how things work exam 1

inertia

A body in motion tends to remain in motion; a body at rest tends to remain at rest.

position

A specific point in space

distance + direction = position

a vector quantity =

magnitude +

magnitude

how much quantity there is

direction

where the quantity is pointing

velocity

measures the rate at which your position is changing with time.

speed

the distance traveled in a certain amount of time

mass

the measure of your inertia, your resistance to changes in velocity

has no direction, so it's not a vector quantity but a scalar quantity

scalar quantity

a quantity that has only an amount.

acceleration

measures the rate at which your velocity is changing with time

newton’s second law of motion

The net force exerted on an object is equal to that object's mass times its acceleration. The acceleration is in the same direction as the net force.

Galilean relativity

The principle that the Laws of Physics are the same for all observers, as long as they are moving at fixed velocities
(We call these inertial reference frames)

Net force is the

• vector sum of all forces on an object.

Acceleration

a change in velocity with time

mass

 measure of object’s inertia

Newton’s Second Law

An object’s acceleration is equal to the net force exerted on it divided by its mass. That acceleration is in the same direction as the net force

Newton’s constant (G)

About  6.7x10^-11 N m² / kg²

gravity on Earth’s surface (g)

About 9.8 m/s²

_____ is the same for all masses

acceleration

A ball’s weight is proportional to its mass:

Force = weight = m g

All things on Earth’s surface accelerate at rate

g

velocity

acceleration * time

Gravitational potential energy

m* g * height

energy is 

conserved

Friction force always

opposes velocity of object.

It depends on:

• how tightly the two surfaces are pressed against one another

• how slippery the surfaces are

• how the surfaces are moving relative to one another

 Static friction is ______ than sliding friction

stronger

Static friction

opposes the start of sliding.

• It varies in amount from zero to a maximum value.

Sliding friction

opposes ongoing sliding.

• It has a constant value proportional to support force.

Static friction’s maximum________sliding friction

exceeds

Only sliding friction wastes energy. Why?

• The two surfaces travel different distances.

• The missing work becomes thermal energy.

• The surfaces also experience wear.

Kinetic

• energy of (relative) motion

Potential

• stored in forces between objects

  • Gravitational

  • Magnetic

  • Electrochemical

  • Nuclear

  • Elastic

  • Electric

  • Chemical

Thermal energy

• disordered motion of atoms

  • Doing work with thermal energy is more difficult

momentum

mass * velocity

An object’s momentum is changed by

changing its velocity -- by pushing on it with a force.

We call the amount of pushing * time

the impusle

momentum is not dependent on

mass

Newton’s 3rd law

The force on an object is always balanced by an opposite force on another object, at each instant of time

Why is momentum conserved?

Impulse = change in momentum = F * t

Momentum is a

conserved vector quantity.

• It can’t be created or destroyed but can be transferred.

• It combines bumper car’s inertia and velocity.

momentum = mass · velocity

Period

interval between two repetitive motion cycle

Frequency

cycles completed per unit of time

f=1/period  → measured in Hz (1/s)

Amplitude

peak distance away from motion’s center

In a good clock, the

period of its timekeeper shouldn’t depend on amplitude

• A harmonic oscillator is a system with

• a stable equilibrium

• a restoring influence that’s proportional to displacement

• Its period is independent of its amplitude!

The farther the pendulum is away from the center,

the larger the restoring force!

F = m g sin(angle)

F is proportional to angle.

Where is the energy  in a pendulum?

At the bottom, kinetic. At the top, gravity potential. It oscillates between them...

A harmonic oscillator always has

• an inertial aspect (e.g.,  a mass)

• a springlike restoring aspect (e.g., a spring)

• A harmonic oscillator’s period decreases as

• its inertial aspect becomes smaller

• its spring-like restoring aspect becomes stiffer

• A pendulum is (almost) a harmonic oscillator.

• For small displacements:

• its restoring force is proportional to displacement

• its period is independent of amplitude

• its period is proportional to sqrt(length/g)
  
  sqrt( m / (m/s²) )  →  sqrt(s²) → s

How are harmonic oscillators used in clocks?

Their motions are gently encouraged and counted

A clock

• supplies energy to keep its harmonic oscillator going

• counts cycles of that oscillator and reports the time

• Common harmonic oscillators used in clocks are

• pendulums

• quartz crystals

• atomic vibrations

A pendulum’s springlike restoring force

• is caused by gravity

• is proportional to the pendulum’s weight

• is therefore proportional to the pendulum’s mass

Increasing a pendulum’s mass

• increases its inertial aspect

• increases its restoring force aspect

• therefore has no effect on its period!

• A quartz crystal is a harmonic oscillator.

• Crystal’s mass provides the inertial aspect.

• Crystal’s body provides the springlike restoring aspect.

As a harmonic oscillator, a quartz crystal’s

• oscillation decay is extremely slow

• fundamental accuracy is extremely high

Quartz is piezoelectric.

• Its mechanical and electrical changes are coupled.

• Its motion can be induced and measured electrically.

