In physics, a force is any interaction that,
when unopposed, will change the motion of
an object. A force can cause an object with
mass to change its velocity (which includes
to begin moving from a state of rest), i.e.,
to accelerate. Force can also be described
intuitively as a push or a pull. A force has
both magnitude and direction, making it a
vector quantity. It is measured in the SI
unit of newtons and represented by the symbol
F.
The original form of Newton's second law states
that the net force acting upon an object is
equal to the rate at which its momentum changes
with time. If the mass of the object is constant,
this law implies that the acceleration of
an object is directly proportional to the
net force acting on the object, is in the
direction of the net force, and is inversely
proportional to the mass of the object.
Concepts related to force include: thrust,
which increases the velocity of an object;
drag, which decreases the velocity of an object;
and torque, which produces changes in rotational
speed of an object. In an extended body, each
part usually applies forces on the adjacent
parts; the distribution of such forces through
the body is the internal mechanical stress.
Such internal mechanical stresses cause no
acceleration of that body as the forces balance
one another. Pressure, the distribution of
many small forces applied over an area of
a body, is a simple type of stress that if
unbalanced can cause the body to accelerate.
Stress usually causes deformation of solid
materials, or flow in fluids.
== Development of the concept ==
Philosophers in antiquity used the concept
of force in the study of stationary and moving
objects and simple machines, but thinkers
such as Aristotle and Archimedes retained
fundamental errors in understanding force.
In part this was due to an incomplete understanding
of the sometimes non-obvious force of friction,
and a consequently inadequate view of the
nature of natural motion. A fundamental error
was the belief that a force is required to
maintain motion, even at a constant velocity.
Most of the previous misunderstandings about
motion and force were eventually corrected
by Galileo Galilei and Sir Isaac Newton. With
his mathematical insight, Sir Isaac Newton
formulated laws of motion that were not improved
for nearly three hundred years. By the early
20th century, Einstein developed a theory
of relativity that correctly predicted the
action of forces on objects with increasing
momenta near the speed of light, and also
provided insight into the forces produced
by gravitation and inertia.
With modern insights into quantum mechanics
and technology that can accelerate particles
close to the speed of light, particle physics
has devised a Standard Model to describe forces
between particles smaller than atoms. The
Standard Model predicts that exchanged particles
called gauge bosons are the fundamental means
by which forces are emitted and absorbed.
Only four main interactions are known: in
order of decreasing strength, they are: strong,
electromagnetic, weak, and gravitational.
High-energy particle physics observations
made during the 1970s and 1980s confirmed
that the weak and electromagnetic forces are
expressions of a more fundamental electroweak
interaction.
== Pre-Newtonian concepts ==
Since antiquity the concept of force has been
recognized as integral to the functioning
of each of the simple machines. The mechanical
advantage given by a simple machine allowed
for less force to be used in exchange for
that force acting over a greater distance
for the same amount of work. Analysis of the
characteristics of forces ultimately culminated
in the work of Archimedes who was especially
famous for formulating a treatment of buoyant
forces inherent in fluids.Aristotle provided
a philosophical discussion of the concept
of a force as an integral part of Aristotelian
cosmology. In Aristotle's view, the terrestrial
sphere contained four elements that come to
rest at different "natural places" therein.
Aristotle believed that motionless objects
on Earth, those composed mostly of the elements
earth and water, to be in their natural place
on the ground and that they will stay that
way if left alone. He distinguished between
the innate tendency of objects to find their
"natural place" (e.g., for heavy bodies to
fall), which led to "natural motion", and
unnatural or forced motion, which required
continued application of a force. This theory,
based on the everyday experience of how objects
move, such as the constant application of
a force needed to keep a cart moving, had
conceptual trouble accounting for the behavior
of projectiles, such as the flight of arrows.
The place where the archer moves the projectile
was at the start of the flight, and while
the projectile sailed through the air, no
discernible efficient cause acts on it. Aristotle
was aware of this problem and proposed that
the air displaced through the projectile's
path carries the projectile to its target.
This explanation demands a continuum like
air for change of place in general.Aristotelian
physics began facing criticism in medieval
science, first by John Philoponus in the 6th
century.
The shortcomings of Aristotelian physics would
not be fully corrected until the 17th century
work of Galileo Galilei, who was influenced
by the late medieval idea that objects in
forced motion carried an innate force of impetus.
Galileo constructed an experiment in which
stones and cannonballs were both rolled down
an incline to disprove the Aristotelian theory
of motion. He showed that the bodies were
accelerated by gravity to an extent that was
independent of their mass and argued that
objects retain their velocity unless acted
on by a force, for example friction.
== Newtonian mechanics ==
Sir Isaac Newton described the motion of all
objects using the concepts of inertia and
force, and in doing so he found they obey
certain conservation laws. In 1687, Newton
published his thesis Philosophiæ Naturalis
Principia Mathematica. In this work Newton
set out three laws of motion that to this
day are the way forces are described in physics.
=== First law ===
Newton's First Law of Motion states that objects
continue to move in a state of constant velocity
unless acted upon by an external net force
(resultant force). This law is an extension
of Galileo's insight that constant velocity
was associated with a lack of net force (see
a more detailed description of this below).
Newton proposed that every object with mass
has an innate inertia that functions as the
fundamental equilibrium "natural state" in
place of the Aristotelian idea of the "natural
state of rest". That is, Newton's empirical
First Law contradicts the intuitive Aristotelian
belief that a net force is required to keep
an object moving with constant velocity. By
making rest physically indistinguishable from
non-zero constant velocity, Newton's First
Law directly connects inertia with the concept
of relative velocities. Specifically, in systems
where objects are moving with different velocities,
it is impossible to determine which object
is "in motion" and which object is "at rest".
The laws of physics are the same in every
inertial frame of reference, that is, in all
frames related by a Galilean transformation.
For instance, while traveling in a moving
vehicle at a constant velocity, the laws of
physics do not change as a result of its motion.
If a person riding within the vehicle throws
a ball straight up, that person will observe
it rise vertically and fall vertically and
not have to apply a force in the direction
the vehicle is moving. Another person, observing
the moving vehicle pass by, would observe
the ball follow a curving parabolic path in
the same direction as the motion of the vehicle.
It is the inertia of the ball associated with
its constant velocity in the direction of
the vehicle's motion that ensures the ball
continues to move forward even as it is thrown
up and falls back down. From the perspective
of the person in the car, the vehicle and
everything inside of it is at rest: It is
the outside world that is moving with a constant
speed in the opposite direction of the vehicle.
Since there is no experiment that can distinguish
whether it is the vehicle that is at rest
or the outside world that is at rest, the
two situations are considered to be physically
indistinguishable. Inertia therefore applies
equally well to constant velocity motion as
it does to rest.
