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parallelogram In Euclidean geometry, a parallelogram is a simple (non- self-intersecting) quadrilateral with two pairs of parallel sides. The opposite or facing sides of a parallelogram are of equal length and the opposite angles of a parallelogram are of eq ...
of forces is a method for solving (or visualizing) the results of applying two
force In physics, a force is an influence that can change the motion of an object. A force can cause an object with mass to change its velocity (e.g. moving from a state of rest), i.e., to accelerate. Force can also be described intuitively as a ...
s to an object. When more than two forces are involved, the geometry is no longer parallelogrammatic, but the same principles apply. Forces, being vectors are observed to obey the laws of vector addition, and so the overall (resultant) force due to the application of a number of forces can be found geometrically by drawing vector arrows for each force. For example, see Figure 1. This construction has the same result as moving F2 so its tail coincides with the head of F1, and taking the net force as the vector joining the tail of F1 to the head of F2. This procedure can be repeated to add F3 to the resultant F1 + F2, and so forth.


Newton's proof


Preliminary: the parallelogram of velocity

Suppose a
particle In the physical sciences, a particle (or corpuscule in older texts) is a small localized object which can be described by several physical or chemical properties, such as volume, density, or mass. They vary greatly in size or quantity, from ...
moves at a uniform rate along a line from A to B (Figure 2) in a given time (say, one
second The second (symbol: s) is the unit of time in the International System of Units (SI), historically defined as of a day – this factor derived from the division of the day first into 24 hours, then to 60 minutes and finally to 60 seconds ea ...
), while in the same time, the line AB moves uniformly from its position at AB to a position at DC, remaining parallel to its original orientation throughout. Accounting for both motions, the particle traces the line AC. Because a displacement in a given time is a measure of
velocity Velocity is the directional speed of an object in motion as an indication of its rate of change in position as observed from a particular frame of reference and as measured by a particular standard of time (e.g. northbound). Velocity i ...
, the length of AB is a measure of the particle's velocity along AB, the length of AD is a measure of the line's velocity along AD, and the length of AC is a measure of the particle's velocity along AC. The particle's motion is the same as if it had moved with a single velocity along AC.


Newton's proof of the parallelogram of force

Suppose two
force In physics, a force is an influence that can change the motion of an object. A force can cause an object with mass to change its velocity (e.g. moving from a state of rest), i.e., to accelerate. Force can also be described intuitively as a ...
s act on a
particle In the physical sciences, a particle (or corpuscule in older texts) is a small localized object which can be described by several physical or chemical properties, such as volume, density, or mass. They vary greatly in size or quantity, from ...
at the origin (the "tails" of the
vector Vector most often refers to: *Euclidean vector, a quantity with a magnitude and a direction *Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism Vector may also refer to: Mathematic ...
s) of Figure 1. Let the lengths of the vectors F1 and F2 represent the velocities the two forces could produce in the particle by acting for a given time, and let the direction of each represent the direction in which they act. Each force acts independently and will produce its particular velocity whether the other force acts or not. At the end of the given time, the particle has ''both'' velocities. By the above proof, they are equivalent to a single velocity, Fnet. By
Newton's second law Newton's laws of motion are three basic laws of classical mechanics that describe the relationship between the motion of an object and the forces acting on it. These laws can be paraphrased as follows: # A body remains at rest, or in mo ...
, this vector is also a measure of the force which would produce that velocity, thus the two forces are equivalent to a single force.


