Force

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For other senses of this word, see force (disambiguation).

In physics, the Newtonian definition of force is that which when acting alone causes a rate of change of momentum.

Contents

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Defining force by what it does

According to Newton's First Law, assuming a non-accelerating, non-rotating and locally flat coordinate system, an object initially not moving will continue to not move. Similarly, an object that is in motion with a given velocity will continue with that velocity.

If there is a change in this motion (or lack thereof), we must deduce it has done so under the influence of a force.

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Describing forces as particle exchanges

On a subatomic level, forces are the result of the exchange of real particles by virtual (mediating) particles. Change of momentum of real particles in those exchanges (as mathematically required by conservation of momentum) is what we observe as a force.

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Fundamental forces

Currently there are three known fundamental forces in nature.

Quantum field theory has been successful in describing the electroweak force, and to a certain extent the strong nuclear force. Gravity on a large scale is described by general relativity.

The three fundamental forces describe every observable phenomenon. All forces, except exchange forces and zero energy pressure force (virtual particles pressure) can be reduced to these fundamental interactions - including magnetism, friction , tension, part of chemical bonding and part of contact force

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Conservative and nonconservative forces

The energy of a system depends on the forces acting on and within that system. A force is called conservative if the path integral of this force (work) over closed loop is zero.

Conservative forces are mathematically equivalent to the gradient of some potential. Examples include gravity, Coulomb force and ideal spring force.

Nonconservative forces include friction, viscosity and drag. Strictly speaking, for any sufficiently sophisticated description, all forces are conservative.

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Quantitative definition

The mathematical definition of force, proposed by Isaac Newton, is:

<math>

\mathbf{F} = \frac{d\mathbf{p}}{dt} </math>

In the case where m does not depend on t this definition results in well known second Newton's law:

<math>

\mathbf{F} = \frac{d\mathbf{p}}{dt}= \frac{d(m\mathbf{v})}{dt} = m\mathbf{a} </math>

where:

<math>\mathbf{F}</math> is the force (being a derivative of a vector is also a vector),
<math>\mathbf{p}</math> is the momentum,
<math>t</math> is the time,
<math>\mathbf{v}</math> is the velocity,
<math>m</math> is the mass, and
<math>\mathbf{a} = \frac{d^2\mathbf{r}}{d^2t}</math> is the acceleration, the second derivative with respect to t of the position vector <math>\mathbf{r}</math>.

If the mass m is measured in kilograms and the acceleration a is measured in metres per second squared, then the unit of force is kilogram × metre/second squared. This unit is called the newton: 1 N = 1 kg x 1 m/s².

This equation is a system of three (by the number of spatial dimensions involved) second-order differential equations with respect to the three-dimensional position vector r which is an unknown function of time. It is often reduced to 2 dimensions (planar motion) or 1 dimension (linear motion and circular mtion) by the proper choice of coordinate system, which is usually aligned with the force and the initial velocity vector. This equation can be solved if F is a known function of r and some of its derivatives and if the mass m is known. Moreover, the boundary conditions are required; for example, the values of the position vector and r and the velocity v at the starting time, say t=0.

To measure force, a known mass (say, 10 kg) is accelerated in free fall and the acceleration is measured (say, 9.8 m/sec^2). Therefore the gravitational force on the mass is equal 98 kg m/sec^2 = 98 Newtons. Knowing the force on the mass, now the mass can be suspended on a spring and let to stretch the spring till the spring stops moving any futher. Lack of motion of the mass on the spring indicates that now mass undegoes exactly equal accelerations down and up. Thus the force of pull of the spring must be exactly equal to the gravitational force 98 Newtons (and directed exactly opposite - up). This way the spring can be calibrated (=added a scale marks in Newtons showing how much force the spring exerts at sertain stretch) and thus used to measure gravitational force on different mass or even other forces. Such spring with scale is usually called spring scale or Newtonmeter and is widely used in measuring various forces in introductory physics classes as well as in research labs. In a similar way balance is calibrated and used to measure forces.

If acceleration and force are known, scale and balance can be used to measure mass. For example, bathroom scale which is intended to be used on surface of Earth (so, acceleration is known to be a=g=9.80 m/s^2) usually is calibrated both in units of force (lb) and in units of mass (kg). (Obviousely, far from surface of Earth (say, on Moon) force scale of bathroom scale will still be correct but the mass scale needs to be re-calibrated according to local value of acceleration g)

Force being time derivative of momentum is obviousely not a fundamental quantity. The two most fundamental theories of nature - quantum electrodynamics and general relativity - do not contain the concept of force at all, because it is redundant to the conservation of momentum.

