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.)
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.
how do you figure out from what direction the net force is exerted?
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.
thank you for explaining this, it really helps.
You're welcome!
To find the net force exerted on the stake by the ropes, we need to determine the vector sum of the individual forces.
First, let's assign directions to the forces:
- The rope pulled toward the east has a force of 10 lb towards the east (positive x-axis).
- The rope pulled toward the south has a force of 20 lb towards the south (negative y-axis).
- The rope pulled toward the west has a force of 30 lb towards the west (negative x-axis).
- The rope pulled toward the north has a force of 40 lb towards the north (positive y-axis).
To find the net force, we will add the forces in the x-direction and the forces in the y-direction separately.
In the x-direction:
- The force towards the east is 10 lb (positive x-component).
- The force towards the west is 30 lb (negative x-component).
Therefore, the net force in the x-direction is 10 lb - 30 lb = -20 lb.
In the y-direction:
- The force towards the south is 20 lb (negative y-component).
- The force towards the north is 40 lb (positive y-component).
Therefore, the net force in the y-direction is 40 lb - 20 lb = 20 lb.
Now, we have the net force in the x-direction (-20 lb) and the net force in the y-direction (20 lb). To find the magnitude and direction of the net force, we can use the Pythagorean theorem.
Magnitude of net force = sqrt((-20 lb)^2 + (20 lb)^2) = sqrt(400 lb^2 + 400 lb^2) = sqrt(1600 lb^2) = 40 lb
The direction of the net force can be found using trigonometry:
Direction angle = arctan((20 lb) / (20 lb)) = arctan(1) ≈ 45°
Therefore, the net force exerted on the stake by the ropes is 40 lb, in the direction approximately 45° counterclockwise from the positive x-axis.