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
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.
1 Development of the concept
2 Pre-Newtonian concepts
3 Newtonian mechanics
3.1 First law
3.3 Third law
Special theory of relativity
5.2 Forces in quantum mechanics
5.3 Feynman diagrams
6 Fundamental forces
6.3 Strong nuclear
6.4 Weak nuclear
7 Non-fundamental forces
7.1 Normal force
7.4 Elastic force
7.5 Continuum mechanics
7.6 Fictitious forces
8 Rotations and torque
8.1 Centripetal force
10.1 Conservative forces
10.2 Nonconservative forces
11 Units of measurement
13 See also
16 Further reading
17 External links
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
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,
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
With modern insights into quantum mechanics and technology that can
accelerate particles close to the speed of light, particle physics has
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.:2–10:79 High-energy particle physics
observations made during the 1970s and 1980s confirmed that the weak
and electromagnetic forces are expressions of a more fundamental
Aristotelian physics and Theory of impetus
Aristotle famously described a force as anything that causes an object
to undergo "unnatural motion"
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
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
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.
Main article: Newton's laws of motion
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.
Main article: Newton's 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
Inertia therefore applies equally well to constant
velocity motion as it does to rest.
Though Sir Isaac Newton's most famous equation is
displaystyle scriptstyle vec F =m vec a
, he actually wrote down a different form for his second law of motion
that did not use differential calculus.
Main article: Newton's second law
A modern statement of Newton's
Second Law is a vector equation:[Note
displaystyle vec F = frac mathrm d vec p mathrm d t
displaystyle vec p
is the momentum of the system, and
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,
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
displaystyle vec v
is the velocity.:9-1,9-2
Newton's second law
Newton's second law is applied to a system of constant mass,[Note
2] m may be moved outside the derivative operator. The equation then
displaystyle vec F =m frac mathrm d vec v mathrm d t
By substituting the definition of acceleration, the algebraic version
Second Law is derived:
displaystyle vec F =m vec a .
Newton never explicitly stated the formula in the reduced form
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
The use of Newton's
Second Law as a definition of force has been
disparaged in some of the more rigorous textbooks,:12-1:59
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
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.
Main article: Newton's 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
displaystyle scriptstyle vec F _ 1,2
is the force of body 1 on body 2 and
displaystyle scriptstyle vec F _ 2,1
that of body 2 on body 1, then
displaystyle vec F _ 1,2 =- vec F _ 2,1 .
This law is sometimes referred to as the action-reaction law, with
displaystyle scriptstyle vec F _ 1,2
called the action and
displaystyle scriptstyle - vec F _ 2,1
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,[Note 3] 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:
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.:19-1
Second and Third Laws, it is possible to show that
the linear momentum of a system is conserved. In a system of two
displaystyle scriptstyle vec p _ 1
is the momentum of object 1 and
displaystyle scriptstyle vec p _ 2
the momentum of object 2, then
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.
displaystyle vec F = frac mathrm d vec p mathrm d t
remains valid because it is a mathematical definition.:855–876
But for relativistic momentum to be conserved, it must be redefined
displaystyle vec p = frac m_ 0 vec v sqrt 1-v^ 2 /c^ 2
displaystyle m_ 0
is the rest mass and
the speed of light.
The relativistic expression relating force and acceleration for a
particle with constant non-zero rest mass
moving in the
displaystyle vec F =left(gamma ^ 3 ma_ x ,gamma ma_ y ,gamma
ma_ z right),
displaystyle gamma = frac 1 sqrt 1-v^ 2 /c^ 2 .
is called the Lorentz factor.
In the early history of relativity, the expressions
displaystyle gamma ^ 3 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
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.
is very small compared to
is very close to 1 and
is a close approximation. Even for use in relativity, however, one can
restore the form of
displaystyle F^ mu =mA^ mu ,
through the use of four-vectors. This relation is correct in
displaystyle F^ mu
is the four-force,
is the invariant mass, and
displaystyle A^ mu
is the four-acceleration.
Free body diagrams of a block on a flat surface and an inclined plane.
Forces are resolved and added together to determine their magnitudes
and the net force.
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 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
There are two kinds of equilibrium: static equilibrium and dynamic
Statics and Static equilibrium
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
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
Isaac Newton expounded his Three Laws of Motion.
Main article: Dynamics (physics)
Galileo Galilei was the first to point out the inherent contradictions
contained in Aristotle's description of forces.
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.
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
Quantum mechanics and Pauli principle
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
similar to classical position variables, i.e.,
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 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.
Main article: Feynman diagrams
Feynman diagram for the decay of a neutron into a proton. The W boson
is between two vertices indicating a repulsion.