Although not the most fundamental quantity in physics, force is an important intermediate concept from which other concepts, such as work, and therefore energy and some others such as pressure, are derived. Force is sometimes confused with stress.

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Properties of force

Because momentum is a vector, then force, being its time derivative, is also a vector - it has magnitude and direction.

Forces can be added together using the parallelogram of force. 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. If the two forces are equal but opposite, then the resultant is zero. This condition is called static equilibrium, with the result that the object remains at rest or moves with a constant velocity.

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.

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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 field U(r) is defined as that field whose gradient is equal and opposite to the force produced at every point:

<math>\textbf{F}=-\nabla U</math>

The derivative of force with respect to time is called yank. Higher-order derivatives are rarely used because in most cases the relationship between second-order, first-order and zero-order derivatives (along with initial and boundary conditions) completely and uniquely defines the behaviour of a physical system.

In most explanations of mechanics, force is usually defined only implicitly, in terms of the equations that work with it. Some physicists, philosophers and mathematicians, such as Ernst Mach, Clifford Truesdell and Walter Noll, have found this problematic and sought a more explicit definition of force.

According to the special theory of relativity, which is important if the speed of the body gets close to the speed of light, the definition of force results in the following. If we choose the coordinate system such that the body is moving along the <math>x</math> direction, the relation between the force and the acceleration is

<math>F_x = \gamma^3 m a_x \, </math>
<math>F_y = \gamma m a_y \, </math>
<math>F_z = \gamma m a_z \, </math>

where

<math>\gamma={1 \over {\sqrt{1-{{v^2} \over\ {c^2}}}}}</math>
<math>v \,</math> is the velocity of the body
<math>c \, </math> is the speed of light.

According to these equations (plus the mathematical definition of energy), an object with nonzero rest mass cannot be accelerated to the speed of light as this would require an infinite amount of work.

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Units of measurement

The SI unit used to measure force is the newton (symbol N), which is equivalent to kg·m·s−2.

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Non-SI units of force and mass

The F=m·a relationship can be used with any consistent units (SI or CGS). If these units are not consistent, a more general form, F=k·m·a, can be used, where the constant k is a conversion factor dependent upon the units being used.

For example, in imperial engineering units, F is measured in "pounds force" or "lbf", m in "pounds mass" or "lb", and a in feet per second squared. In this particular system, one needs to use the more general form above, usually written F=m·a/gc with the constant normally used for this purpose gc = 32.174 lb·ft/(lbf·s2) equal to the reciprocal of the k above.

As with the kilogram, the pound is colloquially used as both a unit of mass and a unit of force. 1 lbf is the force required to accelerate 1 lb at 32.174 ft per second squared, since 32.174 ft per second squared is the standard acceleration due to terrestrial gravity.

Another imperial unit of mass is the slug, defined as 32.174 lb. It is the mass that accelerates by one foot per second squared when a force of one lbf is exerted on it.

When the standard gee (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 gee 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 disintinguish 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.

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Conversions

Below are several coversion factors between various mesurements of force:

  • 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
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Forces in everyday life

Forces are part of everyday life, with examples such as:

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Forces in the laboratory

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Founding experiments

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Instruments to measure forces

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History

Force was first described by Archimedes. Newton is credited to give first mathematical definition to force.

Current fundamental theories (such as quantum mechanics, quantum electrodynamics, general relativity) do not have concept of force - because as seen from the definition, force is redundant to conservation of momentum and energy (see conservation laws).

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See also

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References

  • {{cite book 
| last = Halliday | first = David
| coauthors = Robert Resnick; Kenneth S. Krane
| title = Physics v. 1
| location = New York
| publisher = John Wiley & Sons
| year = 2001
| id = ISBN 0471320579
}}
  • {{cite book 
| last = Serway | first = Raymond A.
| title = Physics for Scientists and Engineers
| location = Philadelphia
| publisher = Saunders College Publishing
| year = 2003
| id = ISBN 0534408427
}}
  • {{cite book 
| last = Tipler | first = Paul
| title = Physics for Scientists and Engineers: Mechanics, Oscillations and Waves, Thermodynamics
| edition = 5th ed.
| publisher = W. H. Freeman
| year = 2004
| id = ISBN 0716708094
}}


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