Newton's
laws of motion
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For other uses, see
Laws
of motion.
Newton's First and Second laws, in
Latin, from the original 1687 Principia
Mathematica.
Newton's laws of motion are three
physical laws
that together laid the foundation for classical
mechanics. They describe the relationship between a body and the
forces acting upon
it, and its motion
in response to said forces. They have been expressed in several
different ways over nearly three centuries,[1]
and can be summarised as follows.
First law:

When viewed in an inertial
reference frame, an object either remains at rest or continues
to move at a constant velocity,
unless acted upon by an external force.[2][3]

Second law:

The vector
sum of the forces
F on an object is equal to the mass
m of that object multiplied by the acceleration
vector a of the object: F = ma.

Third law:

When one body exerts a force on a second body, the second body
simultaneously exerts a force equal in magnitude and opposite in
direction on the first body.

The three laws of motion were first compiled by Isaac
Newton in his Philosophiæ
Naturalis Principia Mathematica (Mathematical Principles
of Natural Philosophy), first published in 1687.[4]
Newton used them to explain and investigate the motion of many
physical objects and systems.[5]
For example, in the third volume of the text, Newton showed that
these laws of motion, combined with his law
of universal gravitation, explained Kepler's
laws of planetary motion.
Overview
Isaac Newton (1643–1727), the physicist who formulated the
laws
Newton's laws are applied to objects
which are idealised as single point masses,[6]
in the sense that the size and shape of the object's body are
neglected to focus on its motion more easily. This can be done when
the object is small compared to the distances involved in its
analysis, or the deformation
and rotation of the body are of no importance. In this way, even a
planet can be idealised as a particle for analysis of its orbital
motion around a star.
In their original form, Newton's laws of
motion are not adequate to characterise the motion of rigid
bodies and deformable
bodies. Leonhard
Euler in 1750 introduced a generalisation of Newton's laws of
motion for rigid bodies called the Euler's
laws of motion, later applied as well for deformable bodies
assumed as a continuum.
If a body is represented as an assemblage of discrete particles, each
governed by Newton's laws of motion, then Euler's laws can be derived
from Newton's laws. Euler's laws can, however, be taken as axioms
describing the laws of motion for extended bodies, independently of
any particle structure.[7]
Newton's
laws hold only with respect to a certain set of frames
of reference called Newtonian
or inertial reference frames. Some authors interpret the first
law as defining what an inertial reference frame is; from this point
of view, the second law only holds when the observation is made from
an inertial reference frame, and therefore the first law cannot be
proved as a special case of the second. Other authors do treat the
first law as a corollary of the second.[8][9]
The explicit concept of an inertial frame of reference was not
developed until long after Newton's death.
In the given interpretation mass,
acceleration,
momentum, and
(most importantly) force
are assumed to be externally defined quantities. This is the most
common, but not the only interpretation of the way one can consider
the laws to be a definition of these quantities.
Newtonian mechanics has been superseded
by special
relativity, but it is still useful as an approximation when the
speeds involved are much slower than the speed
of light.[10]
Newton's first
law
Explanation of Newton's first law and
reference frames. (MIT
Course 8.01)[11]
The first law states that if the net
force (the vector
sum of all forces acting on an object) is zero, then the velocity
of the object is constant. Velocity is a vector
quantity which expresses both the object's speed
and the direction of its motion; therefore, the statement that the
object's velocity is constant is a statement that both its speed and
the direction of its motion are constant.
The first law can be stated mathematically as

