Dynamics
Beginner
Newton's Law
Source: Principles of Physics
Problem
Two forces with magnitudes $F_1$ and $F_2$ act on a body. The angle between the directions of the two forces is $\theta$. The resulting acceleration of the body has a magnitude of $a$.
What is the mass of the body?
$$m = \frac{\sqrt{F_1^2 + F_2^2 + 2F_1F_2\cos\theta}}{a}$$
The net force on the body, $F_{net}$, is the vector sum of the two forces. The magnitude of the net force is found using the law of cosines for vector addition:
$$F_{net} = \sqrt{F_1^2 + F_2^2 + 2F_1F_2\cos\theta}$$According to Newton's second law, the net force is also given by $F_{net} = ma$. We can solve for the mass, $m$:
$$m = \frac{F_{net}}{a}$$Substituting the expression for the magnitude of the net force gives the mass of the body in terms of the given quantities.
$$m = \frac{\sqrt{F_1^2 + F_2^2 + 2F_1F_2\cos\theta}}{a}$$