How to Calculate Equilibrant & Resultant

Updated April 17, 2017

Students encounter the equation f=ma soon after they begin to study physics. If an object experiences a net force, it will experience a corresponding acceleration proportional to the magnitude of that net force. When multiple forces simultaneously act upon the object, you'll need to add the different forces together to get the resultant force. A force identical in magnitude but in the opposite direction is the equilibrant force.

Identify all the different forces acting upon the object. These are vector quantities, which means they will have both magnitude and direction. Some forces, such as air resistance, may be less obvious than others.

Add all the different vectors together, using the rules for vector addition. If all the different forces are in the same direction, or exactly in the opposite direction, then these calculations are elementary. If someone is exerting 4.54kg. of force pushing a lead weight up while gravity is exerting 2.27kg. of pressure pushing it down, the net force is clearly 2.27kg. of force upward. If you are exerting 4.54kg. of force pushing a car north, and someone else is pushing in the same direction with 2.27kg. of force, the net force is 6.8kg. of force in the north direction.

Construct a right triangle to represent two different forces acting at right angles. If you are exerting 0.454kg. of force pushing an object north and someone else is exerting 0.454kg. of pressure pushing the same object east, then two of the sides of the triangle are each 1 unit long. The hypotenuse of this triangle represents the vector addition of the two forces. The Pythagorean theorem tells us that the square of the hypotenuse is equal to the sum of the squares of the two sides, so since each side is 1 unit long and its square is also 1, the hypotenuse equals the square root of 2 or 1.41421356. Elementary trigonometry will also reveal the angle between the hypotenuse and the two sides, which indicates the direction of the resultant force. For this example, that's 45 degrees. The object would experience a resultant force in the northeast direction.

Remember that you can represent any force as the sum of two different vectors. If an object is experiencing two different forces, but they are not at right angles to one another, represent one of the forces as a sum of two vectors, one in the same direction as the other force, and the other at a right angle to it. If one force is pushing an object due north and another is pushing it directly northeast, represent the northeast force as a north vector and an east vector. Simply add the two north vectors and use the Pythagorean theorem to add the east vector.

Take the sum of all the vectors. That's the resultant force. A force equal in quantity but oriented in the exact opposite direction is the equilibrant force.

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About the Author

Andrew Breslin has been writing professionally since 1994. His articles and op-ed pieces have appeared in the "South Florida Sun Sentinel," "St Paul Pioneer Press," "Detroit Free Press," "Charlotte Observer," “Good Medicine,” and others. He studied molecular biology at Westchester University and frequently writes about science and mathematics.