Measurement

The height of a table, the duration of a song, and the mass of a book are physical quantities because each can be measured and compared.

Saying a table is $2$ metres long compares its length with the metre as the chosen unit of length.

A unit is not the same kind of object as the numerical value used to represent a measured quantity. For a scalar quantity, the numerical value is a real number after a unit is chosen. For a vector quantity, the numerical value is a vector after coordinates and a unit are chosen, and the unit tells how to interpret each component.

If length is measured in metres and time is measured in seconds, then a length divided by a time is measured in the product unit $m\cdot s^{-1}$, because division by seconds is multiplication by $s^{-1}$.

Basic Quantities

The time between two clock ticks is a duration, and the order of the ticks tells which event happened first.

A light wave with $5\times {10}^{14}$ cycles each second has frequency $5\times {10}^{14}Hz$.

If a pendulum completes one swing cycle every $2s$, then its period is $2s$.

The width of a doorway and the distance between two points on a table are lengths.

The position of a book on a table can be described by its distance from the table's edges.

A train moving along a track is in motion because its position changes as time passes.

If a car travels $100km$ in $2h$, its average speed is $50km/h$.

This phenomenon can be seen experimentally by giving a bowling ball and a tennis ball similar pushes: the bowling ball's motion changes less.

A raised book has gravitational energy because it can fall and transfer energy to another object on impact.

A hot pan transfers heat to cooler water placed in it.

Defining Constants

Two protons repel each other because they have electric charges of the same kind.

A proton and an electron have opposite charge signs.

The electric charge carried by a proton is positive.

The electric charge carried by an electron is negative.

A neutron is electrically neutral.

In a simple model of a gas, each atom can be treated as one particle moving through space.

Protons, neutrons, and electrons are subatomic particles.

Each hydrogen nucleus contains one proton.

A carbon-12 nucleus contains $6$ neutrons.

A neutral hydrogen atom has one electron bound to its nucleus.

In a carbon-12 atom, the nucleus contains $6$ protons and $6$ neutrons.

A neutral helium atom has a nucleus with $2$ protons and bound electrons around it.

Every atom with $6$ protons is carbon, even if it has a different number of neutrons.

Any atom whose nucleus has $55$ protons is a caesium atom.

Carbon-12 and carbon-14 are isotopes of carbon because both have $6$ protons, but they have different numbers of neutrons.

Radioactive decay is modeled as spontaneous when it occurs without a collision or applied field triggering the particular decay event.

A nucleus that emits an alpha particle undergoes nuclear decay and becomes a different nucleus.

An atom sitting in a room-temperature laboratory with no deliberately applied strong fields is being considered under ordinary conditions for this discussion.

Caesium-133 is stable in the nuclear sense because its nuclei do not spontaneously decay under ordinary conditions.

The atoms used in caesium atomic clocks are caesium-133 atoms.

Sunlight is radiation carrying energy from the Sun to Earth.

A compass needle turns because it is affected by Earth's magnetism.

Light is called electromagnetic because it is described using linked electric and magnetic effects.

A radio antenna creates a changing electromagnetic field around it.

Radio waves, microwaves, visible light, and X-rays are forms of electromagnetic radiation.

An ideal single-frequency laser beam is modeled as periodic radiation.

In a simple quantum model, an electron bound in an atom can occupy only certain atomic energy levels.

When an atom moves from a higher energy level to a lower one, the lost energy can leave as electromagnetic radiation.

The two ground-state levels used in a caesium-133 atomic clock are closely spaced: electromagnetic radiation with frequency $9192631770Hz$ has the energy needed for an energy level transition between them, while visible-light atomic transitions have frequencies hundreds of thousands of times larger.

The electric attraction between an atomic nucleus and an electron is a physical interaction.

The caesium-133 clock transition uses two hyperfine levels of the ground state of a caesium-133 atom.

An isolated caesium-133 atom with no applied external fields is idealized using its unperturbed ground state.

The second is defined by counting periods of the electromagnetic radiation associated with the caesium-133 transition frequency.

Sunlight, laser light, and the light from a lamp are all forms of electromagnetic radiation.

Interplanetary space is often modeled as a vacuum when the small amount of matter there is irrelevant to the calculation.

In a vacuum, light travels at exactly $299792458m/s$ by the definitions of the metre and second.

In a green laser beam, the frequency is the frequency of the associated light radiation: the electromagnetic state repeats at that rate. The beam can be modeled as many photons associated with that same frequency and traveling in nearly the same direction.

A frequency of $5Hz$ is nonzero, while a frequency of $0Hz$ is zero.

The ratio of a circle's circumference to its diameter is dimensionless: measuring both lengths in metres or both lengths in inches gives the same number $\pi$.

The speed of light is a dimensional constant: the same physical speed is written as $299792458m/s$ or about $186000$ miles per second because the length unit changed.

