Question

Four ropes are tied to a stake, and each is pulled toward a compass direction, N, S, E,
or W. A force of 10 lb is applied to the rope pulled toward the east. Forces of 20, 30, and 40 lb are applied
toward the south, west, and north, respectively. What is the net force exerted on the stake by the ropes and in
what direction is it? (Forces are vector quantities. The net force is the vector sum of the individual forces.)

Answer 1

There are two orthogonal directions involved, East (equivalent to +x) and North (equivalent to +y).
West is -x, and South is -y.
Add up individually the forces in the x and y directions to end up with a resultant vector R(Fx, Fy).
Calculate the magnitude using the formula
|R| = sqrt(Fx²+Fy²)
and the angle as tan^-1(Fy/Fx).
Be careful with the angle as the resultant can be in any quadrant.

Answer 2

how do you figure out from what direction the net force is exerted?

Answer 3

East=+x
West=-x
North = +y
South = -y
When you sum the forces, you will end up with two coordinates. Say if it is (-100,200), then it falls in the second quadrant.
Take the arctangent of 200/100 and subtract it from 180 degrees.
If your calculator has atan2() which accepts two parameters, this will be done for you automatically.
A little sketch of the point (-100,200) with respect to the origin will make this clear.

Answer 4

thank you for explaining this, it really helps.

Answer 5

You're welcome!