Borrowed on Sep.11, 2006 for PHY205 by Jason Harlow
Newton's laws of motion
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First and Second laws, in Latin, from the original 1687 edition of the Principia
Newton's Laws of Motion are three physical laws which provide relationships
between the forces acting on a body and the motion of the body, first
formulated by Sir Isaac Newton. Newton's
laws were first published in his work Philosophiae Naturalis
Principia Mathematica (1687). The laws form the basis for classical mechanics.
them to explain many results concerning the motion of physical objects. In the
third volume of the text, he showed that the laws of motion, combined with his law
of universal gravitation, explained Kepler's laws of planetary motion.
The Three Laws of Motion
Laws of Motion describe only the motion of a body as a whole and are valid only
for motions relative to a reference frame. The following are brief modern
formulations of Newton's
three laws of motion:
Objects in motion tend to stay in motion, and objects at rest tend to stay
at rest unless an outside force acts upon them.
The rate of change of the momentum of a body is directly proportional to the
net force acting on it, and the direction of the change in momentum takes place
in the direction of the net force.
To every action (force applied) there is an equal but opposite reaction
(equal force applied in the opposite direction).
It is important to note that these three laws together with his law of
gravitation provide a satisfactory basis for the explanation of motion of
everyday macroscopic objects under everyday conditions. However, when applied
to extremely high speeds or extremely small objects, Newton's laws break down; this was remedied
by Albert Einstein's Special Theory of Relativity for high speeds and by quantum
mechanics for small objects.
Newton's first law: law of inertia
Lex I: Corpus omne perseverare in statu suo quiescendi vel movendi
uniformiter in directum, nisi quatenus a viribus impressis cogitur statum illum
- "An object at rest
will remain at rest unless acted upon by an external and unbalanced force
. An object in motion will remain in motion unless acted upon by an
external and unbalanced force"
This law is also called the law of inertia or Galileo's principle.
The net force on an object is the vector sum of all the forces acting on the
first law says that if this sum is zero, the state of motion of the object does
not change. Essentially, it makes the following two points:
- An object that is not moving
will not move until a force acts upon it.
- An object that is in motion
will not change velocity (including stopping) until a force acts upon it.
The first point seems relatively obvious to most people, but the second may
take some thinking through, because everyone knows that things don't keep
moving forever. If one slides a hockey puck along a table, it doesn't move
forever, it slows and eventually comes to a stop. But according to Newton's laws, this is
because a force is acting on the hockey puck and, sure enough, there is
frictional force between the table and the puck, and that frictional force is
in the direction opposite the movement. It's this force which causes the object
to slow to a stop. In the absence (or virtual absence) of such a force, as on
an air hockey table or ice rink, the puck's motion isn't hindered.
Although the 'Law of Inertia' is commonly attributed to Galileo, Aristotle
wrote the first known description of it:
[N]o one could say why a thing once
set in motion should stop anywhere; for why should it stop here rather
than here? So that a thing will either be at rest or must be moved ad
infinitum, unless something more powerful get in its way.
However, a key difference between Galileo's idea from Aristotle's is that
Galileo realised that force acting on a body determines acceleration,
not velocity. This insight leads to Newton's
First Law - no force means no acceleration, and hence the body will continue to
maintain its velocity.
The 'Law of Inertia' apparently occurred to many different natural
philosophers independently, for example in China the inertia of motion appears
in the 3rd century BC Mo Tzu and René Descartes also formulated the law,
although he did not perform any experiments to confirm it.
There are no perfect demonstrations of the law, as friction usually causes a
force to act on a moving body, and even in outer space relativistic effects or
gravitational forces act, but the law serves to emphasize the elementary causes
of changes in an object's state of motion: forces.
Newton's second law - historical development
In an exact original 1792 translation (from Latin) Newton's Second Law of Motion reads:
"LAW II: 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. — If a 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."
is basically saying that the rate of change in the momentum of an object is
directly proportional to the amount of force exerted upon the object. He also
states that the change in direction of momentum is determined by the angle from
which the force is applied. Interestingly, Newton is restating in his further
explanation another prior idea of Galileo, what we call today the Galilean
transformation or the addition of velocities.
An interesting fact when studying Newton's
Laws of Motion from the Principia is that Newton
himself does not explicitly write formulae for his laws which was common in
scientific writings of that time period. In fact, it is today commonly added
when stating Newton's second law that Newton has said,
"and inversely proportional to the mass of the object." This however
is not found in Newton's
second law as directly translated above. In fact, the idea of mass is not
introduced until the third law.