=== Second law ===
A modern statement of Newton's Second Law
is a vector equation:
F
→
=
d
p
→
d
t
,
{\displaystyle {\vec {F}}={\frac {\mathrm
{d} {\vec {p}}}{\mathrm {d} t}},}
where
p
→
{\displaystyle {\vec {p}}}
is the momentum of the system, and
F
→
{\displaystyle {\vec {F}}}
is the net (vector sum) force. If a body is
in equilibrium, there is zero net force by
definition (balanced forces may be present
nevertheless). In contrast, the second law
states that if there is an unbalanced force
acting on an object it will result in the
object's momentum changing over time.By the
definition of momentum,
F
→
=
d
p
→
d
t
=
d
(
m
v
→
)
d
t
,
{\displaystyle {\vec {F}}={\frac {\mathrm
{d} {\vec {p}}}{\mathrm {d} t}}={\frac {\mathrm
{d} \left(m{\vec {v}}\right)}{\mathrm {d}
t}},}
where m is the mass and
v
→
{\displaystyle {\vec {v}}}
is the velocity.If Newton's second law is
applied to a system of constant mass, m may
be moved outside the derivative operator.
The equation then becomes
F
→
=
m
d
v
→
d
t
.
{\displaystyle {\vec {F}}=m{\frac {\mathrm
{d} {\vec {v}}}{\mathrm {d} t}}.}
By substituting the definition of acceleration,
the algebraic version of Newton's Second Law
is derived:
F
→
=
m
a
→
.
{\displaystyle {\vec {F}}=m{\vec {a}}.}
Newton never explicitly stated the formula
in the reduced form above.Newton's Second
Law asserts the direct proportionality of
acceleration to force and the inverse proportionality
of acceleration to mass. Accelerations can
be defined through kinematic measurements.
However, while kinematics are well-described
through reference frame analysis in advanced
physics, there are still deep questions that
remain as to what is the proper definition
of mass. General relativity offers an equivalence
between space-time and mass, but lacking a
coherent theory of quantum gravity, it is
unclear as to how or whether this connection
is relevant on microscales. With some justification,
Newton's second law can be taken as a quantitative
definition of mass by writing the law as an
equality; the relative units of force and
mass then are fixed.
The use of Newton's Second Law as a definition
of force has been disparaged in some of the
more rigorous textbooks, because it is essentially
a mathematical truism. Notable physicists,
philosophers and mathematicians who have sought
a more explicit definition of the concept
of force include Ernst Mach and Walter Noll.Newton's
Second Law can be used to measure the strength
of forces. For instance, knowledge of the
masses of planets along with the accelerations
of their orbits allows scientists to calculate
the gravitational forces on planets.
=== Third law ===
Whenever one body exerts a force on another,
the latter simultaneously exerts an equal
and opposite force on the first. In vector
form, if
F
→
1
,
2
{\displaystyle \scriptstyle {\vec {F}}_{1,2}}
is the force of body 1 on body 2 and
F
→
2
,
1
{\displaystyle \scriptstyle {\vec {F}}_{2,1}}
that of body 2 on body 1, then
F
→
1
,
2
=
−
F
→
2
,
1
.
{\displaystyle {\vec {F}}_{1,2}=-{\vec {F}}_{2,1}.}
This law is sometimes referred to as the action-reaction
law, with
F
→
1
,
2
{\displaystyle \scriptstyle {\vec {F}}_{1,2}}
called the action and
−
F
→
2
,
1
{\displaystyle \scriptstyle -{\vec {F}}_{2,1}}
the reaction.
Newton's Third Law is a result of applying
symmetry to situations where forces can be
attributed to the presence of different objects.
The third law means that all forces are interactions
between different bodies, and thus that there
is no such thing as a unidirectional force
or a force that acts on only one body.
In a system composed of object 1 and object
2, the net force on the system due to their
mutual interactions is zero:
F
→
1
,
2
+
F
→
2
,
1
=
0.
{\displaystyle {\vec {F}}_{1,2}+{\vec {F}}_{\mathrm
{2,1} }=0.}
More generally, in a closed system of particles,
all internal forces are balanced. The particles
may accelerate with respect to each other
but the center of mass of the system will
not accelerate. If an external force acts
on the system, it will make the center of
mass accelerate in proportion to the magnitude
of the external force divided by the mass
of the system.Combining Newton's Second and
Third Laws, it is possible to show that the
linear momentum of a system is conserved.
In a system of two particles, if
p
→
1
{\displaystyle \scriptstyle {\vec {p}}_{1}}
is the momentum of object 1 and
p
→
2
{\displaystyle \scriptstyle {\vec {p}}_{2}}
the momentum of object 2, then
d
p
→
1
d
t
+
d
p
→
2
d
t
=
F
→
1
,
2
+
F
→
2
,
1
=
0.
{\displaystyle {\frac {\mathrm {d} {\vec {p}}_{1}}{\mathrm
{d} t}}+{\frac {\mathrm {d} {\vec {p}}_{2}}{\mathrm
{d} t}}={\vec {F}}_{1,2}+{\vec {F}}_{2,1}=0.}
Using similar arguments, this can be generalized
to a system with an arbitrary number of particles.
In general, as long as all forces are due
to the interaction of objects with mass, it
is possible to define a system such that net
momentum is never lost nor gained.
== Special theory of relativity ==
In the special theory of relativity, mass
and energy are equivalent (as can be seen
by calculating the work required to accelerate
an object). When an object's velocity increases,
so does its energy and hence its mass equivalent
(inertia). It thus requires more force to
accelerate it the same amount than it did
at a lower velocity. Newton's Second Law
F
→
=
d
p
→
d
t
{\displaystyle {\vec {F}}={\frac {\mathrm
{d} {\vec {p}}}{\mathrm {d} t}}}
remains valid because it is a mathematical
definition. But for relativistic momentum
to be conserved, it must be redefined as:
p
→
=
m
0
v
→
1
−
v
2
/
c
2
,
{\displaystyle {\vec {p}}={\frac {m_{0}{\vec
{v}}}{\sqrt {1-v^{2}/c^{2}}}},}
where
m
0
{\displaystyle m_{0}}
is the rest mass and
c
{\displaystyle c}
the speed of light.
The relativistic expression relating force
and acceleration for a particle with constant
non-zero rest mass
m
{\displaystyle m}
moving in the
x
{\displaystyle x}
direction is:
F
→
=
(
γ
3
m
a
x
,
γ
m
a
y
,
γ
m
a
z
)
,
{\displaystyle {\vec {F}}=\left(\gamma ^{3}ma_{x},\gamma
ma_{y},\gamma ma_{z}\right),}
where
γ
=
1
1
−
v
2
/
c
2
.
{\displaystyle \gamma ={\frac {1}{\sqrt {1-v^{2}/c^{2}}}}.}
is called the Lorentz factor.In the early
history of relativity, the expressions
γ
3
m
{\displaystyle \gamma ^{3}m}
and
γ
m
{\displaystyle \gamma m}
were called longitudinal and transverse mass.
Relativistic force does not produce a constant
acceleration, but an ever-decreasing acceleration
as the object approaches the speed of light.
