State and prove parallelogram law of forces?

The Parallelogram Law of Forces states that if two forces acting at a point are represented in magnitude and direction by the two adjacent sides of a parallelogram, then their resultant is represented in magnitude and direction by the diagonal of the parallelogram that passes through the point.

Proof:

Let's consider two forces P and Q acting on a body simultaneously at a point O, making an angle α. These forces can be represented in magnitude and direction by the two adjacent sides OA and OB of a parallelogram OACB.

According to the Parallelogram Law, the diagonal OC of the parallelogram represents the resultant R of the two forces.

We can calculate the magnitude of R using the law of cosines in triangle OAB:

R^2 = P^2 + Q^2 + 2PQcos(α)

Taking the square root of both sides, we get the magnitude of the resultant force:

R = sqrt(P^2 + Q^2 + 2PQcos(α))

The direction θ of the resultant force can be found using the law of sines:

sin(θ) = Qsin(α) / R

Rearranging for θ, we get:

θ = arcsin(Qsin(α) / R)

This completes the proof of the Parallelogram Law of Forces.