Calculating the resultant force on a body by a combination of forces is a matter of adding the different acting forces componentwise, as discussed in Halliday and Resnick’s “Fundamentals of Physics.” Equivalently, you perform vector addition. Graphically, this means maintaining the angle of the vectors as you move them into position as a chain, one touching its head to the tail of another. Once the chain is completed, draw an arrow from the only tail without a head touching it to the only head without a tail touching it. This arrow is your resultant vector, equal in magnitude and direction to the resultant force. This approach is also known as the “superposition principle.”

Draw a diagram of various forces acting on a 5-kilogram block falling through space. Suppose it has gravity pulling down vertically on it, another force pulling it left with a force of 10 Newton (the SI unit of force), and another force pulling it up and to the right at an angle of 45 degrees with a force of 10?2 Newtons (N).

Sum up the vertical components of the vectors.

In the above example, the gravitational force downward has magnitude F = mg = -5kg x 9.8m/s^2, where g is the gravitational acceleration constant. So its vertical component is -49N, the negative sign indicating that the force pushes downward.

The rightward force has a vertical and horizontal component of 10N each.

The leftward force has no vertical component.

The sum is 39N downward.

Sum up the horizontal components of the vectors.

Continuing with the above example, the left and right vectors contribute 10N in each direction, which cancel each other out to give zero horizontal force.

Use Newton’s second law (F=ma) to determine the acceleration of the body.

The resultant force is therefore 39N downward. For a 5-kg mass, the acceleration is therefore found as follows: 39N = F=ma = 5kg x a, so a = 7.8m/s^2.

#### References

- "Fundamentals of Physics"; David Halliday and Robert Resnick, 1992
- Georgia State University: Vector Operations
- Motor Sports Math: Resultant Force