Note that
γ
{\displaystyle \gamma }
approaches asymptotically an infinite value
and is undefined for an object with a non-zero
rest mass as it approaches the speed of light,
and the theory yields no prediction at that
speed.
If
v
{\displaystyle v}
is very small compared to
c
{\displaystyle c}
, then
γ
{\displaystyle \gamma }
is very close to 1 and
F
=
m
a
{\displaystyle F=ma}
is a close approximation. Even for use in
relativity, however, one can restore the form
of
F
μ
=
m
A
μ
{\displaystyle F^{\mu }=mA^{\mu }\,}
through the use of four-vectors. This relation
is correct in relativity when
F
μ
{\displaystyle F^{\mu }}
is the four-force,
m
{\displaystyle m}
is the invariant mass, and
A
μ
{\displaystyle A^{\mu }}
is the four-acceleration.
== Descriptions ==
Since forces are perceived as pushes or pulls,
this can provide an intuitive understanding
for describing forces. As with other physical
concepts (e.g. temperature), the intuitive
understanding of forces is quantified using
precise operational definitions that are consistent
with direct observations and compared to a
standard measurement scale. Through experimentation,
it is determined that laboratory measurements
of forces are fully consistent with the conceptual
definition of force offered by Newtonian mechanics.
Forces act in a particular direction and have
sizes dependent upon how strong the push or
pull is. Because of these characteristics,
forces are classified as "vector quantities".
This means that forces follow a different
set of mathematical rules than physical quantities
that do not have direction (denoted scalar
quantities). For example, when determining
what happens when two forces act on the same
object, it is necessary to know both the magnitude
and the direction of both forces to calculate
the result. If both of these pieces of information
are not known for each force, the situation
is ambiguous. For example, if you know that
two people are pulling on the same rope with
known magnitudes of force but you do not know
which direction either person is pulling,
it is impossible to determine what the acceleration
of the rope will be. The two people could
be pulling against each other as in tug of
war or the two people could be pulling in
the same direction. In this simple one-dimensional
example, without knowing the direction of
the forces it is impossible to decide whether
the net force is the result of adding the
two force magnitudes or subtracting one from
the other. Associating forces with vectors
avoids such problems.
Historically, forces were first quantitatively
investigated in conditions of static equilibrium
where several forces canceled each other out.
Such experiments demonstrate the crucial properties
that forces are additive vector quantities:
they have magnitude and direction. When two
forces act on a point particle, the resulting
force, the resultant (also called the net
force), can be determined by following the
parallelogram rule of vector addition: the
addition of two vectors represented by sides
of a parallelogram, gives an equivalent resultant
vector that is equal in magnitude and direction
to the transversal of the parallelogram. The
magnitude of the resultant varies from the
difference of the magnitudes of the two forces
to their sum, depending on the angle between
their lines of action. However, if the forces
are acting on an extended body, their respective
lines of application must also be specified
in order to account for their effects on the
motion of the body.
Free-body diagrams can be used as a convenient
way to keep track of forces acting on a system.
Ideally, these diagrams are drawn with the
angles and relative magnitudes of the force
vectors preserved so that graphical vector
addition can be done to determine the net
force.As well as being added, forces can also
be resolved into independent components at
right angles to each other. A horizontal force
pointing northeast can therefore be split
into two forces, one pointing north, and one
pointing east. Summing these component forces
using vector addition yields the original
force. Resolving force vectors into components
of a set of basis vectors is often a more
mathematically clean way to describe forces
than using magnitudes and directions. This
is because, for orthogonal components, the
components of the vector sum are uniquely
determined by the scalar addition of the components
of the individual vectors. Orthogonal components
are independent of each other because forces
acting at ninety degrees to each other have
no effect on the magnitude or direction of
the other. Choosing a set of orthogonal basis
vectors is often done by considering what
set of basis vectors will make the mathematics
most convenient. Choosing a basis vector that
is in the same direction as one of the forces
is desirable, since that force would then
have only one non-zero component. Orthogonal
force vectors can be three-dimensional with
the third component being at right-angles
to the other two.
=== Equilibrium ===
Equilibrium occurs when the resultant force
acting on a point particle is zero (that is,
the vector sum of all forces is zero). When
dealing with an extended body, it is also
necessary that the net torque be zero.
There are two kinds of equilibrium: static
equilibrium and dynamic equilibrium.
==== Static ====
Static equilibrium was understood well before
the invention of classical mechanics. Objects
that are at rest have zero net force acting
on them.The simplest case of static equilibrium
occurs when two forces are equal in magnitude
but opposite in direction. For example, an
object on a level surface is pulled (attracted)
downward toward the center of the Earth by
the force of gravity. At the same time, a
force is applied by the surface that resists
the downward force with equal upward force
(called a normal force). The situation produces
zero net force and hence no acceleration.Pushing
against an object that rests on a frictional
surface can result in a situation where the
object does not move because the applied force
is opposed by static friction, generated between
the object and the table surface. For a situation
with no movement, the static friction force
exactly balances the applied force resulting
in no acceleration. The static friction increases
or decreases in response to the applied force
up to an upper limit determined by the characteristics
of the contact between the surface and the
object.A static equilibrium between two forces
is the most usual way of measuring forces,
using simple devices such as weighing scales
and spring balances. For example, an object
suspended on a vertical spring scale experiences
the force of gravity acting on the object
balanced by a force applied by the "spring
reaction force", which equals the object's
weight. Using such tools, some quantitative
force laws were discovered: that the force
of gravity is proportional to volume for objects
of constant density (widely exploited for
millennia to define standard weights); Archimedes'
principle for buoyancy; Archimedes' analysis
of the lever; Boyle's law for gas pressure;
and Hooke's law for springs. These were all
formulated and experimentally verified before
Isaac Newton expounded his Three Laws of Motion.
==== Dynamic ====
Dynamic equilibrium was first described by
Galileo who noticed that certain assumptions
of Aristotelian physics were contradicted
by observations and logic. Galileo realized
that simple velocity addition demands that
the concept of an "absolute rest frame" did
not exist. Galileo concluded that motion in
a constant velocity was completely equivalent
to rest. This was contrary to Aristotle's
notion of a "natural state" of rest that objects
with mass naturally approached. Simple experiments
showed that Galileo's understanding of the
equivalence of constant velocity and rest
were correct. For example, if a mariner dropped
a cannonball from the crow's nest of a ship
moving at a constant velocity, Aristotelian
physics would have the cannonball fall straight
down while the ship moved beneath it. Thus,
in an Aristotelian universe, the falling cannonball
would land behind the foot of the mast of
a moving ship. However, when this experiment
is actually conducted, the cannonball always
falls at the foot of the mast, as if the cannonball
knows to travel with the ship despite being
separated from it. Since there is no forward
horizontal force being applied on the cannonball
as it falls, the only conclusion left is that
the cannonball continues to move with the
same velocity as the boat as it falls. Thus,
no force is required to keep the cannonball
moving at the constant forward velocity.Moreover,
any object traveling at a constant velocity
must be subject to zero net force (resultant
force). This is the definition of dynamic
equilibrium: when all the forces on an object
balance but it still moves at a constant velocity.
