to move to the right.
3. The restoring force is in the opposite direction. It stops the ruler and moves it back toward equilibrium again.
4. Now the ruler has momentum to the left.
5. In the absence of damping (caused by frictional forces), the ruler reaches its original position. From there, the motion will repeat itself.

Linear Restoring Force

“Mass on a Horizontal Spring”

The restoring force exerted on the mass by the spring:� F = −k x (Hooke’s Law)

F = m a (Newton’s Second Law)

Combine and solve for acceleration:

“Mass on a Horizontal Spring”

• Acceleration is the rate of change of velocity.
• Velocity is the rate of change of x.
• So, this equation is saying to us:

“The slope of the slope of x equals a negative constant times x.”

“Mass on a Horizontal Spring”

Chapter 2 Review

Google Sheets: Numerical Integration

• Let’s split up the motion of a mass on a spring into many segments, each over a small time Δt in which things that are not constant are approximately constant.
• If acceleration is approximately constant over a small time interval, Δt, then you can estimate the new velocity after that small time with the equation of constant acceleration: v_f ≈ v_i + a Δt.
• If velocity is approximately constant over a small time interval, Δt, then you can estimate the new position after that small time with the equation of constant velocity: x_f ≈ x_i + v Δt.
• Now that you have a new value of x, compute a new value of a, etc.
• This can be done with a Google Sheets spreadsheet! Let’s do it!

[ https://docs.google.com/spreadsheets/d/1gaZrFYz6GnrL7wPbiJFWwW721QLOFon1OrFl9bczlrM ]

“Simple harmonic motion”

• Simple harmonic motion (SHM) is motion that can be described by the following equation:

​

• It is a mathematical model of motion.

• Notice that x = +A at t = 0. If an object is at x = 0 at t = 0, you can either adjust the cos function by adding –(π/2) or use the sine function:

Vocabulary: “Amplitude”

• Amplitude The amplitude A of a vibration is the magnitude of the maximum displacement of the vibrating object from its equilibrium position.

Your book calls the amplitude X, I will call it A.

Equilibrium

• A spring with a mass attached to it is stretched and released. When the spring returns to equilibrium, will the mass be moving?
• Answer: Yes! It is not accelerating, but it is moving at that moment. Inertia then carries it past equilibrium to the other side. After passing equilibrium, the acceleration is opposite the velocity, so it slows down, eventually turning around.
• To analyze the motion, we always define x = 0, the origin, to be the equilibrium position of the mass.

Vocabulary: “Simple harmonic motion”

• Simple harmonic motion (SHM) is motion that can be described by the following equation:

​

​

​

• It is a mathematical model of motion.

S.H.M. notes.

• The period, T, is set by the properties of the system. In the case of a mass m attached to a spring of spring-constant k, the period is always

• A is set by the initial conditions: x₀ (initial position) and v₀ (initial velocity).

The Body Mass Measurement Device (BMMD) on the International Space Station is a chair of mass = 32 kg which has a oscillation period of 1.2 s when empty. When an astronaut sits on the chair, the oscillation period is more than 1.2 s.

The Body Mass Measurement Device chair (mass = 32 kg) has a oscillation period of 1.2 s when empty. When an astronaut sits on the chair, the period changes to 2.1 s. Determine the mass of the astronaut.

�Energy of Oscillating Systems��As a cart-spring system oscillates back and forth, the energy of the system continuously changes from all elastic to all kinetic.

Relationship between the amplitude of the vibration and the cart's maximum speed

• The equation can be rearranged to give:

​

​

​

• This makes sense conceptually:
• When the mass of the cart is large, it should move slowly.
• If the spring is stiff, the cart will move more rapidly.

Today’s Preclass Survey

What is the net torque on this pendulum? (Assume the rotation axis is the point where the string is attached to the ceiling.)

Suppose we restrict a pendulum’s oscillations to small angles (< 10°). Then we may use the small angle approximation sin θ ≈ θ, where θ is measured in radians. The net torque on the mass is

But the rotational inertia of a point mass m a distance L from the rotation axis is I = mL², so

So the simple harmonic motion equation for θ as a function of time is:

Leg swinging frequency

• When you walk, your arms and legs swing back and forth. These motions repeat themselves.

​

​

​

​

​

​

​

• The back-and-forth motion of an object that passes through the same positions is an important feature of vibrational motion.

Leg swinging frequency

• Your leg can be modeled as a simple pendulum, with length equal to the distance between your hip joint (rotation axis) and the centre of mass, L = 0.5 m.
• In this case, the frequency is:

• Longer legs have lower swinging frequencies.
• Giraffes take fewer steps per second than humans because of their long legs.
• Small dogs take more steps per second than humans because of their short legs.

Luke and Leia have a combined mass of 120 kg and both grasp a rope of length 30 m that is attached to a beam above them. The beam is half-way across a 10 m horizontal gap, and they want to swing across. If they start from rest and swing down and up, just reaching the other side, how long does this take?