Bernoulli's proof for perpendicular vectors

We model forces as Euclidean vectors or members of \mathbb^2 . Our first assumption is that the resultant of two forces is in fact another force, so that for any two forces \mathbf, \mathbf \in \mathbb^2 there is another force \mathbf \oplus \mathbf \in \mathbb^2 . Our final assumption is that the resultant of two forces doesn't change when rotated. If R: \mathbb^2\to \mathbb^2 is any rotation (any orthogonal map for the usual vector space structure of \mathbb^2 with \det R = 1), then for all forces \mathbf, \mathbf\in \mathbb^2 R \left( \mathbf \oplus \mathbf \right) = R \left(\mathbf \right) \oplus R \left(\mathbf \right) Consider two perpendicular forces \mathbf_1 of length a and \mathbf_2 of length b , with x being the length of \mathbf_1 \oplus \mathbf_2 . Let \mathbf_1 := \tfrac \left( \mathbf_1\oplus \mathbf_2 \right) and \mathbf_2 := \tfrac R(\mathbf_2), where R is the rotation between \mathbf_1 and \mathbf_1 \oplus \mathbf_2 , so \mathbf = \tfrac R \left(\mathbf_1 \right) . Under the invariance of the rotation, we get \mathbf_1=\fracR^ \left(\mathbf_1 \right) = \fracR^ \left(\mathbf_1\oplus\mathbf_2 \right)=\fracR^ \left(\mathbf_1 \right)\oplus\fracR^ \left(\mathbf_2 \right)=\mathbf_1 \oplus \mathbf_2 Similarly, consider two more forces \mathbf_1 := -\mathbf_2 and \mathbf_2 := \tfrac \left( \mathbf_1 \oplus \mathbf_2 \right) . Let T be the rotation from \mathbf_1 to \mathbf_1 : \mathbf_1 = \tfrac T\left(\mathbf_1\right) , which by inspection makes \mathbf_2 =\tfrac T\left(\mathbf_2\right) . \mathbf_2 = \fracT^\left(\mathbf_2\right) = \fracT^\left(\mathbf_1\oplus\mathbf_2\right)=\fracT^\left(\mathbf_1\right)\oplus\fracT^\left(\mathbf_2\right)=\mathbf_1\oplus\mathbf Applying these two equations \mathbf_1\oplus \mathbf_2 = \left(\mathbf_1 \oplus \mathbf_2 \right) \oplus \left(\mathbf_1\oplus \mathbf \right) = \left(\mathbf_1 \oplus \mathbf_2 \right) \oplus \left(-\mathbf_2\oplus \mathbf_2 \right) = \mathbf_1 \oplus \mathbf_2 Since \mathbf_1 and \mathbf_2 both lie along \mathbf_1 \oplus \mathbf_2 , their lengths are equal x= \left, \mathbf_1 \oplus \mathbf_2 \ = \left, \mathbf_1 \oplus \mathbf_2 \= \tfrac+\tfrac x = \sqrt which implies that \mathbf_1 \oplus \mathbf_2 = a \mathbf_1 \oplus b \mathbf_2 has length \sqrt , which is the length of a \mathbf_1 + b \mathbf_2 . Thus for the case where \mathbf_1 and \mathbf_2 are perpendicular, \mathbf_1 \oplus \mathbf_2 = \mathbf_1 + \mathbf_2 . However, when combining our two sets of auxiliary forces we used the associativity of \oplus . Using this additional assumption, we will form an additional proof below.


Algebraic proof of the parallelogram of force

We model forces as Euclidean vectors or members of \mathbb^2 . Our first assumption is that the resultant of two forces is in fact another force, so that for any two forces \mathbf, \mathbf \in \mathbb^2 there is another force \mathbf \oplus \mathbf \in \mathbb^2 . We assume commutativity, as these are forces being applied concurrently, so the order shouldn't matter \mathbf \oplus \mathbf = \mathbf \oplus \mathbf . Consider the map


Controversy

The mathematical proof of the parallelogram of force is not generally accepted to be mathematically valid. Various proofs were developed (chiefly ''Duchayla's'' and '' Poisson's''), and these also caused objections. That the parallelogram of force was true was not questioned, but ''why'' it was true. Today the parallelogram of force is accepted as an empirical fact, non-reducible to Newton's first principles.


See also

* Newton's ''Mathematical Principles of Natural Philosophy'', Axioms or Laws of Motion, Corollary I, at
Wikisource Wikisource is an online digital library of free-content textual sources on a wiki, operated by the Wikimedia Foundation. Wikisource is the name of the project as a whole and the name for each instance of that project (each instance usually re ...
*
Vector (geometric) In mathematics, physics, and engineering, a Euclidean vector or simply a vector (sometimes called a geometric vector or spatial vector) is a geometric object that has magnitude (or length) and direction. Vectors can be added to other vectors ...
*
Net force Net Force may refer to: * Net force, the overall force acting on an object * ''NetForce'' (film), a 1999 American television film * Tom Clancy's Net Force, a novel series * Tom Clancy's Net Force Explorers, a young adult novel series {{disam ...


References

{{DEFAULTSORT:Parallelogram Of Force Force Vector calculus