A simple case of dynamic equilibrium occurs
in constant velocity motion across a surface
with kinetic friction. In such a situation,
a force is applied in the direction of motion
while the kinetic friction force exactly opposes
the applied force. This results in zero net
force, but since the object started with a
non-zero velocity, it continues to move with
a non-zero velocity. Aristotle misinterpreted
this motion as being caused by the applied
force. However, when kinetic friction is taken
into consideration it is clear that there
is no net force causing constant velocity
motion.
=== Forces in quantum mechanics ===
The notion "force" keeps its meaning in quantum
mechanics, though one is now dealing with
operators instead of classical variables and
though the physics is now described by the
Schrödinger equation instead of Newtonian
equations. This has the consequence that the
results of a measurement are now sometimes
"quantized", i.e. they appear in discrete
portions. This is, of course, difficult to
imagine in the context of "forces". However,
the potentials V(x,y,z) or fields, from which
the forces generally can be derived, are treated
similarly to classical position variables,
i.e.,
V
(
x
,
y
,
z
)
→
V
^
(
x
^
,
y
^
,
z
^
)
{\displaystyle V(x,y,z)\to {\hat {V}}({\hat
{x}},{\hat {y}},{\hat {z}})}
.
This becomes different only in the framework
of quantum field theory, where these fields
are also quantized.
However, already in quantum mechanics there
is one "caveat", namely the particles acting
onto each other do not only possess the spatial
variable, but also a discrete intrinsic angular
momentum-like variable called the "spin",
and there is the Pauli exclusion principle
relating the space and the spin variables.
Depending on the value of the spin, identical
particles split into two different classes,
fermions and bosons. If two identical fermions
(e.g. electrons) have a symmetric spin function
(e.g. parallel spins) the spatial variables
must be antisymmetric (i.e. they exclude each
other from their places much as if there was
a repulsive force), and vice versa, i.e. for
antiparallel spins the position variables
must be symmetric (i.e. the apparent force
must be attractive). Thus in the case of two
fermions there is a strictly negative correlation
between spatial and spin variables, whereas
for two bosons (e.g. quanta of electromagnetic
waves, photons) the correlation is strictly
positive.
Thus the notion "force" loses already part
of its meaning.
=== Feynman diagrams ===
In modern particle physics, forces and the
acceleration of particles are explained as
a mathematical by-product of exchange of momentum-carrying
gauge bosons. With the development of quantum
field theory and general relativity, it was
realized that force is a redundant concept
arising from conservation of momentum (4-momentum
in relativity and momentum of virtual particles
in quantum electrodynamics). The conservation
of momentum can be directly derived from the
homogeneity or symmetry of space and so is
usually considered more fundamental than the
concept of a force. Thus the currently known
fundamental forces are considered more accurately
to be "fundamental interactions". When particle
A emits (creates) or absorbs (annihilates)
virtual particle B, a momentum conservation
results in recoil of particle A making impression
of repulsion or attraction between particles
A A' exchanging by B. This description applies
to all forces arising from fundamental interactions.
While sophisticated mathematical descriptions
are needed to predict, in full detail, the
accurate result of such interactions, there
is a conceptually simple way to describe such
interactions through the use of Feynman diagrams.
In a Feynman diagram, each matter particle
is represented as a straight line (see world
line) traveling through time, which normally
increases up or to the right in the diagram.
Matter and anti-matter particles are identical
except for their direction of propagation
through the Feynman diagram. World lines of
particles intersect at interaction vertices,
and the Feynman diagram represents any force
arising from an interaction as occurring at
the vertex with an associated instantaneous
change in the direction of the particle world
lines. Gauge bosons are emitted away from
the vertex as wavy lines and, in the case
of virtual particle exchange, are absorbed
at an adjacent vertex.The utility of Feynman
diagrams is that other types of physical phenomena
that are part of the general picture of fundamental
interactions but are conceptually separate
from forces can also be described using the
same rules. For example, a Feynman diagram
can describe in succinct detail how a neutron
decays into an electron, proton, and neutrino,
an interaction mediated by the same gauge
boson that is responsible for the weak nuclear
force.
== Fundamental forces ==
All of the known forces of the universe are
classified into four fundamental interactions.
The strong and the weak forces are nuclear
forces that act only at very short distances,
and are responsible for the interactions between
subatomic particles, including nucleons and
compound nuclei. The electromagnetic force
acts between electric charges, and the gravitational
force acts between masses. All other forces
in nature derive from these four fundamental
interactions. For example, friction is a manifestation
of the electromagnetic force acting between
atoms of two surfaces, and the Pauli exclusion
principle, which does not permit atoms to
pass through each other. Similarly, the forces
in springs, modeled by Hooke's law, are the
result of electromagnetic forces and the Pauli
exclusion principle acting together to return
an object to its equilibrium position. Centrifugal
forces are acceleration forces that arise
simply from the acceleration of rotating frames
of reference.The fundamental theories for
forces developed from the unification of different
ideas. For example, Sir. Isaac Newton unified,
with his universal theory of gravitation,
the force responsible for objects falling
near the surface of the Earth with the force
responsible for the falling of celestial bodies
about the Earth (the Moon) and around the
Sun (the planets). Michael Faraday and James
Clerk Maxwell demonstrated that electric and
magnetic forces were unified through a theory
of electromagnetism. In the 20th century,
the development of quantum mechanics led to
a modern understanding that the first three
fundamental forces (all except gravity) are
manifestations of matter (fermions) interacting
by exchanging virtual particles called gauge
bosons. This standard model of particle physics
assumes a similarity between the forces and
led scientists to predict the unification
of the weak and electromagnetic forces in
electroweak theory, which was subsequently
confirmed by observation. The complete formulation
of the standard model predicts an as yet unobserved
Higgs mechanism, but observations such as
neutrino oscillations suggest that the standard
model is incomplete. A Grand Unified Theory
that allows for the combination of the electroweak
interaction with the strong force is held
out as a possibility with candidate theories
such as supersymmetry proposed to accommodate
some of the outstanding unsolved problems
in physics. Physicists are still attempting
to develop self-consistent unification models
that would combine all four fundamental interactions
into a theory of everything. Einstein tried
and failed at this endeavor, but currently
the most popular approach to answering this
question is string theory.
=== Gravitational ===
What we now call gravity was not identified
as a universal force until the work of Isaac
Newton. Before Newton, the tendency for objects
to fall towards the Earth was not understood
to be related to the motions of celestial
objects. Galileo was instrumental in describing
the characteristics of falling objects by
determining that the acceleration of every
object in free-fall was constant and independent
of the mass of the object. Today, this acceleration
due to gravity towards the surface of the
Earth is usually designated as
g
→
{\displaystyle \scriptstyle {\vec {g}}}
and has a magnitude of about 9.81 meters per
second squared (this measurement is taken
from sea level and may vary depending on location),
and points toward the center of the Earth.
