A force is a vector represented by an arrow on a force diagram. The arrow points in the direction of the force and its length indicates the magnitude (size) of the force. Vector addition must be used to add a group of forces together to get the total (resultant) force. For forces lying in the same plane, the trigonometric ratios cosine (cos) and sine (sin) are used to find the horizontal (x) and vertical (y) components for each force. Adding these components gives the final x- and y-components of the total force, from which the magnitude of the resultant force is calculated.

- Skill level:
- Challenging

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### Things you need

- Understanding of trigonometry and vector math
- Scientific calculator
- Paper and pencil

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## Instructions

- 1
Write down the magnitude (size) of each force and the angle the force makes with the horizontal. Use zero degrees to indicate a horizontal force pointing to the right. Measure non-zero angles by starting at zero degrees and rotating away from the horizontal in a anticlockwise direction. For example, forces pointing straight up, horizontally to the left and straight down have the respective angles of 90, 180 and 270 degrees.

For our example, we will find the magnitude of the resultant force arising from three individual forces: F1, F2 and F3. Their respective magnitudes are 20, 40 and 15 pounds and their respective angles, A1, A2 and A3, are 30, 160 and 230 degrees.

- 2
Calculate the x-component (Fx = F cos A ) and y-component (Fy = F sin A) for each force. Use a scientific calculator to find the cos and sin values of each angle. Record the Fx and Fy values for each force.

In our example, the components for F1 are F1x = F1 cos A1 = 20 cos 30 = 20 (0.8660) = 7.86kg. and F1y = F1 sin A1 = 20 sin 30 = 20 ( 0.5) = 4.54kg. Similarly, F2x = 40 cos 160 = - 17.1kg., F2y = 40 sin 160 = 6.21kg., F3x = 15 cos 230 = - 4.37kg. and F3y = 15 sin 230 = - 5.21kg.

- 3
Add all of the x-components together to get the total x-component (Rx) and add all the y-components to get the total y-component (Ry) of the resultant force.

In our example, we get Rx = F1x + F2x + F3x = 17.32 + (- 37.59) + (- 9.64 ) = -13.6kg and Ry = F1y + F2y +F3y = 10.00 + 13.68 + (- 11.49) = 5.53kg.

- 4
Calculate Rx(Rx) + Ry(Ry) and take the square root of the result to get the magnitude of the resultant force. The resultant has the same units as your force vectors.

In general, the magnitude (R) of the resultant is R = square root (Rx-squared + Ry-squared).

For our example, we get Rx(Rx) +Ry(Ry) = (-29.91)(-29.91) +(12.19)(12.19) = 1043.20. The square root is 32.30, so the magnitude of the resultant force is 14.7kg.

#### Tips and warnings

- The angle of the resultant can be found by calculating A = inverse tan of Ry/Rx. If A is positive and Rx and Ry are positive, then the angle of the resultant force is A. If A is positive, but Rx and Ry are both negative, the angle is A +180 degrees. If A is negative with a positive Rx and a negative Ry, the angle is A + 360. If A is negative with a negative Rx and a positive Ry, the angle is A + 180 degrees. For our example, A = -22.2 and the direction of the resultant force is -22.2 + 180 = 157.8 degrees.
- Make sure your calculator is set to degrees (usually the default) and not radians for the angle. Check this by calculating sin 90 on your calculator. If the answer is 1, your calculator is set to degrees. If the answer is 0.893996... your calculator is set to radians. Consult your instruction manual to determine how to change from radian to degree units.
- If the forces do not all lie in the same plane, each force has three components and two angles are needed to define each force direction. Adding these types of forces is beyond the scope of this article.