Consequently,
This is known as uniform motion. An object continues
to do whatever it happens to be doing unless a force is exerted upon
it. If it is at rest, it continues in a state of rest (demonstrated
when a tablecloth is skilfully whipped from under dishes on a
tabletop and the dishes remain in their initial state of rest). If an
object is moving, it continues to move without turning or changing
its speed. This is evident in space probes that continually move in
outer space. Changes in motion must be imposed against the tendency
of an object to retain its state of motion. In the absence of net
forces, a moving object tends to move along a straight line path
indefinitely.
Newton placed the first law of motion to establish frames
of reference for which the other laws are applicable. The first
law of motion postulates the existence of at least one frame
of reference called a Newtonian or inertial
reference frame, relative to which the motion of a particle not
subject to forces is a straight line at a constant speed.[8][12]
Newton's first law is often referred to as the law
of inertia. Thus, a condition necessary for the uniform
motion of a particle relative to an inertial reference frame is that
the total net force
acting on it is zero. In this sense, the first law can be restated
as:
In every material universe,
the motion of a particle in a preferential reference frame Φ is
determined by the action of forces whose total vanished for all times
when and only when the velocity of the particle is constant in Φ.
That is, a particle initially at rest or in uniform motion in the
preferential frame Φ continues in that state unless compelled by
forces to change it.[13]
Newton's laws are valid only in an
inertial
reference frame. Any reference frame that is in uniform motion
with respect to an inertial frame is also an inertial frame, i.e.
Galilean
invariance or the principle
of Newtonian relativity.[14]
Newton's
second law
Explanation of Newton's second law,
using gravity as an example. (MIT
OCW)[15]
The second law states that the net
force on an object is equal to the rate of change (that is, the
derivative)
of its linear
momentum p in an inertial
reference frame:

The second law can also be stated in terms of an object's
acceleration. Since Newton's second law is only valid for
constantmass systems,[16][17][18]
mass can be taken outside the differentiation
operator by the constant
factor rule in differentiation. Thus,

where F is the net force applied, m is the mass of the
body, and a is the body's acceleration. Thus, the net force
applied to a body produces a proportional acceleration. In other
words, if a body is accelerating, then there is a force on it.
Consistent with the first
law, the time derivative of the momentum is nonzero when the
momentum changes direction, even if there is no change in its
magnitude; such is the case with uniform
circular motion. The relationship also implies the conservation
of momentum: when the net force on the body is zero, the momentum
of the body is constant. Any net force is equal to the rate of change
of the momentum.
Any mass that is gained or lost by the system will cause a change
in momentum that is not the result of an external force. A different
equation is necessary for variablemass systems (see below).
Newton's second law requires modification if the effects of
special
relativity are to be taken into account, because at high speeds
the approximation that momentum is the product of rest mass and
velocity is not accurate.
Impulse
An
impulse
J occurs when a force F acts over an interval of time
Δt, and it is given by[19][20]

Since force is the time derivative of momentum, it follows that

This relation between impulse and
momentum is closer to Newton's wording of the second law.[21]
Impulse is a concept
frequently used in the analysis of collisions and impacts.[22]
Variablemass
systems
Main article: Variablemass
system
Variablemass systems, like a rocket burning fuel and ejecting spent
gases, are not closed
and cannot be directly treated by making mass a function of time in
the second law;[17]
that is, the following formula is wrong:[18]

The falsehood of this formula can be seen by noting that it does not
respect Galilean
invariance: a variablemass object with F = 0 in one
frame will be seen to have F ≠ 0 in another frame.[16]
The correct equation of motion for a body whose mass m varies
with time by either ejecting or accreting mass is obtained by
applying the second law to the entire, constantmass system
consisting of the body and its ejected/accreted mass; the result
is[16]