This observation means that the force of gravity
on an object at the Earth's surface is directly
proportional to the object's mass. Thus an
object that has a mass of
m
{\displaystyle m}
will experience a force:
F
→
=
m
g
→
{\displaystyle {\vec {F}}=m{\vec {g}}}
For an object in free-fall, this force is
unopposed and the net force on the object
is its weight. For objects not in free-fall,
the force of gravity is opposed by the reaction
forces applied by their supports. For example,
a person standing on the ground experiences
zero net force, since a normal force (a reaction
force) is exerted by the ground upward on
the person that counterbalances his weight
that is directed downward.Newton's contribution
to gravitational theory was to unify the motions
of heavenly bodies, which Aristotle had assumed
were in a natural state of constant motion,
with falling motion observed on the Earth.
He proposed a law of gravity that could account
for the celestial motions that had been described
earlier using Kepler's laws of planetary motion.Newton
came to realize that the effects of gravity
might be observed in different ways at larger
distances. In particular, Newton determined
that the acceleration of the Moon around the
Earth could be ascribed to the same force
of gravity if the acceleration due to gravity
decreased as an inverse square law. Further,
Newton realized that the acceleration of a
body due to gravity is proportional to the
mass of the other attracting body. Combining
these ideas gives a formula that relates the
mass (
m
⊕
{\displaystyle \scriptstyle m_{\oplus }}
) and the radius (
R
⊕
{\displaystyle \scriptstyle R_{\oplus }}
) of the Earth to the gravitational acceleration:
g
→
=
−
G
m
⊕
R
⊕
2
r
^
{\displaystyle {\vec {g}}=-{\frac {Gm_{\oplus
}}{{R_{\oplus }}^{2}}}{\hat {r}}}
where the vector direction is given by
r
^
{\displaystyle {\hat {r}}}
, is the unit vector directed outward from
the center of the Earth.In this equation,
a dimensional constant
G
{\displaystyle G}
is used to describe the relative strength
of gravity. This constant has come to be known
as Newton's Universal Gravitation Constant,
though its value was unknown in Newton's lifetime.
Not until 1798 was Henry Cavendish able to
make the first measurement of
G
{\displaystyle G}
using a torsion balance; this was widely reported
in the press as a measurement of the mass
of the Earth since knowing
G
{\displaystyle G}
could allow one to solve for the Earth's mass
given the above equation. Newton, however,
realized that since all celestial bodies followed
the same laws of motion, his law of gravity
had to be universal. Succinctly stated, Newton's
Law of Gravitation states that the force on
a spherical object of mass
m
1
{\displaystyle m_{1}}
due to the gravitational pull of mass
m
2
{\displaystyle m_{2}}
is
F
→
=
−
G
m
1
m
2
r
2
r
^
{\displaystyle {\vec {F}}=-{\frac {Gm_{1}m_{2}}{r^{2}}}{\hat
{r}}}
where
r
{\displaystyle r}
is the distance between the two objects' centers
of mass and
r
^
{\displaystyle \scriptstyle {\hat {r}}}
is the unit vector pointed in the direction
away from the center of the first object toward
the center of the second object.This formula
was powerful enough to stand as the basis
for all subsequent descriptions of motion
within the solar system until the 20th century.
During that time, sophisticated methods of
perturbation analysis were invented to calculate
the deviations of orbits due to the influence
of multiple bodies on a planet, moon, comet,
or asteroid. The formalism was exact enough
to allow mathematicians to predict the existence
of the planet Neptune before it was observed.
Mercury's orbit, however, did not match that
predicted by Newton's Law of Gravitation.
Some astrophysicists predicted the existence
of another planet (Vulcan) that would explain
the discrepancies; however no such planet
could be found. When Albert Einstein formulated
his theory of general relativity (GR) he turned
his attention to the problem of Mercury's
orbit and found that his theory added a correction,
which could account for the discrepancy. This
was the first time that Newton's Theory of
Gravity had been shown to be inexact.Since
then, general relativity has been acknowledged
as the theory that best explains gravity.
In GR, gravitation is not viewed as a force,
but rather, objects moving freely in gravitational
fields travel under their own inertia in straight
lines through curved space-time – defined
as the shortest space-time path between two
space-time events. From the perspective of
the object, all motion occurs as if there
were no gravitation whatsoever. It is only
when observing the motion in a global sense
that the curvature of space-time can be observed
and the force is inferred from the object's
curved path. Thus, the straight line path
in space-time is seen as a curved line in
space, and it is called the ballistic trajectory
of the object. For example, a basketball thrown
from the ground moves in a parabola, as it
is in a uniform gravitational field. Its space-time
trajectory is almost a straight line, slightly
curved (with the radius of curvature of the
order of few light-years). The time derivative
of the changing momentum of the object is
what we label as "gravitational force".
=== Electromagnetic ===
The electrostatic force was first described
in 1784 by Coulomb as a force that existed
intrinsically between two charges. The properties
of the electrostatic force were that it varied
as an inverse square law directed in the radial
direction, was both attractive and repulsive
(there was intrinsic polarity), was independent
of the mass of the charged objects, and followed
the superposition principle. Coulomb's law
unifies all these observations into one succinct
statement.Subsequent mathematicians and physicists
found the construct of the electric field
to be useful for determining the electrostatic
force on an electric charge at any point in
space. The electric field was based on using
a hypothetical "test charge" anywhere in space
and then using Coulomb's Law to determine
the electrostatic force. Thus the electric
field anywhere in space is defined as
E
→
=
F
→
q
{\displaystyle {\vec {E}}={{\vec {F}} \over
{q}}}
where
q
{\displaystyle q}
is the magnitude of the hypothetical test
charge.
Meanwhile, the Lorentz force of magnetism
was discovered to exist between two electric
currents. It has the same mathematical character
as Coulomb's Law with the proviso that like
currents attract and unlike currents repel.
Similar to the electric field, the magnetic
field can be used to determine the magnetic
force on an electric current at any point
in space. In this case, the magnitude of the
magnetic field was determined to be
B
=
F
I
ℓ
{\displaystyle B={F \over {I\ell }}}
where
I
{\displaystyle I}
is the magnitude of the hypothetical test
current and
ℓ
{\displaystyle \scriptstyle \ell }
is the length of hypothetical wire through
which the test current flows. The magnetic
field exerts a force on all magnets including,
for example, those used in compasses. The
fact that the Earth's magnetic field is aligned
closely with the orientation of the Earth's
axis causes compass magnets to become oriented
because of the magnetic force pulling on the
needle.
