First
law:
In an inertial frame of reference , an object
either remains at rest or continues to move
at a constant velocity , unless acted upon by
a force . [2][3]
Second
law:
In an inertial frame of reference, the vector
sum of the forces F on an object is equal to
the mass m of that object multiplied by the
acceleration a of the object: F = ma . (It is
assumed here that the mass m is constant –
see below .)
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.
A fourth law is often also described in the
bibliography, which states that forces add up like
vectors, that is, that forces obey the principle of
superposition. [6][7][8]
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, [9] 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 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. [10]
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 holds only 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. [11]
[12] 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 .[13]
Laws
Newton's first law
Main article: Inertia
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 when the
mass is a non-zero constant, as,
Consequently,
An object that is at rest will stay at rest unless a
force acts upon it.
An object that is in motion will not change its
velocity unless a force acts upon it.
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 continuously 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. [11][14]
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:
Newton's first and second 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. [16]
Newton's second law
The second law states that the rate of change of
momentum of a body is directly proportional to the
force applied, and this change in momentum takes
place in the direction of the applied force.
The second law can also be stated in terms of an
object's acceleration. Since Newton's second law is
valid only for constant-mass systems, [17][18][19] m
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. An
application of this notation is the derivation of G
Subscript C.
Consistent with the first law, the time derivative of
the momentum is non-zero 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 variable-mass systems (see below ).
Newton's second law is an approximation that is
increasingly worse at high speeds because of
relativistic effects.
Impulse
An impulse J occurs when a force F acts over an
interval of time Δt , and it is given by [20][21]
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. [22]
Impulse is a concept frequently used in the analysis
of collisions and impacts. [23]
Variable-mass systems
Main article: Variable-mass system
Variable-mass 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; [18] that is, the following
formula is wrong: [19]
The falsehood of this formula can be seen by noting
that it does not respect Galilean invariance : a
variable-mass object with F = 0 in one frame will be
seen to have F ≠ 0 in another frame. [17] 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,
constant-mass system consisting of the body and its
ejected/accreted mass; the result is[17]
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 d m /d t on
the left-hand 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 = m a .
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 magnitudes of both forces are equal,
but they have opposite directions, as
dictated by Newton's third law.
The third law states that all forces between two
objects exist in equal magnitude and opposite
direction: 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 in
magnitude and opposite in direction: F A = − F B .[24]
The third law means that all forces are interactions
between different bodies,[25][26] or different regions
within one body, and thus that there is no such thing
as a force that is not accompanied by an equal and
opposite force. In some situations, the magnitude
and direction of the forces are determined entirely by
one of the two bodies, say Body A; the force exerted
by Body A on Body B is called the "action", and the
force exerted by Body B on Body A is called the
"reaction". This law is sometimes referred to as the
action-reaction law, with F A called the "action" and
F B the "reaction". In other situations the magnitude
and directions of the forces are determined jointly by
both bodies and it isn't necessary to identify one
force as the "action" and the other as 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 First and Second laws,
in Latin, from the original 1687
Principia Mathematica
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
Johannes Kepler had 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. [30][31]
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,[32] this is understood, in modern
terms, as an equivalent of:
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.
Motte's 1729 translation of Newton's Latin continued
with Newton's commentary on the second law of
motion, reading:
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. [33]
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:
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 ;[35] 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 the 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 non-relativistically 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 = d p /d t , where F and p
are four-vectors .
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 non-classical 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 4-momentum in inertial motion
via curved space-time results in what we call
gravitational force in general relativity theory. The
application of the space derivative (which is a
momentum operator in quantum mechanics) to the
overlapping wave functions of a pair of fermions
(particles with half-integer spin ) results in shifts of
maxima of compound wavefunction away from each
other, which is observable as the "repulsion" of the
fermions.
Newton stated the third law within a world-view 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. (In particular, this refers to Bell's
theorem – that no local model can reproduce the
predictions of quantum theory.) Despite only being
an approximation, in modern engineering and 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 not every
physical quantity is conserved over time, thus
disproving the validity of inducing the opposite
metaphysical view from Newton's laws. Hence, a
"steady-state" worldview based solely on Newton's
laws and the conservation laws does not take
entropy into account.
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