In mathematical terms, the differential equation can be written as:
where F is force, m is mass, v is velocity, t is time and k is the constant
of proportionality. The product of the mass and velocity is the momentum of the
If mass of an object in question is known to be constant and using the
definition of acceleration, this differential equation can be rewritten as:
where a is the acceleration.
Using only SI Units for the definition of Newton, the constant of proportionality is
unity (1). Hence:
However, it has been a common convention to describe Newton's second law in the mathematical
formula F = ma where F is Force, a is
acceleration and m is mass. This is actually a combination of laws two and
three of Newton
expressed in a very useful form. This formula in this form did not even begin
to be used until the 18th century, after Newton's
death, but it is implicit in his laws.
Law of Motion states: "LAW III: To every action there is always opposed an
equal reaction: or the mutual actions of two bodies upon each other are always
equal, and directed to contrary parts. -- Whatever draws or presses another is
as much drawn or pressed by that other. If you press a stone with your finger,
the finger is also pressed by the stone. If a horse draws a stone tied to a
rope, the horse (if I may so say) will be equally drawn back towards the stone:
for the distended rope, by the same endeavour to relax or unbend itself, will
draw the horse as much towards the stone, as it does the stone towards the
horse, and will obstruct the progress of the one as much as it advances that of
the other. If a body impinge upon another, and by its force change the motion
of the other, that body also (because of the equality of the mutual pressure)
will undergo an equal change, in its own motion, toward the contrary part. The
changes made by these actions are equal, not in the velocities but in the
motions of the bodies; that is to say, if the bodies are not hindered by any
other impediments. For, because the motions are equally changed, the changes of
the velocities made toward contrary parts are reciprocally proportional to the
bodies. This law takes place also in attractions, as will be proved in the next
The explanation of mass is expressed here for the first time in the words
"reciprocally proportional to the bodies" which have now been
traditionally added to Law 2 as "inversely proportional to the mass of the
object." This is because Newton
in his definition 1 had already stated that when he said "body" he
meant "mass". Thus we arrive at F=ma. When the formula F=ma is taken
into account, Law II can be also interpreted as a quantitative restatement of
Law I, where mass also acts as a measurement of inertia.
Newton's third law: law of reciprocal actions
third law. The skaters' forces on each other are equal in magnitude, and in
Lex III: Actioni contrariam semper et æqualem esse reactionem: sive
corporum duorum actiones in se mutuo semper esse æquales et in partes
- All forces occur in pairs,
and these two forces are equal in magnitude and opposite in direction.
The third law follows mathematically from the law of conservation of
As shown in the diagram opposite, the skaters' forces on each other are
equal in magnitude, and opposite in direction. Although the forces are equal,
the accelerations are not: the less massive skater will have a greater
acceleration due to Newton's
second law. If a basketball hits the ground, the basketball's force on the
Earth is the same as Earth's force on the basketball. However, due to the ball's
much smaller mass, Newton's
second law predicts that its acceleration will be much greater than that of the
Earth. Not only do planets accelerate toward stars, but stars also accelerate
The two forces in Newton's third law are of
the same type, e.g., if the road exerts a forward frictional force on an
accelerating car's tires, then it is also a frictional force that Newton's third law
predicts for the tires pushing backward on the road.
Importance and range of
laws were verified by experiment and observation for over 200 years,
and they are excellent approximations at the scales and speeds of everyday
laws of motion, together with his law of universal gravitation and the
mathematical techniques of calculus, provided for the first time a unified
quantitative explanation for a wide range of physical phenomena.
In quantum mechanics concepts such as force, momentum, and position are
defined by linear operators that operate on the quantum state. At speeds that
are much lower than the speed of light, Newton's
laws are just as exact for these operators as they are for classical objects.
At speeds comparable to the speed of light, the second law holds in the
original form F = dp / dt,
which says that the force is the derivative of the momentum of the object with
respect to time, but some of the newer versions of the second law (such as the
constant mass approximation above) do not hold at relativistic velocities.
- Marion, Jerry and Thornton,
Stephen. Classical Dynamics of Particles and Systems. Harcourt College Publishers, 1995. ISBN
- Fowles, G. R. and Cassiday,
G. L. Analytical Mechanics (6ed). Saunders College
Publishing, 1999. ISBN