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In
physics, 'force' is anything that can cause a
massive body to
accelerate. It may be experienced as a lift, a push, or a pull. The acceleration of the body is proportional to the
vector sum of all forces acting on it (known as
net force or resultant force). In an extended body, force may also cause
rotation,
deformation, or an increase in
pressure for the body. Rotational effects are determined by the
torques, while deformation and pressure are determined by the
stresses that the forces create.
Net force is mathematically equal to the rate of change of the
momentum of the body on which it acts. Since
momentum is a
vector quantity (has both a magnitude and direction), force also is a vector quantity.
The concept of force has formed part of
statics and
dynamics since ancient times. Ancient contributions to statics culminated in the work of
Archimedes in the
3rd century BC, which still forms part of modern physics. In contrast,
Aristotle's dynamics incorporated intuitive misunderstandings of the role of force which were eventually corrected in the
17th century, culminating in the work of
Isaac Newton. Following the development of
quantum mechanics it is now understood that particles influence each another through
fundamental interactions, making force a useful concept only on the macroscopic level. Only four
fundamental interactions are known:
strong,
electromagnetic,
weak (unified into one
electroweak interaction in 1970s), and
gravitational (in order of decreasing strength).
History
Aristotle and his followers believed that it was the ''natural state'' of objects on
Earth to be motionless and that they tended towards that state if left alone. He distinguished between the innate tendency of objects to find their "natural place" (e.g. for heavy bodies to fall), which lead to "natural motion", and unnatural or forced motion, which required continued application of a force. But this
theory, although based on the everyday experience of how objects move (e.g. a horse and cart), had severe trouble accounting for projectiles, such as the flight of arrows. Several theories were discussed over the centuries, and the late medieval idea that objects in forced motion carried an innate force of
impetus was influential on the work of
Galileo. Galileo constructed an experiment in which stones and cannonballs were both rolled down an incline to disprove the
Aristotelian theory of motion early in the
17th century. He showed that the bodies were accelerated by gravity to an extent which was independent of their
mass and argued that objects retain their
velocity unless acted on by a force - usually
friction.
Isaac Newton is recognised as having argued explicitly for the first time that, in general, a constant force causes a constant rate of change (
time derivative) of momentum. In essesse, he gave the first (and the only) mathematical definition of the quantity force itself - as being the time-derivative of momentum: F = dp/dt.
In
1784 Charles Coulomb discovered the
inverse square law of interaction between
electric charges using a
torsion balance, which was the second fundamental force. The weak and strong forces were discovered in the
20th century.
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 QED). Thus currently known
fundamental forces are more accurately called “
fundamental interactions”.
Types of force
Although there are apparently many types of forces in the Universe, they are all based on four fundamental forces. The strong and weak forces only act at very short distances and are responsible for holding certain
nucleons and compound
nuclei together. The electromagnetic force acts between
electric charges and the gravitational force acts between
masses. The
Pauli exclusion principle is responsible for the tendency of
atoms not to overlap each other, and is thus responsible for the "stiffness" or "rigidness" of matter, but this also depends on the electromagnetic force which binds the constituents of every atom.
All other forces are based on these four. 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 allow atoms to pass through each other. The forces in
springs modeled by
Hooke's law are also the result of electromagnetic forces and the exclusion principle acting together to return the object to its equilibrium position.
Centrifugal forces are acceleration forces which arise simply from the acceleration of rotating
frames of reference.
There is currently some debate to whether there are five forces not four, due to the discovery of
dark energy, which could be just an energy of vacuum fluctuations, or it could be a new type of energy resulting in a repulsive force.
The modern quantum mechanical view of the first three fundamental forces (all except gravity) is that particles of matter (
fermions) do not directly interact with each other but rather by exchange of
virtual particles (
bosons). This exchange results in what we call
electromagnetic interaction (
Coulomb force is one example of electromagnetic interaction).
In
general relativity, gravitation is not viewed as a force. Rather, objects moving freely in gravitational fields simply undergo
inertial motion along a
straight line in the
curved space-time - defined as the shortest space-time path between two space-time points. This straight line 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 shape as it is in a uniform gravitational field. Its space-time trajectory (when the extra ct dimension is added) 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 body is what we label as "gravitational force".force is one of the best things to ever be mad
Examples
★ A heavy object is in free fall. Its momentum changes as dp/dt = mdv/dt = ma =mg (if the mass m is constant), thus we call the quantity mg a "gravitational force" acting on the object. This is the
definition of
weight (w=mg) of an object.