where u is the velocity of the escaping or incoming mass
relative to the body. From this equation one can derive the equation
of motion for a varying mass system, for example, the Tsiolkovsky
rocket equation. Under some conventions, the quantity u dm/dt
on the lefthand side, which represents the advection
of momentum, is
defined as a force (the force exerted on the body by the changing
mass, such as rocket exhaust) and is included in the quantity F.
Then, by substituting the definition of acceleration, the equation
becomes F = ma.
Newton's third
law
An illustration of Newton's third law
in which two skaters push against each other. The first skater on the
left exerts a normal force N_{12} on the second skater
directed towards the right, and the second skater exerts a normal
force N_{21} on the first skater directed towards the
left.
The magnitude of both forces are equal, but they have
opposite directions, as dictated by Newton's third law.
A description of Newton's third law and
contact forces[23]
The third law states that all forces exist in pairs: if one object A
exerts a force F_{A} on a second object B,
then B simultaneously exerts a force F_{B}
on A, and the two forces are equal and opposite: F_{A}
= −F_{B}.[24]
The third law means that all forces are interactions
between different bodies,[25][26]
and thus that there is no such thing as a unidirectional force or a
force that acts on only one body. This law is sometimes referred to
as the actionreaction
law, with F_{A} called the "action"
and F_{B} the "reaction". The action
and the reaction are simultaneous, and it does not matter which is
called the action and which is called reaction; both
forces are part of a single interaction, and neither force exists
without the other.[24]
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).
From a conceptual standpoint, Newton's
third law is seen when a person walks: they push against the floor,
and the floor pushes against the person. Similarly, the tires of a
car push against the road while the road pushes back on the tires—the
tires and road simultaneously push against each other. In swimming, a
person interacts with the water, pushing the water backward, while
the water simultaneously pushes the person forward—both the
person and the water push against each other. The reaction forces
account for the motion in these examples. These forces depend on
friction; a person or car on ice, for example, may be unable to exert
the action force to produce the needed reaction force.[27]
History
Newton's 1st Law
From the original Latin
of Newton's Principia:
“

Lex I: Corpus omne
perseverare in statu suo quiescendi vel movendi uniformiter in
directum, nisi quatenus a viribus impressis cogitur statum illum
mutare.

”

Translated to English, this reads:
“

Law
I: Every body persists in its state of being at rest or of moving
uniformly straight forward, except insofar as it is compelled to
change its state by force impressed.[28]

”

The ancient Greek philosopher Aristotle
had the view that all objects have a natural place in the universe:
that heavy objects (such as rocks) wanted to be at rest on the Earth
and that light objects like smoke wanted to be at rest in the sky and
the stars wanted to remain in the heavens. He thought that a body was
in its natural state when it was at rest, and for the body to move in
a straight line at a constant speed an external agent was needed
continually to propel it, otherwise it would stop moving. Galileo
Galilei, however, realised that a force is necessary to change
the velocity of a body, i.e., acceleration, but no force is needed to
maintain its velocity. In other words, Galileo stated that, in the
absence of a force, a moving object will continue moving. The
tendency of objects to resist changes in motion was what Galileo
called inertia. This insight was refined by Newton, who made
it into his first law, also known as the "law of inertia"—no
force means no acceleration, and hence the body will maintain its
velocity. As Newton's first law is a restatement of the law of
inertia which Galileo had already described, Newton appropriately
gave credit to Galileo.
The law of inertia apparently occurred
to several different natural philosophers and scientists
independently, including Thomas
Hobbes in his Leviathan.[29]
The 17th century philosopher and mathematician René
Descartes also formulated the law, although he did not perform
any experiments to confirm it.[citation
needed]
Newton's 2nd Law
Newton's original Latin reads:
“

Lex II: Mutationem motus
proportionalem esse vi motrici impressae, et fieri secundum lineam
rectam qua vis illa imprimitur.

”

This was translated quite closely in Motte's 1729 translation as:
“

Law II: The alteration of
motion is ever proportional to the motive force impress'd; and is
made in the direction of the right line in which that force is
impress'd.

