## Background on Concurrent Forces

Joe and his father are trying to pull a big rock out of the ground. They know that if they apply a force of 1400 N to the rock, it will move. However, neither Joe nor his father can apply this much force alone.

To try and move the rock, they tie two ropes to the rock and both pull at the same time with equal forces of 1000 N each, but they don't pull in the same direction. Joe pulls his rope due East, while his father pulls due North. Will they be able to move the rock?

To answer this question, we need to know how to calculate the resultant of these two applied forces. The **resultant force** is the single force that could be applied to an object (in this case, the rock), that would have the same effect as all the applied forces combined.

Because both forces pass through the same point at the center of the rock, we say that these two forces are concurrent. **Concurrent forces** are forces that all act on the same point. To find the resultant of two concurrent forces, you can't always simply add the two amounts of force together. Why not? Because forces are vectors!

A **vector** is any quantity with a magnitude and a direction, and the direction is the really critical part here. If Joe and his father were pulling in the same direction, then we could easily find the resultant force by simply adding their two forces together. Similarly, if they were pulling in opposite directions, the resultant force would be zero because we would subtract the force acting to the left (in the negative direction) from the force acting to the right (in the positive direction).

However, in this case, since the forces are perpendicular, we will have to use another technique to find the resultant. One way to find the resultant of two forces is to use the parallelogram rule. To do this, first draw the two force vectors with their tails at the same point, and then draw a parallelogram with those two force vectors as two of the sides of the parallelogram. The resultant will be the diagonal of this parallelogram.

You can use some geometry and trigonometry to find the magnitude and direction of the resultant. To find the magnitude of this resultant force, you can use the Pythagorean theorem. Be careful with this, because it's only valid if there is a right angle in the triangle!

You can find the angle that the resultant force acts using the tangent function, which is equal to the side of the triangle opposite to the angle divided by the side adjacent to the angle.

Let's walk through this. So, the resultant force equals the square root of 1000 N squared plus another 1000 N squared. The resultant force is 1414 N. Since we know the rock will move if a force of 1400 N is applied to it, we now know that Joe and his father will be able to move the rock!