Newton’s second law of motion


Newton’s second law of motion

s 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 is a measure of resistance to acceleration.
    (More generally, mass is a measure of resistance to change.)
  • Mass is a scalar quantity associated with matter.
  • When a system is composed of several objects it is the total mass that matters.
  • The SI unit of mass is the kilogram [kg].
mass (kg)object
~1053observable universe
1.5 ~ 2× 1042milky way
~ 8× 1041largest known black hole
> 6× 1030stellar black hole
2.8 ~ 6× 1030neutron star
1.99× 1030sun
1.90× 1027jupiter
5.97× 1024earth
6.42× 1023mars
7.35× 1022moon
1.31× 1022pluto
1.35× 1021earth’s hydrosphere
5.14× 1018earth’s atmosphere
1.84× 1015earth’s biosphere
~ 150,000blue whale
~ 5,000african elephant
~ 1,500passenger car
635world’s heaviest man
100the author
7.72world’s smallest woman
3 ~ 7bowling ball
0.16billiard ball
~ 3× 10−6snowflake
2.18× 10−8planck mass
~ 10−12bacterium
~ 10−15virus
3.9529260× 10−25uranium 238 atom
3.054× 10−25top quark
1.6749274× 10−27neutron
1.6735327× 10−27hydrogen 1 atom
1.6726218× 10−27proton
9.1093836× 10−31electron
< 3.6× 10−36neutrino (upper limit)
< 1.8× 10−53photon (upper limit)
Mass of selected objects

So what is a force?


  • A force is an interaction that causes acceleration. More generally, a force is an interaction that causes a change.
  • Force is a vector quantity associated with an interaction.
  • When several forces act on a system it is the net, external force that matters. Since force is a vector quantity, use geometry instead of arithmetic when combining forces.
  • For a force to accelerate an object it must come from outside it. External force. Can’t pull yourself up by your own bootstraps. Anyone who says you can is engaging in hyperbole.
  • The SI unit of force is the newton [N = kg m/s2]. Rule of thumb: one newton is approximately equal to a quarter pound
force (N)event, process, phenomena
10−14langevin forces of brownian motion
10−11molecular motors consuming ATP
10−10breaking noncovalent bonds (denaturing proteins)
1009breaking covalent bonds
1006adhesive force of bacteria
0.030threshold of touch sensation; index, middle, pinky
0.040threshold of touch sensation; ring finger
256average dog bite
980weight of the author
2,200peak foot force, 75 kg human, running
140,000peak foot force, 10,000 kg asian elephant, running
Selected Forces

The concept of inertia comes in many forms.

of change
to change
×rate of
change of…
second law
F =mdv
of inertia
τ =Idω
Fx =ηdx/dz
ΔT =Rdq
V =Rdq
V =LdI

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