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 (
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".:199–128 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
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.
Main article: Fundamental interaction
All of the forces in the universe are based on four fundamental
interactions. The strong and 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 the
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 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.:12-11:359
The fundamental theories for forces developed from the unification of
disparate ideas. For example,
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 the Sun (the planets).
Michael Faraday and
James Clerk Maxwell
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
Grand Unified Theory
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
The four fundamental forces of nature
Mass - Energy
(not yet observed)
W+ W− Z0
Strength in the scale of quarks:
Strength in the scale of
Main article: Gravity
Images of a freely falling basketball taken with a stroboscope at 20
flashes per second. The distance units on the right are multiples of
about 12 millimeters. The basketball starts at rest. At the time
of the first flash (distance zero) it is released, after which the
number of units fallen is equal to the square of the number of
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
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
will experience a force:
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
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 (
displaystyle scriptstyle m_ oplus
) and the radius (
displaystyle scriptstyle R_ oplus
) of the Earth to the gravitational acceleration:
displaystyle vec g =- frac Gm_ oplus R_ oplus ^ 2
where the vector direction is given by
displaystyle hat r
, is the unit vector directed outward from the center of the
In this equation, a dimensional constant
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
using a torsion balance; this was widely reported in the press as a
measurement of the mass of the Earth since knowing
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
Newton's Law of Gravitation states that
the force on a spherical object of mass
displaystyle m_ 1
due to the gravitational pull of mass
displaystyle m_ 2
displaystyle vec F =- frac Gm_ 1 m_ 2 r^ 2 hat r
is the distance between the two objects' centers of mass and
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
Instruments like GRAVITY provide a powerful probe for gravity force
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
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
Main article: Electromagnetic force
The electrostatic force was first described in 1784 by Coulomb as a
force that existed intrinsically between two charges.:519 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
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.:4-6 to 4-8
Thus the electric field anywhere in space is defined as
displaystyle vec E = vec F over q
is the magnitude of the hypothetical test charge.
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
displaystyle B= F over Iell
is the magnitude of the hypothetical test current and
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:
displaystyle vec F =q( vec E + vec v times vec B )
displaystyle scriptstyle vec F
is the electromagnetic force,
is the magnitude of the charge of the particle,
displaystyle scriptstyle vec E
is the electric field,
displaystyle scriptstyle vec v
is the velocity of the particle that is crossed with the magnetic
displaystyle scriptstyle vec B
The origin of electric and magnetic fields would not be fully
explained until 1864 when
James Clerk Maxwell
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
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.[Note
Main article: Strong interaction
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:940 is the force responsible for the
structural integrity of atomic nuclei while the weak nuclear
force:951 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.
Main article: Weak interaction
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.
Some forces are consequences of the fundamental ones. In such
situations, idealized models can be utilized to gain physical insight.
FN represents the normal force exerted on the object.
Main article: 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.:93 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.
Main article: 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
The static friction force (
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
displaystyle mu _ mathrm sf
) multiplied by the normal force (
displaystyle F_ N
). In other words, the magnitude of the static friction force
satisfies the inequality:
displaystyle 0leq F_ mathrm sf leq mu _ mathrm sf F_ mathrm
The kinetic friction force (
displaystyle F_ mathrm kf
) is independent of both the forces applied and the movement of the
object. Thus, the magnitude of the force equals:
displaystyle F_ mathrm kf =mu _ mathrm kf F_ mathrm N ,
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
Main article: Tension (physics)
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.
Elasticity (physics) and Hooke's law
Fk is the force that responds to the load on the spring
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
displaystyle Delta x
is the displacement, the force exerted by an ideal spring equals:
displaystyle vec F =-kDelta vec x
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.
When the drag force (
displaystyle F_ d
) associated with air resistance becomes equal in magnitude to the
force of gravity on a falling object (
displaystyle F_ g
), the object reaches a state of dynamic equilibrium at terminal
Main articles: Pressure, Drag (physics), and Stress (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:
displaystyle frac vec F V =- vec nabla P
is the volume of the object in the fluid and
is the scalar function that describes the pressure at all locations
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
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:
displaystyle vec F _ mathrm d =-b vec v ,
is a constant that depends on the properties of the fluid and the
dimensions of the object (usually the cross-sectional area), and
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
displaystyle sigma = frac F 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
Main article: 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
In general relativity, gravity becomes a fictitious force that arises
in situations where spacetime deviates from a flat geometry. As an
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
Relationship between force (F), torque (τ), and momentum vectors (p
and L) in a rotating system.