If a photon has energy $E$ and its associated electromagnetic radiation has nonzero frequency $f$, then dividing $E$ by $f$ gives the same physical constant $h$ for every photon. The frequency is not a frequency of the photon moving around a path; it is the frequency of the electromagnetic radiation associated with the photon. To assign a numerical value to $h$, the energy and frequency must be measured using chosen units. Since energy is measured in joules and $1J=1kg\cdot m^2\cdot s^{-2}$, the numerical value of $h$ depends on the kilogram, metre, and second.

Different photons can have different energies and frequencies while keeping the same ratio. A photon associated with frequency $1Hz$ has energy $$E=hf=(6.62607015\times {10}^{-34}J\cdot s)\left(1s^{-1}\right)=6.62607015\times {10}^{-34}J,$$ so $E/f=6.62607015\times {10}^{-34}J\cdot s$. A photon associated with frequency ${10}^{14}Hz$ has energy $$E=(6.62607015\times {10}^{-34}J\cdot s)\left({10}^{14}{s}^{-1}\right)=6.62607015\times {10}^{-20}J,$$ and again $E/f=6.62607015\times {10}^{-34}J\cdot s$. A photon associated with frequency $5\times {10}^{14}Hz$ has energy $$E=3.313035075\times {10}^{-19}J,$$ and dividing by $5\times {10}^{14}{s}^{-1}$ gives the same ratio $h$.

The same Planck constant can therefore be written with different numerical values. Using joules and seconds, $$h=6.62607015\times {10}^{-34}J\cdot s.$$ Since $1J={10}^{7}erg$, the same constant is $$h=6.62607015\times {10}^{-27}erg\cdot s.$$ Since $1kJ=1000J$, it is also $$h=6.62607015\times {10}^{-37}kJ\cdot s.$$ Using electronvolts for energy gives approximately $$h\approx 4.135667696\times {10}^{-15}eV\cdot s.$$ These are not different physical constants; they are the same constant expressed with different energy units.

Base Units

A stopwatch reading of $3s$ measures a duration of three seconds.

A metre stick is a physical object made to approximate one metre of length.

The physical value of the Planck constant is not being changed. Instead, the size of the unit $kg$ is chosen so that, when $h$ is measured using kilograms, metres, and seconds, its numerical value is exactly $6.62607015\times {10}^{-34}$. This is like choosing a length unit around a particular table so that the table measures exactly $1$ table-unit: the table does not change, but the unit has been defined by using it as the reference.

The particular number $6.62607015\times {10}^{-34}$ was chosen so that the newly defined kilogram would have the same practical size as the kilogram already used in measurements. Before the kilogram was fixed this way, experiments measured the numerical value of $h$ using the existing mass scale. The chosen number is that measured value, rounded and then made exact, so ordinary masses did not suddenly change size when the definition changed.

A $1kg$ calibration mass is an object whose mass approximates one kilogram.

Derived Quantities

A car moving north at $50km/h$ and a car moving east at $50km/h$ have the same speed but different velocities.

An object whose speed increases by $2m/s$ every second has acceleration $2m/s^2$.

Pushing a cart applies a force that can change the cart's motion.

Pushing a box forward across the floor does work on the box when the force and motion point in the same direction.

Derived Units

A $60Hz$ signal repeats $60$ times each second.

When mass is measured in kilograms and acceleration is measured in metres per second squared, the corresponding force unit is $kg\cdot m\cdot s^{-2}$. In a one-dimensional example, a $2kg$ object accelerating at $3m/s^2$ corresponds to a force measurement of $2\cdot 3=6$ in that product unit, so the force is $6kg\cdot m\cdot s^{-2}$ in the direction of the acceleration.

A force measured as $\overrightarrow{F}=(\mathrm{1,1,1})$ in newtons has three component values, each measured in $N$. Its magnitude is $\sqrt{3}N$, since $$\Vert \overrightarrow{F}\Vert =\sqrt{1^2+1^2+1^2}N=\sqrt{3}N.$$ If this force acts on a $1kg$ mass, then $\overrightarrow{F}=m\overrightarrow{a}$, so $$\Vert \overrightarrow{F}\Vert =\Vert m\overrightarrow{a}\Vert =m\Vert \overrightarrow{a}\Vert =1kg\cdot \Vert \overrightarrow{a}\Vert .$$ Since $\Vert \overrightarrow{F}\Vert =\sqrt{3}N=\sqrt{3}kg\cdot m\cdot s^{-2}$, we get $\Vert \overrightarrow{a}\Vert =\sqrt{3}m\cdot s^{-2}$. Because $m$ is positive, $\overrightarrow{a}$ points in the same direction as $\overrightarrow{F}$.

Lifting a small object can transfer about one joule of energy if the force and distance multiply to $1N\cdot m$.

• Pushing an object with a force of $1N$ through $1m$ in the direction of the push transfers $1N\cdot 1m=1J$ of energy.
• Pushing a box with a force of $10N$ through $3m$ transfers $10N\cdot 3m=30J$ of energy.
• Lifting a $2kg$ object upward by $0.5m$ near Earth's surface requires a force of about $19.6N$, so the energy transferred is about $19.6N\cdot 0.5m=9.8J$.
• Pulling a sled with a force of $40N$ through $25m$ in the direction of the pull transfers $40N\cdot 25m=1000J$ of energy.