Through combining the definition of electric
current as the time rate of change of electric
charge, a rule of vector multiplication called
Lorentz's Law describes the force on a charge
moving in a magnetic field. The connection
between electricity and magnetism allows for
the description of a unified electromagnetic
force that acts on a charge. This force can
be written as a sum of the electrostatic force
(due to the electric field) and the magnetic
force (due to the magnetic field). Fully stated,
this is 
the law:
F
→
=
q
(
E
→
+
v
→
×
B
→
)
{\displaystyle {\vec {F}}=q({\vec {E}}+{\vec
{v}}\times {\vec {B}})}
where
F
→
{\displaystyle \scriptstyle {\vec {F}}}
is the electromagnetic force,
q
{\displaystyle q}
is the magnitude of the charge of the particle,
E
→
{\displaystyle \scriptstyle {\vec {E}}}
is the electric field,
v
→
{\displaystyle \scriptstyle {\vec {v}}}
is the velocity of the particle that is crossed
with the magnetic field (
B
→
{\displaystyle \scriptstyle {\vec {B}}}
).
The origin of electric and magnetic fields
would not be fully explained until 1864 when
James Clerk Maxwell unified a number of earlier
theories into a set of 20 scalar equations,
which were later reformulated into 4 vector
equations by Oliver Heaviside and Josiah Willard
Gibbs. These "Maxwell Equations" fully described
the sources of the fields as being stationary
and moving charges, and the interactions of
the fields themselves. This led Maxwell to
discover that electric and magnetic fields
could be "self-generating" through a wave
that traveled at a speed that he calculated
to be the speed of light. This insight united
the nascent fields of electromagnetic theory
with optics and led directly to a complete
description of the electromagnetic spectrum.However,
attempting to reconcile electromagnetic theory
with two observations, the photoelectric effect,
and the nonexistence of the ultraviolet catastrophe,
proved troublesome. Through the work of leading
theoretical physicists, a new theory of electromagnetism
was developed using quantum mechanics. This
final modification to electromagnetic theory
ultimately led to quantum electrodynamics
(or QED), which fully describes all electromagnetic
phenomena as being mediated by wave–particles
known as photons. In QED, photons are the
fundamental exchange particle, which described
all interactions relating to electromagnetism
including the electromagnetic force.
=== Strong nuclear ===
There are two "nuclear forces", which today
are usually described as interactions that
take place in quantum theories of particle
physics. The strong nuclear force is the force
responsible for the structural integrity of
atomic nuclei while the weak nuclear force
is responsible for the decay of certain nucleons
into leptons and other types of hadrons.The
strong force is today understood to represent
the interactions between quarks and gluons
as detailed by the theory of quantum chromodynamics
(QCD). The strong force is the fundamental
force mediated by gluons, acting upon quarks,
antiquarks, and the gluons themselves. The
(aptly named) strong interaction is the "strongest"
of the four fundamental forces.
The strong force only acts directly upon elementary
particles. However, a residual of the force
is observed between hadrons (the best known
example being the force that acts between
nucleons in atomic nuclei) as the nuclear
force. Here the strong force acts indirectly,
transmitted as gluons, which form part of
the virtual pi and rho mesons, which classically
transmit the nuclear force (see this topic
for more). The failure of many searches for
free quarks has shown that the elementary
particles affected are not directly observable.
This phenomenon is called color confinement.
=== Weak nuclear ===
The weak force is due to the exchange of the
heavy W and Z bosons. Its most familiar effect
is beta decay (of neutrons in atomic nuclei)
and the associated radioactivity. The word
"weak" derives from the fact that the field
strength is some 1013 times less than that
of the strong force. Still, it is stronger
than gravity over short distances. A consistent
electroweak theory has also been developed,
which shows that electromagnetic forces and
the weak force are indistinguishable at a
temperatures in excess of approximately 1015
kelvins. Such temperatures have been probed
in modern particle accelerators and show the
conditions of the universe in the early moments
of the Big Bang.
== Non-fundamental forces ==
Some forces are consequences of the fundamental
ones. In such situations, idealized models
can be utilized to gain physical insight.
=== Normal force ===
The normal force is due to repulsive forces
of interaction between atoms at close contact.
When their electron clouds overlap, Pauli
repulsion (due to fermionic nature of electrons)
follows resulting in the force that acts in
a direction normal to the surface interface
between two objects. The normal force, for
example, is responsible for the structural
integrity of tables and floors as well as
being the force that responds whenever an
external force pushes on a solid object. An
example of the normal force in action is the
impact force on an object crashing into an
immobile surface.
=== Friction ===
Friction is a surface force that opposes relative
motion. The frictional force is directly related
to the normal force that acts to keep two
solid objects separated at the point of contact.
There are two broad classifications of frictional
forces: static friction and kinetic friction.
The static friction force (
F
s
f
{\displaystyle F_{\mathrm {sf} }}
) will exactly oppose forces applied to an
object parallel to a surface contact up to
the limit specified by the coefficient of
static friction (
μ
s
f
{\displaystyle \mu _{\mathrm {sf} }}
) multiplied by the normal force (
F
N
{\displaystyle F_{N}}
). In other words, the magnitude of the static
friction force satisfies the inequality:
0
≤
F
s
f
≤
μ
s
f
F
N
.
{\displaystyle 0\leq F_{\mathrm {sf} }\leq
\mu _{\mathrm {sf} }F_{\mathrm {N} }.}
The kinetic friction force (
F
k
f
{\displaystyle F_{\mathrm {kf} }}
) is independent of both the forces applied
and the movement of the object. Thus, the
magnitude of the force equals:
F
k
f
=
μ
k
f
F
N
,
{\displaystyle F_{\mathrm {kf} }=\mu _{\mathrm
{kf} }F_{\mathrm {N} },}
where
μ
k
f
{\displaystyle \mu _{\mathrm {kf} }}
is the coefficient of kinetic friction. For
most surface interfaces, the coefficient of
kinetic friction is less than the coefficient
of static friction.
=== Tension ===
Tension forces can be modeled using ideal
strings that are massless, frictionless, unbreakable,
and unstretchable. They can be combined with
ideal pulleys, which allow ideal strings to
switch physical direction. Ideal strings transmit
tension forces instantaneously in action-reaction
pairs so that if two objects are connected
by an ideal string, any force directed along
the string by the first object is accompanied
by a force directed along the string in the
opposite direction by the second object. By
connecting the same string multiple times
to the same object through the use of a set-up
that uses movable pulleys, the tension force
on a load can be multiplied. For every string
that acts on a load, another factor of the
tension force in the string acts on the load.
However, even though such machines allow for
an increase in force, there is a corresponding
increase in the length of string that must
be displaced in order to move the load. These
tandem effects result ultimately in the conservation
of mechanical energy since the work done on
the load is the same no matter how complicated
the machine.
=== Elastic force ===
An elastic force acts to return a spring to
its natural length. An ideal spring is taken
to be massless, frictionless, unbreakable,
and infinitely stretchable. Such springs exert
forces that push when contracted, or pull
when extended, in proportion to the displacement
of the spring from its equilibrium position.
