mechanics Mechanics () is the area of physics concerned with the relationships between force, matter, and motion among Physical object, physical objects. Forces applied to objects may result in Displacement (vector), displacements, which are changes of ...

, an equilibrant force is a

force In physics, a force is an influence that can cause an Physical object, object to change its velocity unless counterbalanced by other forces. In mechanics, force makes ideas like 'pushing' or 'pulling' mathematically precise. Because the Magnitu ...

which brings a body into

mechanical equilibrium In classical mechanics, a particle is in mechanical equilibrium if the net force on that particle is zero. By extension, a physical system made up of many parts is in mechanical equilibrium if the net force on each of its individual parts is ze ...

. According to

Newton's second law Newton's laws of motion are three physical laws that describe the relationship between the motion of an object and the forces acting on it. These laws, which provide the basis for Newtonian mechanics, can be paraphrased as follows: # A body re ...

, a body has zero

acceleration In mechanics, acceleration is the Rate (mathematics), rate of change of the velocity of an object with respect to time. Acceleration is one of several components of kinematics, the study of motion. Accelerations are Euclidean vector, vector ...

when the

vector sum 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. Euclidean vectors can be added and scal ...

of all the forces acting upon it is zero: :

\sum \mathbf F = m \mathbf a; \quad \sum \mathbf F = 0 \ \  \Rightarrow \ \  \mathbf a = 0

Therefore, an equilibrant force is equal in magnitude and opposite in direction to the

resultant In mathematics, the resultant of two polynomials is a polynomial expression of their coefficients that is equal to zero if and only if the polynomials have a common root (possibly in a field extension), or, equivalently, a common factor (over th ...

of all the other forces acting on a body. The term has been attested since the late 19th century.

Example

Suppose that two known forces, which are going to represented as vectors, A and B are pushing an object and an unknown equilibrant force, C, is acting to maintain that object in a fixed position. Force A points to the west and has a magnitude of 10 N and is represented by the vector <-10, 0>N. Force B points to the south and has a magnitude of 8.0 N and is represented by the vector <0, -8>N. Since these forces are vectors, they can be added by using the parallelogram rule or

vector addition Vector most often refers to: * Euclidean vector, a quantity with a magnitude and a direction * Disease vector, an agent that carries and transmits an infectious pathogen into another living organism Vector may also refer to: Mathematics a ...

. This addition will look like A + B = <-10, 0>N + <0, -8>N = <-10, -8>N which is the vector representation of the resultant force. By the

Pythagorean theorem In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposite t ...

, the magnitude of the resultant force is -10)² + (-8)²sup>1/2 ≈ 12.8 N, which is also the magnitude of the equilibrant force. The angle of the equilibrant force can be found by

trigonometry Trigonometry () is a branch of mathematics concerned with relationships between angles and side lengths of triangles. In particular, the trigonometric functions relate the angles of a right triangle with ratios of its side lengths. The fiel ...

to be approximately 51 degrees north of east. Because the angle of the equilibrant force is opposite of the resultant force, if 180 degrees are added or subtracted to the resultant force's angle, the equilibrant force's angle will be known. Multiplying the resultant force vector by a -1 will give the correct equilibrant force vector: <-10, -8>N x (-1) = <10, 8>N = C.

References

External links

Equilibrium
Force {{classicalmechanics-stub