Question

In a net force question including two applied forces at two different angles, how would one calculate the resulting net force and angle?

Answer 1

use phasor diagrams(arithmetic method) or resolve components in X and Y directions(graphical method)...

Answer 2

Good old vector addition.

Let \(F_1\) have a magnitude of A and makes an angle \(\theta\) will the x-axis.
Let \(F_2\) have a magnitude of B and makes an angle \(\alpha\) will the x-axis.

Realize that\[F_1 = A \cos(\theta) \hat i + A \sin(\theta) \hat j\]\[F_2 = B \cos(\alpha) \hat i + B \sin(\alpha) \hat j\]

Summing the vectors yields\[ \vec F_m = \vec F_1 + \vec F_2 = (A \cos(\theta) + B \cos(\alpha)) \hat i + (A \sin(\theta) + B \sin(\alpha)) \hat j\]

This new vector \(F_m\) has magnitude\[|F_m| = \sqrt{(A \cos(\theta) + B \cos(\alpha))^2 + (A \sin(\theta) + B \sin(\alpha))^2}\]

The angle of \(F_m\) is defined as\[\phi = \arctan \left ( A \sin(\theta) + B \sin(\alpha) \over A \cos(\theta ) + B \cos(\alpha) \right)\]

Answer 3

The other good old method would be use cosine formula.