Force Vectors
Before you can solve force vectors, you need to know more about right triangles and trigonometry.
(2 isn’t the actual angle measure)
Force vectors are almost the same as the triangle above, but at the same time they’re different. Vectors have both magnitude and direction. Vectors are given a variable, and they are written with an arrow in handwritten notation like so:
(They have an arrow above the letter)
The magnitude is the length as the line segment, so the magnitude of the vector above is 3. The angle between a reference axis and the arrow’s line of action is the direction, so the direction of the force vector above is 30O counterclockwise from the x-axis (the positive x-axis). The sense is indicated by the direction of the tip of the arrow on the force vector, so the sense for the force vector above would be upward and to the right.
These are the senses
Solving a force vector is a lot like finding the sides to a right triangle; it would help if you imagine a force vector having two sides: an x-direction side and a y-direction side, when the force vector is the hypotenuse. For example:
FX and FY will be negative if the force vector was facing left and down, but they aren’t in the picture above because the force-vector is facing right and up. If the force-vector was facing right and downward, FY will be negative and FX will be positive. If the force-vector was facing left and upward, then FX will be negative and FY will be positive.
So now that you know the basics, here’s an example:
So now that you know the basics, here’s an example:
Let’s say the red line is the force vector and the black lines are the x/y-axis. Let’s say the force-vector has a magnitude of 5lb and is 37o from the y-axis. Solve for FX and FY.
Earlier we learned that FY equals F multiplied by Sin (Ѳ). So if F=5lb, then:
FX=3lb because we just simply solve using trigonometry. We know:
Here’s another example:
Two children are dragging their teddy bear across the ground with the same magnitude. If two children are dragging their big teddy bear on the ground, but are moving away from the y-axis (37o), what is the resultant force (Child #2 is B and Child #1 is A)? What you do first is you solve for FX and FY for one force-vector (because they have the same angles). So:
FY is negative because the force-vectors are both moving in the downward direction, so:
Now you solve for FX.
Now here is the tricky part, FX for Child #2 is -3lb because Child #2’s force-vector sense is to the left, but FX for Child #2 is 3lb because the force-vector’s sense is to the right. They have the same FY force because both of the forces are moving downward with the same angle and magnitude. So how do you calculate the resultant forces?
There are two equations:
There are two equations:
What they mean is the sum of the forces in the x-direction, and the sum of the forces in the y-direction. So you have to add the components for each vector for the x/y-direction. So:
The big teddy bear isn’t moving anywhere along the x-axis because the sum of the forces in the x-direction equaled zero. Now:
The big teddy bear is moving down the y-axis because the two children are pulling down the y-axis. The bear is moving -8lb down the y-axis because the two children are pulling it down the y-axis.
Conclusion
Force-vectors are important in engineering, without understanding force-vectors engineers wouldn’t make structures that stabilize and support towers, bridges, and other buildings.