The quartz tuning fork in a quartz clock is

• kept vibrating by giving it energy electronically

• observed and its vibrations counted electronically

• insensitive to gravity, temperature, pressure, and acceleration

Quartz’s slow oscillation decay
gives it

• a very precise period.

The most accurate clocks today are “atomic”

• Electrons in atoms
  vibrate at precise rates,
  when isolated and cold

• A group of Cs atoms
  is held in a radio cavity
  at 9,192,631,770 Hz

NIST-F2 accurate to
1 second in 300 million years, synchronizes GPS

• A tight string has

• stable equilibrium shape: a straight line

• mass that provides an inertial aspect

• tension and length to provide springlike restoring aspect

A tight string is a harmonic oscillator

• It vibrates about its equilibrium shape.

• Pitch is independent of its amplitude/volume!

A string has a

• fundamental vibrational mode.

A string has a fundamental vibrational mode.

• A string vibrates up and down as a single arc.

• 1 displacement antinode at string’s center

• 2 displacement nodes, 1 node at each end of string

Its fundamental pitch (frequency of vibration) is

• proportional to sqrt(tension)

• proportional to 1/length

• proportional to 1/sqrt(mass)

Why does a vibrating string sound like a string?

It has specific harmonics that define its sound

A string can vibrate as

• 2 half-strings (2 antinodes)

• 3 third-strings (3 antinodes)

• and more higher-order modes

Higher-order vibrational modes

• produce overtones (over the fundamental pitch)

• String’s overtones are harmonics: integer multiples

First overtone involves 2 half-strings:

• 2 × the fundamental pitch: 2^nd harmonic

• one octave above the fundamental frequency

• Second overtone involves 3 third-strings:

• 3 × the fundamental pitch: 3^rd harmonic

• an octave and a fifth above the fundamental

Humans can hear between

~20 and ~20,000 Hz

Plucking a string

• transfers energy all at once.

Bowing a string

• transfers energy gradually.

  • Bow does a little work on the string every cycle.

Energy accumulates via resonant energy transfer.

A string will exhibit sympathetic vibration when

• another object vibrates at string’s resonant frequency

• resonant energy transfer goes from object to string

Why do stringed instruments need surfaces?

Surfaces project sound much better than strings.

In air, sound consists of density fluctuations.

• Air has a stable equilibrium: uniform density.

• Disturbances from uniform density make air vibrate

Vibrating strings don’t project sound well

Air flows easily around narrow vibrating strings.

Surfaces project sound much better

• Air can’t flow easily around vibrating surfaces.

• Air is substantially compressed or rarefied: sound.

Why does a drum sound particularly different?

Its overtones are not harmonics.

Most 1-dimensional instruments

• can vibrate at half, third, quarter length, etc.

• have harmonic overtones

Most 2- or 3- dimensional instruments

• have complicated higher-order vibrations

• have non-harmonic overtones.

superposition

Waves add to each other: “superposition”…

This is how all the harmonics can exist all at once.

This is also why you can hear two people talking at the same time – their sound waves simply “add” and pass right through each other!

Interference

Two waves when “adding” to each other interfere, if they have the same frequencies…

If waves have the same “phase”

they “interfere constructively”

If waves have the opposite “phase”

they cancel each other out – “interfere destructively” 🡪 how noise-canceling headphones work!

If they have nearly the same frequencies

they form “beats”, at a frequency twice the difference in frequencies…

It oscillates between constructive and destructive interference.

What is vibrating in a wind instrument?

Air in a tube is a harmonic oscillator

Air in a tube

• It has a stable equilibrium arrangement: uniform density

• Mass provides an inertial aspect.

• Pressure and length provide spring-like restoring aspect.

Air in a tube is a harmonic oscillator.

• It vibrates about its equilibrium arrangement.

• Pitch is independent of its amplitude/volume!

• Air column has a fundamental vibrational mode.

• Air column vibrates up and down as a single object.

• 1 pressure anti-node at air column’s center

• 2 pressure nodes, 1 node at each open end of column

Air’s fundamental pitch is

• proportional to sqrt(pressure)

• proportional to 1/length

• and proportional to 1/sqrt(density)

• Air column has a fundamental vibrational mode.

• Air column vibrates up and down as a single object.

• 1 pressure anti-node at air column’s closed end

• 1 pressure node at air column’s open end

The air column in an open-closed pipe vibrates

• like half the air column in an open-open pipe

• at half the frequency of an open-open pipe

In an open-open pipe, the overtones are at

• 2 × the fundamental (2 pressure antinodes)

• 3 × the fundamental (3 pressure antinodes)

• and all integer harmonics

In an open-closed pipe, the overtones are at

• 3 × the fundamental (2 antinodes)

• 5 × the fundamental (3 antinodes)

• and all odd-integer harmonics

Why does adding water to the bottle raise the pitch of that tone?

The water shortens the column of moving air inside the bottle and increases the frequency of its fundamental vibrational mode.

How does sound travel through air?

Air exhibits longitudinal traveling waves.

Basic modes of finite objects are

standing waves.

Standing wave

nodes and antinodes don’t move