This linear relationship was described by
Robert Hooke in 1676, for whom Hooke's law
is named. If
Δ
x
{\displaystyle \Delta x}
is the displacement, the force exerted by
an ideal spring equals:
F
→
=
−
k
Δ
x
→
{\displaystyle {\vec {F}}=-k\Delta {\vec {x}}}
where
k
{\displaystyle k}
is the 
spring constant (or force constant), which
is particular to the spring. The minus sign
accounts for the tendency of the force to
act in opposition to the applied load.
=== Continuum mechanics ===
Newton's laws and Newtonian mechanics in general
were first developed to describe how forces
affect idealized point particles rather than
three-dimensional objects. However, in real
life, matter has extended structure and forces
that act on one part of an object might affect
other parts of an object. For situations where
lattice holding together the atoms in an object
is able to flow, contract, expand, or otherwise
change shape, the theories of continuum mechanics
describe the way forces affect the material.
For example, in extended fluids, differences
in pressure result in forces being directed
along the pressure gradients as follows:
F
→
V
=
−
∇
→
P
{\displaystyle {\frac {\vec {F}}{V}}=-{\vec
{\nabla }}P}
where
V
{\displaystyle V}
is the volume of the object in the fluid and
P
{\displaystyle P}
is the scalar function that describes the
pressure at all locations in space. Pressure
gradients and differentials result in the
buoyant force for fluids suspended in gravitational
fields, winds in atmospheric science, and
the lift associated with aerodynamics and
flight.A specific instance of such a force
that is associated with dynamic pressure is
fluid resistance: a body force that resists
the motion of an object through a fluid due
to viscosity. For so-called "Stokes' drag"
the force is approximately proportional to
the velocity, but opposite in direction:
F
→
d
=
−
b
v
→
{\displaystyle {\vec {F}}_{\mathrm {d} }=-b{\vec
{v}}\,}
where:
b
{\displaystyle b}
is a constant that depends on the properties
of the fluid and the dimensions of the object
(usually the cross-sectional area), and
v
→
{\displaystyle \scriptstyle {\vec {v}}}
is the velocity of the object.More formally,
forces in continuum mechanics are fully described
by a stress–tensor with terms that are roughly
defined as
σ
=
F
A
{\displaystyle \sigma ={\frac {F}{A}}}
where
A
{\displaystyle A}
is the relevant cross-sectional area for the
volume for which the stress-tensor is being
calculated. This formalism includes pressure
terms associated with forces that act normal
to the cross-sectional area (the matrix diagonals
of the tensor) as well as shear terms associated
with forces that act parallel to the cross-sectional
area (the off-diagonal elements). The stress
tensor accounts for forces that cause all
strains (deformations) including also tensile
stresses and compressions.
=== Fictitious forces ===
There are forces that are frame dependent,
meaning that they appear due to the adoption
of non-Newtonian (that is, non-inertial) reference
frames. Such forces include the centrifugal
force and the Coriolis force. These forces
are considered fictitious because they do
not exist in frames of reference that are
not accelerating. Because these forces are
not genuine they are also referred to as "pseudo
forces".In general relativity, gravity becomes
a fictitious force that arises in situations
where spacetime deviates from a flat geometry.
As an extension, Kaluza–Klein theory and
string theory ascribe electromagnetism and
the other fundamental forces respectively
to the curvature of differently scaled dimensions,
which would ultimately imply that all forces
are fictitious.
== Rotations and torque ==
Forces that cause extended objects to rotate
are associated with torques. Mathematically,
the torque of a force
F
→
{\displaystyle \scriptstyle {\vec {F}}}
is defined relative to an arbitrary reference
point as the cross-product:
τ
→
=
r
→
×
F
→
{\displaystyle {\vec {\tau }}={\vec {r}}\times
{\vec {F}}}
where
r
→
{\displaystyle \scriptstyle {\vec {r}}}
is the position vector of the force application
point relative to the reference point.Torque
is the rotation equivalent of force in the
same way that angle is the rotational equivalent
for position, angular velocity for velocity,
and angular momentum for momentum. As a consequence
of Newton's First Law of Motion, there exists
rotational inertia that ensures that all bodies
maintain their angular momentum unless acted
upon by an unbalanced torque. Likewise, Newton's
Second Law of Motion can be used to derive
an analogous equation for the instantaneous
angular acceleration of the rigid body:
τ
→
=
I
α
→
{\displaystyle {\vec {\tau }}=I{\vec {\alpha
}}}
where
I
{\displaystyle I}
is the moment of inertia of the body
α
→
{\displaystyle \scriptstyle {\vec {\alpha
}}}
is the angular acceleration of the body.This
provides a definition for the moment of inertia,
which is the rotational equivalent for mass.
In more advanced treatments of mechanics,
where the rotation over a time interval is
described, the moment of inertia must be substituted
by the tensor that, when properly analyzed,
fully determines the characteristics of rotations
including precession and nutation.
Equivalently, the differential form of Newton's
Second Law provides an alternative definition
of torque:
τ
→
=
d
L
→
d
t
,
{\displaystyle {\vec {\tau }}={\frac {\mathrm
{d} {\vec {L}}}{\mathrm {dt} }},}
where
L
→
{\displaystyle \scriptstyle {\vec {L}}}
is the angular momentum of the particle.Newton's
Third Law of Motion requires that all objects
exerting torques themselves experience equal
and opposite torques, and therefore also directly
implies the conservation of angular momentum
for closed systems that experience rotations
and revolutions through the action of internal
torques.
=== Centripetal force ===
For an object accelerating in circular motion,
the unbalanced force acting on the object
equals:
F
→
=
−
m
v
2
r
^
r
{\displaystyle {\vec {F}}=-{\frac {mv^{2}{\hat
{r}}}{r}}}
where
m
{\displaystyle m}
is the mass of the object,
v
{\displaystyle v}
is the velocity of the object and
r
{\displaystyle r}
is the distance to the center of the circular
path and
r
^
{\displaystyle \scriptstyle {\hat {r}}}
is the unit vector pointing in the radial
direction outwards from the center. This means
that the unbalanced centripetal force felt
by any object is always directed toward the
center of the curving path. Such forces act
perpendicular to the velocity vector associated
with the motion of an object, and therefore
do not change the speed of the object (magnitude
of the velocity), but only the direction of
the velocity vector. The unbalanced force
that accelerates an object can be resolved
into a component that is perpendicular to
the path, and one that is tangential to the
path. This yields both the tangential force,
which accelerates the object by either slowing
it down or speeding it up, and the radial
(centripetal) force, which changes its direction.
== Kinematic integrals ==
Forces can be used to define a number of physical
concepts by integrating with respect to kinematic
variables. For example, integrating with respect
to time gives the definition of impulse:
I
→
=
∫
t
1
t
2
F
→
d
t
,
{\displaystyle {\vec {I}}=\int _{t_{1}}^{t_{2}}{{\vec
{F}}\mathrm {d} t},}
which by Newton's Second Law must be equivalent
to the change in momentum (yielding the Impulse
momentum theorem).