★ A heavy object on a table is pulled (attracted) downward toward the floor by the force of gravity (i.e., its weight). At the same time, the table resists the downward force with equal upward force (called the
normal force), resulting in zero net force, and no acceleration. (If the object is a person, he actually feels the normal force acting on him from below.)
★ A heavy object on a table is gently pushed in a sideways direction by a finger. However, it doesn't move because the force of the finger on the object is now opposed by a ''new'' force of
static friction, generated between the object and the table surface. This newly generated force ''exactly'' balances the force exerted on the object by the finger, and ''again'' no acceleration occurs. The static friction increases or decreases automatically. If the force of the finger is increased (up to a point), the opposing sideways force of static friction 'increases' exactly to the point of perfect opposition.
★ A heavy object on a table is pushed by a finger hard enough that static friction cannot generate sufficient force to match the force exerted by the finger, and the object starts sliding across the surface. If the finger is moved with a constant velocity, it needs to apply a force that exactly cancels the force of
kinetic friction from the surface of the table and then the object moves with the same constant velocity. Here it seems to the naive observer that application of a force produces a velocity (rather than an acceleration). However, the velocity is constant only because the force of the finger and the kinetic friction cancel each other. Without friction, the object would continually accelerate in response to a constant force.
★ A heavy object reaches the edge of the table and falls. Now the object, subjected to the constant force of its weight, but freed of the normal force and friction forces from the table, gains in velocity in direct proportion to the time of fall, and thus (before it reaches velocities where air resistance forces becomes significant compared to gravity forces) its rate of ''gain'' in momentum and velocity is constant. These facts were first discovered by
Galileo.
★ A heavy object is suspended on a
spring scale. Because object is not moving (so time derivative of its
momentum is zero) then along with acceleration of free fall g it has to experience equal and oppositely directed acceleration a=-g due to the action of the spring. This acceleration multiplied by the mass of the object is what we label as "spring reaction force" which is obviousely equal and opposite to object's weight mg. Knowing the mass (say, 1 kg) and the acceleration of free fall (say, 9.80 m/s^2) we can calibrate spring scale by a mark "9.8 N". Attaching various masses (2 kg, 3 kg ...) we can calibrate spring scale and then use this calibrated scale to measure many other forces (friction, reaction forces, elecric force, magnetic force, etc).
Quantitative definition
We have an intuitive grasp of the notion of force, since forces can be directly perceived as a push or pull. As with other physical concepts (e.g.
temperature), the intuitive notion is quantified using
operational definitions that are consistent with direct perception, but more precise. Historically, forces were first quantitatively investigated in conditions of
static equilibrium where several forces cancelled each other out. Such experiments prove the crucial properties that forces are additive
vector quantities: they have
magnitude and
direction. So, when two forces act on an object, the resulting force, the ''resultant'', is the
vector sum of the original forces. This is called the principle of
superposition. 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. As with all vector addition this results in a
parallelogram rule: the addition of two vectors represented by sides of a parallelogram, gives an equivalent resultant vector which is equal in magnitude and direction to the transversal of the parallelogram.
As well as being added, forces can also be broken down (or 'resolved'). For example, a horizontal force pointing northeast can be split into two forces, one pointing north, and one pointing east. Summing these component forces using vector addition yields the original force. Force vectors can also be three-dimensional, with the third (vertical) component at right-angles to the two horizontal components.
The simplest case of static equilibrium is when two forces are equal in magnitude but opposite in direction. This remains the most usual way of measuring forces, using simple devices such as
weighing scales and
spring balances. Using such tools, several quantitative force laws were discovered: that the force of gravity is proportional to volume for objects made of a given material (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: all these were all formulated and experimentally verified before
Isaac Newton expounded his three laws of motion.
Force is sometimes defined using Newton's second law, as the product of
mass times
acceleration :
:
(or, more generally, as the rate of change of momentum). This approach is disparaged by the large majority of textbooks.
