Question

Two people pull on a stubborn mule. One person pulls at a 70 degree angle with 50 lbs of force, and the other person pulls at a 100 degree angle with 40 lbs of force. Find the single force that is equivalent to the two forces. Find a third force that a person would need to exert to make the resultant force equal to zero. Explain.

Answer 1

First, we need to take the dot product of the two vectors. \[F_1 \cdot F_2 = |F_1||F_2| \cos(\theta_2 - \theta_1)\]

Second, we need to find the components of each force and add them. \[F_x = F_1 \cos(\theta_1) + F_2 \cos(\theta_2)\]\[F_y = F_1 \sin(\theta_1) + F_2 \sin(\theta_2)\]The third force must be equal but opposite to these. The angle at which this force acts\[\tan^{-1} \left( \left \vert F_y \over F_x \right \vert \right )\]

Answer 2

Case A:
F1=50, a1=70 degrees
F2=40, a2=100 degrees.
All angles measured with respect to the x-axis.
Equivalent force has the same x- and y-components, so
Fx = 50cos(70)+40cos(100)
Fy = 50sin(70)+40sin(100)
angle of equivalent force = arctan(Fy/Fx)
magnitude of equivalent force = sqrt(Fx^2+Fy^2)

Case B is similar to case A, with the angle 100 degrees replaced by -100 degrees.