Mass on Spring versus Pendulum

Simple Harmonic Motion (SHM)

• If the net force on an object is a linear restoring force (ie a mass on a spring, or a pendulum with small oscillations), then the position as a function of time is related to cosine:

• Cosine is a function that goes forever, but in real life, due to friction or drag, all oscillations eventually slow down.

Damping

• The phenomenon of decreasing vibration amplitude and increasing period is called damping.

​

​

​

​

​

• Damping is a useful aspect of the design of vehicles and bridges.

Same mass, same spring, Different viscosity of fluids!

Three Classes of Damping

1. An under-damped system continues to vibrate for many periods.

Underdamped Oscillator

Three Classes of Damping

2. In an overdamped system, the oscillating system takes a long time to return to the equilibrium position, if it ever does.

​

​

3. In a critically damped system, the oscillating object returns to equilibrium in the shortest time possible without producing vibrations.

Well-adjusted Mountain Bike shocks do this.

Driven Oscillations and Resonance

• Consider an oscillating system that, when left to itself, oscillates at a frequency f₀. We call this the natural frequency of the oscillator.

​

• Suppose that this system is subjected to a periodic external force of frequency f_ext. This frequency is called the driving frequency. Driven systems oscillate at f_ext.

​

• The amplitude of oscillations is generally not very high if  f_ext differs much from f₀.

​

• As f_ext gets closer and closer to f₀, the amplitude of the oscillation rises dramatically.

Externally Driven Oscillations

Resonance!

Energy transfer through resonance

• Resonance caused the collapse of the Tacoma Narrows Bridge in Washington only four months after its completion.

Waves

• Two of the five senses depend on waves in order to work: which two?
• Answer: Sight and Sound!
• Sound is a pressure wave which travels through the air.
• Light is a wave in the electric and magnetic fields.

• An oscillation is a periodic linear motion of a particle about an equilibrium position.

• When many particles oscillate and carry energy through space, this is a wave. A wave extends from one place to another.
• Examples are:
• water waves
• light, which is an electromagnetic wave
• sound

[image from https://webspace.utexas.edu/cokerwr/www/index.html/waves.html ©1999 by Daniel A. Russell ]

Waves

Image from https://www.thehappycatsite.com/cat-drinking-a-lot-of-water/

• When an object vibrates, it also disturbs the medium surrounding it.
• When a cat’s tongue touches the surface of the water, the vibrating tongue (the source) sends ripples (waves) across the bowl.
• The medium here is the water.

Waves and Wave Fronts

• A wave front is the locus of all crest points at which the disturbance of a wave is at a maximum.
• Spherical wave fronts of sound spread out uniformly in all directions from a point source.
• Electromagnetic waves in vacuum also spread out as shown here.

​

Transverse waves

• Medium vibrates perpendicularly to direction of energy transfer
• Side-to-side movement

Example:

• Vibrations in stretched strings of musical instruments

[image from http://www.maths.gla.ac.uk/~fhg/waves/waves1.html ]

Longitudinal waves

• Medium vibrates parallel to direction of energy transfer
• Backward and forward movement consists of
• compressions (wave compressed)
• rarefactions (stretched region between compressions)

Example: sound waves in solid, liquid, gas

[image from http://www.maths.gla.ac.uk/~fhg/waves/waves1.html ]

Longitudinal Waves

• Sound is a longitudinal wave.
• Compression regions travel at the speed of sound.
• In a compression region, the density and pressure of the air is higher than the average density and pressure.

Snapshot Graph

• A graph that shows the wave’s displacement as a function of position at a single instant of time is called a snapshot graph.

• For a wave on a string, a snapshot graph is literally a picture of the wave at this instant.

One-Dimensional Waves

• The figure shows a sequence of snapshot graphs as a wave pulse moves.
• These are like successive frames from a movie.
• Notice that the wave pulse moves forward distance �Δx = vΔt during the time interval Δt.
• That is, the wave moves with constant speed.

History Graph

• A graph that shows the wave’s displacement as a function of time at a single position in space is called a history graph.

• This graph tells the history of that particular point in the medium.
• Note that for a wave moving from left to right, the shape of the history graph is reversed compared to the snapshot graph.

“Cosine” is one shape a wave can have!

• The Period T in seconds is the time for one complete vibration of a point in the medium anywhere along the wave’s path.
• The Frequency f in Hz (s^–1) f = 1/T, is the number of vibrations per second of a point in the medium as the wave passes.
• The Amplitude A is the maximum distance of a point of the medium from its equilibrium position as the wave passes.
• The Wave Speed v in m/s is the distance a disturbance travels in a time interval divided by that time interval.

“Sinusoidal” Wave on a String

• Shown is a snapshot graph of a wave on a string with vectors showing the velocity of the string at various points.
• As the wave moves along x, the velocity of a particle on �the string is in the �y-direction.

y

Sinusoidal Wave Animation

• Here is an animated GIF image of a traveling sinusoidal-wave
• The wave pattern remains the same shape, it just travels from left to right
• All the individual particles, anywhere along the string, oscillate up and down as the wave pattern passes.

Properties of Sinsoidal Waves

• Wavelength λ is the distance over which a wave repeats in space.
• Period T is the time for a complete oscillation of the wave at a fixed position.
• Frequency f is the number of wave cycles per unit time: f = 1/T
• Amplitude A is the maximum value of the wave disturbance.