[1] By making this a definition of force, all empirical content is removed from the law. In fact, the
in this equation represents the net (vector sum) force; in static equilibrium this is zero by definition, but (balanced) forces are present nevertheless. Instead, Newton's law is meaningful because it asserts the proportionality of two quantities which can be defined without reference to it. Thus, the intuitive Aristotelian belief that a net force is required to keep an object moving with constant velocity (therefore zero acceleration) is objectively wrong and not just a consequence of a poor choice definition. With rather more justification, Newton's second law can be taken as a quantitative definition of ''mass''; certainly, by writing the law as an equality, the relative units of force and mass are fixed.
Given the empirical success of Newton's law, it is sometimes used to measure the strength of forces (for instance, using astronomical orbits to determine gravitational forces). Nevertheless, the force and the (mass times acceleration) used to measure it remain distinct concepts.
The definition of force is sometimes regarded as problematic, since it must either ultimately be referred to our intuitive understanding of our direct perceptions, or be defined implicitly through a set of self-consistent mathematical formulae. Notable physicists, philosophers and mathematicians who have sought a more explicit definition include
Ernst Mach,
Clifford Truesdell and
Walter Noll.
[2]
Force in special relativity
In the
special theory of relativity mass and
energy are equivalent (as can be seen by calculating the work required to accelerate a body). When an object's velocity increases so does its energy and hence its mass equivalent (inertia). It thus requires a greater force to accelerate it the same amount than it did at a lower velocity. The definition
remains valid. But in order to be conserved, momentum must be redefined as:
:
where
:
is the velocity and
:
is the
speed of light.
The relativistic expression relating force and acceleration for a particle with non-zero
rest mass moving in the
direction is:
:
:
:
where the
Lorentz factor
:
Here a constant force does not produce a constant acceleration, but an ever decreasing acceleration as the object approaches the speed of light. Note that
is
undefined for an object with a non zero
rest mass at the speed of light, and the theory yields no prediction at that speed.
One can however restore the form of
:
for use in relativity through the use of
four-vectors. This relation is correct in relativity when
is the
four-force, m is the
invariant mass, and
is the
four-acceleration.
Force and potential
Instead of a force, the mathematically equivalent concept of a
potential energy field can be used for convenience. For instance, the gravitational force acting upon a body can be seen as the action of the
gravitational field that is present at the body's location. Restating mathematically the definition of
energy (via definition of
work), a potential
scalar field is defined as that field whose
gradient is equal and opposite to the force produced at every point:
:
Forces can be classified as
conservative or nonconservative. Conservative forces are equivalent to the
gradient of a
potential.
Conservative forces
Main articles: 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, therefore, have models which are dependent on a position often given as a
radial vector eminating from
spherically symmetric potentials. Examples of this follow:
For gravity:
:
where
is the
gravitational constant,
is the mass of object ''n''.
For electrostatic forces:
:
where
is
electric permittivity of free space,
is the
electric charge of object ''n''.
For spring forces:
:
where
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 which 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 which 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 non-conservative 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 used to measure force is the
newton (symbol N), which is equivalent to kg·m·s
−2. The earlier
CGS unit is the
dyne. The relationship 'F'=''m''·'a' can be used with either of these. In
Imperial engineering units, if ''F'' is measured in "
pounds force" or "lbf", and ''a'' in feet per second squared, then ''m'' must be measured in
slugs. Similarly, if mass is measured in
pounds mass, and ''a'' in feet per second squared, the force must be measured in
poundals. The units of
slugs and
poundals are specifically designed to avoid a constant of proportionality in this equation.
A more general form 'F'=''k''·''m''·'a' is needed if consistent units are not used. Here, the constant ''k'' is a conversion factor dependent upon the units being used.
When the standard
''g'' (an acceleration of 9.80665 m/s²) is used to define pounds force, the mass in pounds is numerically equal to the weight in pounds force. However, even at sea level on Earth, the actual acceleration of free fall is quite variable, over 0.53% more at the poles than at the equator. Thus, a mass of 1.0000 lb at ''sea level'' at the equator exerts a force due to gravity of 0.9973 lbf, whereas a mass of 1.000 lb at ''sea level'' at the poles exerts a force due to gravity of 1.0026 lbf. The normal average sea level acceleration on Earth (World Gravity Formula 1980) is 9.79764 m/s², so on average at ''sea level'' on Earth, 1.0000 lb will exerts a force of 0.9991 lbf.