Main article: Torque
Forces that cause extended objects to rotate are associated with
torques. Mathematically, the torque of a force
displaystyle scriptstyle vec F
is defined relative to an arbitrary reference point as the
displaystyle vec tau = vec r times vec F
displaystyle scriptstyle vec r
is the position vector of the force application point relative to the
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:
displaystyle vec tau =I vec alpha
is the moment of inertia of the body
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:
displaystyle vec tau = frac mathrm d vec L mathrm dt
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.
Main article: Centripetal force
For an object accelerating in circular motion, the unbalanced force
acting on the object equals:
displaystyle vec F =- frac mv^ 2 hat r r
is the mass of the object,
is the velocity of the object and
is the distance to the center of the circular path and
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
Main articles: Impulse, Mechanical work, and Power (physics)
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:
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::13-3
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
Power P is the rate of change dW/dt of the work W, as the trajectory
is extended by a position change
displaystyle scriptstyle d vec x
in a time interval dt::13-2
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 ,
displaystyle vec v text = text d vec x / text d t
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
displaystyle scriptstyle U( vec r )
is defined as that field whose gradient is equal and opposite to the
force produced at every point:
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.
Main article: Conservative force
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
displaystyle scriptstyle vec r
emanating from spherically symmetric potentials. Examples of this
displaystyle vec F =- frac Gm_ 1 m_ 2 vec r r^ 3
is the gravitational constant, and
displaystyle m_ n
is the mass of object n.
For electrostatic forces:
displaystyle vec F = frac q_ 1 q_ 2 vec r 4pi epsilon _
0 r^ 3
displaystyle epsilon _ 0
is electric permittivity of free space, and
displaystyle q_ n
is the electric charge of object n.
For spring forces:
displaystyle vec F =-k vec r
is the spring constant.
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
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
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
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
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.
Units of force
≡ 1 kg⋅m/s2
= 105 dyn
≈ 0.10197 kp
≈ 0.22481 lbf
≈ 7.2330 pdl
= 10−5 N
≡ 1 g⋅cm/s2
≈ 1.0197 × 10−6 kp
≈ 2.2481 × 10−6 lbf
≈ 7.2330 × 10−5 pdl
= 9.80665 N
= 980665 dyn
≡ gn ⋅ (1 kg)
≈ 2.2046 lbf
≈ 70.932 pdl
≈ 4.448222 N
≈ 444822 dyn
≈ 0.45359 kp
≡ gn ⋅ (1 lb)
≈ 32.174 pdl
≈ 0.138255 N
≈ 13825 dyn
≈ 0.014098 kp
≈ 0.031081 lbf
≡ 1 lb⋅ft/s2
The value of gn as used in the official definition of the
kilogram-force is used here for all gravitational units.
See also Ton-force.
See force gauge, spring scale, load cell
Orders of magnitude (force)
^ Newton's Principia Mathematica actually used a finite difference
version of this equation based upon impulse. See Impulse.
^ "It is important to note that we cannot derive a general expression
Newton's second law
Newton's second law for variable mass systems by treating the mass
in F = dP/dt = d(Mv) as a variable. [...] We can use F = dP/dt to
analyze variable mass systems only if we apply it to an entire system
of constant mass having parts among which there is an interchange of
mass." [Emphasis as in the original] (Halliday, Resnick & Krane
2001, p. 199)
^ "Any single force is only one aspect of a mutual interaction between
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Wikimedia Commons has media related to Forces (physics).
Look up force in Wiktionary, the free dictionary.
Video lecture on Newton's three laws by
Walter Lewin from MIT
A Java simulation on vector addition of forces
Force demonstrated as any influence on an object that changes the
object's shape or motion (video)
Classical mechanics SI units
distance: d, position: r, s, x, displacement
angle: θ, angular displacement: θ
solid angle: Ω
speed: v, velocity: v
kinematic viscosity: ν,
specific angular momentum: h
angular speed: ω, angular velocity: ω
angular acceleration: α
angular jerk: ζ
moment of inertia: I
momentum: p, impulse: J
kg m s−1, N s
action: 𝒮, actergy: ℵ
kg m2 s−1, J s
angular momentum: L, angular impulse: ΔL
kg m2 s−1
action: 𝒮, actergy: ℵ
kg m2 s−1, J s
force: F, weight: Fg
kg m s−2, N
energy: E, work: W
kg m2 s−2, J
torque: τ, moment: M
kg m2 s−2, N m
energy: E, work: W
kg m2 s−2, J
kg m s−3, N s−1
kg m2 s−3, W
kg m2 s−3, N m s−1
kg m2 s−3, W