”

According to modern ideas of how Newton
was using his terminology,[30]
this is understood, in modern terms, as an equivalent of:
The change of momentum of a body is proportional to
the impulse impressed on the body, and happens along the straight
line on which that impulse is impressed.
This may be expressed by the formula F = p', where p' is the time
derivative of the momentum p. This equation can be seen clearly in
the Wren Library
of Trinity
College, Cambridge, in a glass case in which Newton's manuscript
is open to the relevant page. Interestingly enough, the equation F =
p' remains true in the context of Special
relativity.
Motte's 1729 translation of Newton's Latin continued with Newton's
commentary on the second law of motion, reading:
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.
The sense or senses in which Newton used
his terminology, and how he understood the second law and intended it
to be understood, have been extensively discussed by historians of
science, along with the relations between Newton's formulation and
modern formulations.[31]
Newton's 3rd Law
“

Lex III: Actioni
contrariam semper et æqualem esse reactionem: sive corporum
duorum actiones in se mutuo semper esse æquales et in partes
contrarias dirigi.

”

Translated to English, this reads:
“

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.

”

Newton's Scholium
(explanatory comment) to this law:
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 impinges upon another, and by its force changes
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, as 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 scholium.[32]
In the above, as usual, motion is Newton's name for
momentum, hence his careful distinction between motion and velocity.
Newton used the third law to derive the
law of conservation
of momentum;[33]
from a deeper perspective, however, conservation of momentum is the
more fundamental idea (derived via Noether's
theorem from Galilean
invariance), and holds in cases where Newton's third law appears
to fail, for instance when force
fields as well as particles carry momentum, and in quantum
mechanics.
Importance
and range of validity
Newton's laws were verified by experiment and observation for over
200 years, and they are excellent approximations at the scales and
speeds of everyday life. Newton's 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.
These three laws hold to a good approximation for macroscopic
objects under everyday conditions. However, Newton's laws (combined
with universal gravitation and classical
electrodynamics) are inappropriate for use in certain
circumstances, most notably at very small scales, very high speeds
(in special
relativity, the Lorentz
factor must be included in the expression for momentum along with
rest mass and
velocity) or very strong gravitational fields. Therefore, the laws
cannot be used to explain phenomena such as conduction of electricity
in a semiconductor,
optical properties of substances, errors in nonrelativistically
corrected GPS systems
and superconductivity.
Explanation of these phenomena requires more sophisticated physical
theories, including general
relativity and quantum
field theory.
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,
where F and p are fourvectors.
Relationship to the conservation laws
In modern physics, the laws
of conservation of momentum,
energy, and angular
momentum are of more general validity than Newton's laws, since
they apply to both light and matter, and to both classical and
nonclassical physics.
This can be stated simply, "Momentum, energy and angular
momentum cannot be created or destroyed."
Because force is the time derivative of momentum, the concept of
force is redundant and subordinate to the conservation of momentum,
and is not used in fundamental theories (e.g., quantum
mechanics, quantum
electrodynamics, general
relativity, etc.). The standard
model explains in detail how the three fundamental forces known
as gauge forces
originate out of exchange by virtual
particles. Other forces such as gravity
and fermionic
degeneracy pressure also arise from the momentum conservation.
Indeed, the conservation of 4momentum
in inertial motion via curved
spacetime results in what we call gravitational
force in general
relativity theory. Application of space derivative (which is a
momentum
operator in quantum mechanics) to overlapping wave
functions of pair of fermions
(particles with halfinteger spin)
results in shifts of maxima of compound wavefunction away from each
other, which is observable as "repulsion" of fermions.
Newton stated the third law within a worldview that assumed
instantaneous action at a distance between material particles.
However, he was prepared for philosophical criticism of this action
at a distance, and it was in this context that he stated the
famous phrase "I
feign no hypotheses". In modern physics, action at a
distance has been completely eliminated, except for subtle effects
involving quantum
entanglement.[citation
needed] However in modern engineering in all practical
applications involving the motion of vehicles and satellites, the
concept of action at a distance is used extensively.
The discovery of the Second
Law of Thermodynamics by Carnot in the 19th century showed that
every physical quantity is not conserved over time, thus disproving
the validity of inducing the opposite metaphysical view from Newton's
laws. Hence, a "steadystate" worldview based solely on
Newton's laws and the conservation laws does not take entropy
into account.
See also