The equivalence 1 lb = 0.453 592 37 kg is always true, by definition, anywhere in the universe. If you use the standard
''g'' which is official for defining kilograms force to define pounds force as well, then the same relationship will hold between pounds-force and kilograms-force (an old non-SI unit is still used). If a different value is used to define pounds force, then the relationship to kilograms force will be slightly different—but in any case, that relationship is also a constant anywhere in the universe. What is not constant throughout the universe is the amount of force in terms of pounds-force (or any other force units) which 1 lb will exert due to gravity.
By analogy with the slug, there is a rarely used unit of mass called the "metric slug". This is the mass that accelerates at one metre per second squared when pushed by a force of one
kgf. An item with a mass of 10 kg has a mass of 1.01972661 metric slugs (= 10 kg divided by 9.80665 kg per metric slug). This unit is also known by various other names such as the
hyl, TME (from a German acronym), and mug (from metric slug).
Another unit of force called the
poundal (pdl) is defined as the force that accelerates 1 lbm at 1 foot per second squared. Given that 1 lbf = 32.174 lb times one foot per second squared, we have 1 lbf = 32.174 pdl.
The
kilogram-force is a unit of force that was used in various fields of science and technology. In 1901, the
CGPM improved the definition of the kilogram-force, adopting a standard acceleration of gravity for the purpose, and making the kilogram-force equal to the force exerted by a mass of 1 kg when accelerated by 9.80665 m/s². The kilogram-force is not a part of the modern
SI system, but is still used in applications such as:
★ Thrust of
jet and
rocket engines
★ Spoke tension of
bicycles
★ Draw weight of
bows
★
Torque wrenches in units such as "meter kilograms" or "kilogram centimetres" (the kilograms are rarely identified as units of force)
★ Engine torque output (kgf·m expressed in various word orders, spellings, and symbols)
★ Pressure gauges in "kg/cm²" or "kgf/cm²"
In colloquial, non-scientific usage, the "kilograms" used for "weight" are almost always the proper SI units for this purpose. They are units of mass, not units of force.
The symbol "kgm" for kilograms is also sometimes encountered. This might occasionally be an attempt to distinguish kilograms as units of mass from the "kgf" symbol for the units of force. It might also be used as a symbol for those obsolete torque units (kilogram-force metres) mentioned above, used without properly separating the units for kilogram and metre with either a space or a centered dot.
Conversions
Below are several conversion factors between various measurements of force:
★ 1 dyne = 10
-5 newtons
★ 1 kgf (kilopond kp) = 9.80665 newtons
★ 1
metric slug = 9.80665 kg
★ 1 lbf = 32.174 poundals
★ 1 slug = 32.174 lb
★ 1 kgf = 2.2046 lbf
See also
References
1. e.g. Lectures on Physics, Vol 1, Feynman, R. P., Leighton, R. B., Sands, M., , , Addison-Wesley, 1963, Sect 12.1; An introduction to mechanics, Kleppner, D., Kolenkow, R. J., , , McGraw-Hill, , Sect 2.1; Sears & Zemansky's University Physics, Young, H.D., Freedman, R.A., , , Pearson Addison-Wesley, 2004, Sect 4.3.
2. e.g. W. Noll, “On the Concept of Force”, in part B of Walter Noll's website..
★
Encyclopedia of Physics, p 443,, , Sybil, Parker, McGraw-Hill, 1993, ISBN 0-07-051400-3
★
Classical Mechanics p 28,, , H.C., Corbell, Dover publications, 1994, ISBN 0-486-68063-0
★
Physics v. 1, , David, Halliday, John Wiley & Sons, 2001, ISBN 0-471-32057-9
★
Physics for Scientists and Engineers, , Raymond A., Serway, Saunders College Publishing, 2003, ISBN 0-534-40842-7
★
Physics for Scientists and Engineers: Mechanics, Oscillations and Waves, Thermodynamics, , Paul, Tipler, W. H. Freeman, 2004, ISBN 0-7167-0809-4
★
Concepts of Physics Vol 1., , H.C., Verma, Bharti Bhavan, 2004, ISBN 81-7709-187-5
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