Similarly, integrating with respect to position
gives a definition for the work done by a
force:
W
=
∫
x
→
1
x
→
2
F
→
⋅
d
x
→
,
{\displaystyle W=\int _{{\vec {x}}_{1}}^{{\vec
{x}}_{2}}{{\vec {F}}\cdot {\mathrm {d} {\vec
{x}}}},}
which is equivalent to changes in kinetic
energy (yielding the work energy theorem).Power
P is the rate of change dW/dt of the work
W, as the trajectory is extended by a position
change
d
x
→
{\displaystyle \scriptstyle {d}{\vec {x}}}
in a time interval dt:
d
W
=
d
W
d
x
→
⋅
d
x
→
=
F
→
⋅
d
x
→
,
so
P
=
d
W
d
t
=
d
W
d
x
→
⋅
d
x
→
d
t
=
F
→
⋅
v
→
,
{\displaystyle {\text{d}}W\,=\,{\frac {{\text{d}}W}{{\text{d}}{\vec
{x}}}}\,\cdot \,{\text{d}}{\vec {x}}\,=\,{\vec
{F}}\,\cdot \,{\text{d}}{\vec {x}},\qquad
{\text{ so }}\quad P\,=\,{\frac {{\text{d}}W}{{\text{d}}t}}\,=\,{\frac
{{\text{d}}W}{{\text{d}}{\vec {x}}}}\,\cdot
\,{\frac {{\text{d}}{\vec {x}}}{{\text{d}}t}}\,=\,{\vec
{F}}\,\cdot \,{\vec {v}},}
with
v
→
=
d
x
→
/
d
t
{\displaystyle {{\vec {v}}{\text{ }}={\text{
d}}{\vec {x}}/{\text{d}}t}}
the velocity.
== Potential energy ==
Instead of a force, often the mathematically
related concept of a potential energy field
can be used for convenience. For instance,
the gravitational force acting upon an object
can be seen as the action of the gravitational
field that is present at the object's location.
Restating mathematically the definition of
energy (via the definition of work), a potential
scalar field
U
(
r
→
)
{\displaystyle \scriptstyle {U({\vec {r}})}}
is defined as that field whose gradient is
equal and opposite to the force produced at
every point:
F
→
=
−
∇
→
U
.
{\displaystyle {\vec {F}}=-{\vec {\nabla }}U.}
Forces can be classified as conservative or
nonconservative. Conservative forces are equivalent
to the gradient of a potential while nonconservative
forces are not.
=== Conservative forces ===
A conservative force that acts on a closed
system has an associated mechanical work that
allows energy to convert only between kinetic
or potential forms. This means that for a
closed system, the net mechanical energy is
conserved whenever a conservative force acts
on the system. The force, therefore, is related
directly to the difference in potential energy
between two different locations in space,
and can be considered to be an artifact of
the potential field in the same way that the
direction and amount of a flow of water can
be considered to be an artifact of the contour
map of the elevation of an area.Conservative
forces include gravity, the electromagnetic
force, and the spring force. Each of these
forces has models that are dependent on a
position often given as a radial vector
r
→
{\displaystyle \scriptstyle {\vec {r}}}
emanating from spherically symmetric potentials.
Examples of this follow:
For gravity:
F
→
g
=
−
G
m
1
m
2
r
2
r
^
{\displaystyle {\vec {F}}_{g}=-{\frac {Gm_{1}m_{2}}{r^{2}}}{\hat
{r}}}
where
G
{\displaystyle G}
is the gravitational constant, and
m
n
{\displaystyle m_{n}}
is the mass of object n.
For electrostatic forces:
F
→
e
=
q
1
q
2
4
π
ϵ
0
r
2
r
^
{\displaystyle {\vec {F}}_{e}={\frac {q_{1}q_{2}}{4\pi
\epsilon _{0}r^{2}}}{\hat {r}}}
where
ϵ
0
{\displaystyle \epsilon _{0}}
is electric permittivity of free space, and
q
n
{\displaystyle q_{n}}
is the electric charge of object n.
For spring forces:
F
→
x
=
−
k
r
→
{\displaystyle {\vec {F}}_{x}=-k{\vec {r}}}
where
k
{\displaystyle k}
is the spring constant.
=== Nonconservative forces ===
For certain physical scenarios, it is impossible
to model forces as being due to gradient of
potentials. This is often due to macrophysical
considerations that yield forces as arising
from a macroscopic statistical average of
microstates. For example, friction is caused
by the gradients of numerous electrostatic
potentials between the atoms, but manifests
as a force model that is independent of any
macroscale position vector. Nonconservative
forces other than friction include other contact
forces, tension, compression, and drag. However,
for any sufficiently detailed description,
all these forces are the results of conservative
ones since each of these macroscopic forces
are the net results of the gradients of microscopic
potentials.The connection between macroscopic
nonconservative forces and microscopic conservative
forces is described by detailed treatment
with statistical mechanics. In macroscopic
closed systems, nonconservative forces act
to change the internal energies of the system,
and are often associated with the transfer
of heat. According to the Second law of thermodynamics,
nonconservative forces necessarily result
in energy transformations within closed systems
from ordered to more random conditions as
entropy increases.
== Units of measurement ==
The SI unit of force is the newton (symbol
N), which is the force required to accelerate
a one kilogram mass at a rate of one meter
per second squared, or kg·m·s−2. The corresponding
CGS unit is the dyne, the force required to
accelerate a one gram mass by one centimeter
per second squared, or g·cm·s−2. A newton
is thus equal to 100,000 dynes.
The gravitational foot-pound-second English
unit of force is the pound-force (lbf), defined
as the force exerted by gravity on a pound-mass
in the standard gravitational field of 9.80665
m·s−2. The pound-force provides an alternative
unit of mass: one slug is the mass that will
accelerate by one foot per second squared
when acted on by one pound-force.An alternative
unit of force in a different foot-pound-second
system, the absolute fps system, is the poundal,
defined as the force required to accelerate
a one-pound mass at a rate of one foot per
second squared. The units of slug and poundal
are designed to avoid a constant of proportionality
in Newton's Second Law.
The pound-force has a metric counterpart,
less commonly used than the newton: the kilogram-force
(kgf) (sometimes kilopond), is the force exerted
by standard gravity on one kilogram of mass.
The kilogram-force leads to an alternate,
but rarely used unit of mass: the metric slug
(sometimes mug or hyl) is that mass that accelerates
at 1 m·s−2 when subjected to a force of
1 kgf. The kilogram-force is not a part of
the modern SI system, and is generally deprecated;
however it still sees use for some purposes
as expressing aircraft weight, jet thrust,
bicycle spoke tension, torque wrench settings
and engine output torque. Other arcane units
of force include the sthène, which is equivalent
to 1000 N, and the kip, which is equivalent
to 1000 lbf.
See also Ton-force.
== 
Force measurement ==
See force gauge, spring scale, load cell
== 
See also ==
Orders of magnitude (force)
Parallel force
== Notes