Wave Speed

• Wave speed is the rate at which the wave pattern propagates.
• Wave speed, wavelength, period, and frequency are related:

From today’s Preclass Survey

Waves in Two and Three Dimensions

• Particles cannot occupy the same space. They collide.

Particles and Waves

[Animations from http://www.physicsclassroom.com/mmedia/newtlaws/mb.cfm and http://www.acs.psu.edu/drussell/demos/superposition/superposition.html ]

• Waves pass right through each other. They interfere.

Waves in Two and Three Dimensions

If two or more waves combine at a given point, the resulting disturbance is the sum of the disturbances of the individual waves.

The Superposition Principle

Standing Waves on a String

Reflections at the ends of the string cause waves of equal amplitude and wavelength to travel in opposite directions along the string, which results in a standing wave.

Standing Waves

• When two sine waves, identical in every way, but traveling in opposite directions, combine, you get a standing wave!

Standing Waves Vocabulary

You can draw a standing wave as a set of 5 overlapping snapshots:

What is really happening is this:

1.

2.

3.

Fundamental, or First harmonic

Second harmonic

Third harmonic

½ wavelength

½ wavelength

½ wavelength

Etc

Standing Waves

• Standing Waves are a form of RESONANCE.
• In resonance, there is a natural frequency, and a driving frequency.
• Natural frequency is set by the properties of the system.
• You choose the driving frequency.
• If you match the driving frequency to the natural frequency, the amplitude goes very high. (This is how most musical instruments work.)
• With standing waves, there is more than one natural frequency!

Let’s Crank The Numbers!!!

Once you know the fundamental frequency, f₁, for a particular standing wave, then the formula to find the frequencies of the harmonics is:

�𝑓_𝑛 = 𝑛 × 𝑓₁

​

Here n is a whole number which labels each harmonic.

Fundamental, or First harmonic

Second harmonic, or First Overtone

Third harmonic, or Second Overtone

n = 1. f₁ = 20 Hz.

n = 2. f₂ = 40 Hz.

n = 3. f₃ = 60 Hz.

Notice that in each case, 𝑓_𝑛 = 𝑛 × 𝑓₁

Beats

• Periodic variations in the loudness of sound due to interference
• Occur when two waves of similar, but not equal frequencies are superposed.
• Provide a comparison of frequencies
• Frequency of beats is equal to the difference between the frequencies of the two waves.

[image from http://hyperphysics.phy-astr.gsu.edu/hbase/sound/beat.html ]

[Demonstration]

time

Air Pressure

time

time

+

=

“in phase”

Amplitude = 2 × original wave

time

Air Pressure

time

time

+

=

“completely out of phase”

Amplitude = 0

time

Air Pressure

time

time

+

=

“out of phase”

Amplitude = between 0 and 2 × original wave

Beats

time

[s]

+

=

time

[s]

time

[s]

220 Hz

221 Hz

• The first 0.1 seconds.

​

• Both tones start “in phase”

​

• There are 22 full oscillations of the first tone.

​

• There are 22.1 full oscillations of the second tone.

Beats

time

[s]

+

=

time

[s]

time

[s]

220 Hz

221 Hz

• The first 0.5 seconds.

​

• Both tones start “in phase”

​

• There are 110 full oscillations of the first tone.

​

• There are 110.5 full oscillations of the second tone.

​

• They end up “out of phase”

Beats

time

[s]

+

=

time

[s]

time

[s]

220 Hz

221 Hz

• The first 2.5 seconds.

​

• There are so many oscillations that the wiggly line just looks like a solid colour!

​

• The blue bumps are called “beats”

Beats

time

[s]

+

=

time

[s]

time

[s]

220 Hz

233 Hz

• The first 0.5 seconds of two different tones.

​

• Both tones start “in phase”

​

• Every 1/13^th of a second, the red curve has 16.92 oscillations, and the green curve has 17.92 oscillations.

​

• So they end up in phase again.

Beat and beat frequencies

Eq.16.73

Beats

• Applications
• Piano tuning by listening to the disappearance of beats from a known frequency and a piano key
• Tuning instruments in an orchestra by listening for beats between instruments and piano tone

Wave Intensity

• A speaker emits a certain power in Watts. (Joules per second)
• The wave fronts are spheres (or hemispheres) each with a certain surface area in metres².

Wave intensity

• The intensity of a wave is the average power it carries per unit area.
• If the waves spread out uniformly in all directions and no energy is absorbed, the intensity I at any distance r from a wave source is inversely proportional to r².

Sound Intensity

Outer ear

Auditory canal

Eardrum

• Our eardrum has a certain area, which “catches” a certain amount of power.
• So, it is the intensity that reaches our head which determines how loud a sound is.
• That’s why further sound sources seem quieter to us!

Wave Intensity

• Wave intensity is the power crossing�a unit perpendicular area.
• In a plane wave, the intensity remains�constant.
• A spherical wave spreads in three�dimensions, so its intensity drops�as the inverse square of the distance�from its source: