Question

Find the angle between the forces given the magnitude of their resultant. (Hint: Write force 1 as a vector in the direction of the positive x-axis and force 2 as a vector at an angle with the positive x-axis. Round your answer to one decimal place.)

Force 1 Force 2 Resultant Force
50 pounds 75 pounds 100 pounds

Answer 1

75.5°

Explanation:

You want the angle between a 50 pound force and a 75 pound force given that their resultant is a 100 pound force. Force 1 is aligned with the +x axis, and Force 2 is in the first quadrant.

Law of Cosines

The forces and their resultant form a triangle such that the angle opposite the resultant is the supplement of the angle between the forces. We can find the angle opposite the resultant using the law of cosines:

c² = a² +b² -2ab·cos(C)

cos(C) = (a² +b² -c²)/(2ab)

C = arccos(a² +b² -c²)/(2ab) = arccos((50² +75² -100²)/(2·50·75))

C = arccos(-1875/7500) = arccos(-1/4) ≈ 104.48°

The angle between the vectors is the supplement of this:

angle12 = 180° -104.48° = 75.52° ≈ 75.5°

The angle between the forces is about 75.5°.