Question 615495: Two forces act on a body at an angle of 100 degrees. The forces are 30 and 40 pounds.
Find magnitude of the resultant force to the nearest tenth and the angle formed by the greater of the two forces and the resultant force to the nearest degree.
I remember learning this in class but totally forgot any sort of equation or such to answer it. Be great if you could point me in the right direction, thanks :)
Found 2 solutions by jsmallt9, Edwin McCravy:
Answer by jsmallt9(3758)   (Show Source):
You can put this solution on YOUR website! Drawing a diagram will probably help:- Draw two vectors starting at the same point (let's call this point B for "body") and forming an angle of about 100 degrees. (It doesn't have to be exact.)
- Label the length of one vector as 30 and the other as 40.
- Label the endpoint of the 30 pound force as A and the endpoint of the 40 pound force as C. This makes the 30 pound force vector BA and the 40 pound force BC.
- The resultant force is the sum of these two vectors. Complete a parallelogram using the two vectors we already have as consecutive sides. Let's name the unnamed vertex of this parallelogram as D.
- Since opposite sides of a parallelogram are always congruent, label the length of CD as 30 (since it is opposite BA) and label the length of AD as 40 (since it is opposite BC).
- Since consecutive angles in a parallelogram are always supplementary, label the angle at A as 80 degrees, the angle at D as 100 degrees and the angle at C as 80 degrees.
- Draw the diagonal BD. This is the vector for the resultant force.
With this diagram we can now figure out how to solve the problem.
The magnitude of the resultant force is the length of BD. BD is a side of a triangle in this diagram. Just use the Law of Cosines with the sides 30 and 40 and the angle of 80 degrees to find the length of BD.
The angle between the greater force and the resultant force is angle DBC. This angle is an angle in a triangle. You can use the Law of Sines or the Law of Cosines to find the angle. (I recommend the Law of Cosines because finding angles with the Law of Sines can have complications. For example if you get sin(x) = 0.5 then x could be either 30 degrees or 150 degrees. Unless you can rule out one of these, then there may be two solutions. The Law of Cosines does not have this problem when finding angles.)
Answer by Edwin McCravy(20060)  (Show Source):
You can put this solution on YOUR website!
Here's the complete solution:
Draw a little rectangle for the body, and sticking out of the body,
draw two vectors, one 3 units long for the 3 lb vector, and one 4
units long for the 4 lb vector, so that the angle between the two
vectors is 100°:
Next we complete the parallelogram, Through each arrowhead, draw
a line parallel to the other vector. I'll draw them in green:
The resultant of the two forces is a vector which is the diagonal of
the parallelogram from the body to the point where the two green lines
intersect, which I will draw in red
Two adjacent angles of a parallelogram are supplementary so the angle
on the lower right is 180°-100° or 80°.
To find the magnitude of R, we use the law of cosines:
R² = a² + b² - 3·a·b·cos(phi)
R² = 3² + 4² - 2(3)(4)cos(80°)
R² = 20.83244374
R = 4.564257194 or to the nearest tenth of a pound,
R = 4.6 lb.
The angle between the 4 lb. vector and the resultant is either
of the two angles labeled q below:
We'll get the q at the top right
since we're working in that triangle.
 = 
R·sin(q) = 3·sin(80°)
sin(q) =  .
sin(q) =  .
sin(q) = 0.6472955255
Use the inverse sine function on your calculator:
q = 40.33800507°
or to the nearest degree, 40°.
Edwin
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