Force and Mass

Glenn Elert

Force and Mass

Discussion

introduction

Discuss…

different forces on same object (result?)
different objects with same forces (result?)
different objects with same acceleration (how?)

Consequences…

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
a ∝	m

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
a =	m

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

history

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?

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

Mass…

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 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?

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

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)
10⁻⁰⁹	breaking covalent bonds
10⁻⁰⁶	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

cause
of change

=

resistance
to change

×

rate of
change of…

newton's
second law

force

mass

velocity

F =	m	dv
		dt

rotational
dynamics

torque

moment
of inertia

angular
velocity

τ =	I	dω
		dt

newtonian
fluids

shearing
stress

viscosity

shear

F_x	=	η	dx/dz
A			dt

thermal
conduction

temperature
gradient

r-factor

heat

∆T =	R	dq
		dt

ohm's
law

potential
difference

electrical
resistance

charge

V =	R	dq
		dt

faraday's
law

potential
difference

inductance

current

V =	L	dI
		dt

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.

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.