The Physics
Opus in profectus

Force and Mass

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Newton's second law of motion states that acceleration is directly proportional to net force when mass is constant…

a ∝  F  

and that acceleration is inversely proportional to mass when net force is constant…

a ∝  1  

and that net force is directly proportional to mass when acceleration is constant…

F ∝  m  

This is more compactly written as an equation that combines these relationships…

a =  F  

For a variety of reasons, Newton's second law of motion is often written with net force as the subject of the equation like this…

F = ma


Lex. II.   Law II.
Mutationem motus proportionalem eſſe vi motrici impreſſæ, & fieri ſecundum lineaum rectam qua vis illa imprimitur.   The alteration of motion is ever proportional to the motive force impressed; and is made in the direction of the right line in which that force is impressed.
Si vis aliqua motum quemvis generet; dupla duplum, tripla triplum generabit, ſive ſimul & ſemel, ſive gradatim & ſucceſſive impreſſa fuerit. Et hic motus quoniam in eandem ſemper plagam cum vi generatrice determinatur, ſi corpus antea movebatur, motui ejus vel conſpiranti additur, vel contrario ſubducitur, vel obliquo oblique adjicitur, & cum eo ſecundum utriuſque determinationem componitur.   If any force generates a motion, a double force will generate double the motion, a triple force triple the motion, whether that force be impressed altogether and at once, or gradually and successively. And this motion being always directed the same way with the generating force, if the body moved before, is added to or subtracted from the former motion, according as they directly conspire with or are directly contrary to each other; or obliquely joined, when they are oblique, so as to produce a new motion compounded from the determination of both.

Newton also defined what he called "the quantity of matter" and "the quantity of motion". We now call them "mass" and "momentum", respectively.

Definitio. I.   Definition I.
Quantitas materiæ est mensura ejusdem orta ex illius densitate et magnitudine conjunctim.   The quantity of matter is the measure of the same, arising from its density and bulk conjunctly.
Aer densitate duplicata, in spatio etiam duplicato, sit quadruplus; in triplicato sextuplus. Idem intelige de nive & pulveribus per compressionem vel liquesactionem condensatis. Et par eft ratio corporum omnium, quæ per caufas quascunque diversimode condensantur. Medii interea, si quod fuerit, interstitia partium libere pervadentis, hic nullam rationem habeo. Hanc autem quantitatem sub nomine corporis vel masse in sequentibus passim intelligo. Innotescit ea per corporis cujusque pondus: Nam ponderi proportionalem esse reperi per experimenta pendulorum accuratissime instituta, uti posthac docebitur.   Thus air of double density, in a double space, is quadruple in quantity; in a triple space, sextuple in quantity. The same thing is to be understood of snow, and fine dust or powders, that are condensed by compression or liquefaction; and of all bodies that are by any caused whatever differently condensed. I have no regard in this place to a medium, if any such there is, that freely pervades the interstices between the parts of bodies. It is this quantity that I mean hereafter everywhere under the name of body or mass. And the same is known by the weight of each body; for it is proportional to the weight, as I have found by experiments on pendulums, very accurately made, which shall be shewn hereafter.
Definitio. II.   Definition II.
Quantitas motus est mensura ejusdem orta ex velocitate et quantite materiæ conjunctim.   The quantity of motion is the measure of the same, arising from the velocity and the quantity of matter conjunctly.
Motus totius est summa motuum in partibus singulis; ideoque in corpore duplo majore, æ quali cum velocitate, duplus est, & dupla cum velocitate quadruplus.   The motion of the whole is the sum of the motions of all the parts; and therefore in a body double in quantity, with equal velocity, the motion is double; with twice the velocity it is quadruple.

So what is mass?

  1. pick an object to be the standard unit mass
  2. push mass with reproducible force (or use the principle of action-reaction)
  3. measure its acceleration
  4. push an unknown mass with the same force
  5. measure new acceleration
  6. mass is inversely proportional to acceleration


Mass of selected objects
mass (kg) object
~1053 observable universe
~1042 Milky Way
8 × 1040 largest black hole (S5 0014+81)
6 × 1032 most massive star (R136a1)
2 × 1031 smallest black hole (XTE J1650−500)
2.8~6 × 1030 neutron star
1.99 × 1030 Sun
1.90 × 1027 Jupiter
5.97 × 1024 Earth
6.42 × 1023 Mars
7.35 × 1022 Moon
1.31 × 1022 Pluto
1.35 × 1021 Earth's hydrosphere
5.14 × 1018 Earth's atmosphere
1.84 × 1015 Earth's biosphere
~150,000 blue whale
~5,000 African elephant
~1,750 passenger car
635 world's heaviest man
100 the author
7.72 world's smallest woman
3~7 bowling ball
0.16 billiard ball
~3 × 10−6 snowflake
2.18 × 10−8 Planck mass
~10−12 bacterium
~10−15 virus
3.9529264 × 10−25 uranium 238 atom
3.0784789 × 10−25 top quark
1.6749275 × 10−27 neutron
1.6735328 × 10−27 hydrogen 1 atom
1.6726219 × 10−27 proton
9.1093837 × 10−31 electron
<2.0 × 10−36 neutrino (upper limit)
<1.8 × 10−54 photon (upper limit)

So what is a force?


Selected forces (smallest to largest)
force (N) device, event, phenomenon, process
10−14 langevin forces of brownian motion
10−11 molecular motors consuming ATP
10−10 breaking noncovalent bonds (denaturing proteins)
1009 breaking covalent bonds
1006 adhesive force of bacteria
0.032 threshold of touch sensation; index, middle, pinky
0.044 threshold of touch sensation; ring finger
4.436–4.460 weight of 1 lb on Earth
4.44822 weight of 1 lb in standard gravity
9.780–9.832 weight of 1 kg on Earth
9.80665 weight of 1 kg in standard gravity
32 French press coffee plunger
256 average dog bite
980 weight of the author
2,200 peak foot force, 75 kg human, running
140,000 peak foot force, 10,000 kg asian elephant, running

The concept of inertia comes in many forms.

Analogous applications of Newton's second law of motion
of change
= resistance
to change
× rate of
change of…
second law
force mass velocity
F =  m  dv
torque moment
of inertia
τ =  I  dω
viscosity shear
Fx  =  η  dx/dz
A dt
r-factor heat
T =  R  dq
V =  R  dq
inductance current
V =  L  dI