Find one of two forces, given their resultant

Question

The resultant of two forces acting on a rock is 107N and it makes an angle of 38 degrees, and 40 minutes with the first force of 51N. Find the magnitude of the second force and the angle it makes with the resultant force.

Can I solve this using "law of Cosine"?

Answer 1

Hints. Draw a sketch to see that this Physics problem reduces to the trigonometric problem of solving a triangle given the sides $a=107\mathrm N$ and $b=51\mathrm N$, and the angle $\gamma=38^\circ\ 40'$ between these two sides, which is the case SAS of Solution of triangles (Wikipedia).

Can i solve this using "law of Cosine"?

Yes, it's a possibility. You can apply the law of cosines twice:

1. to find the third side $c$ (the magnitude of the second force), and
2. to determine the second angle $\beta$ (the angle the second force makes with the resultant force).

Answer 2

An easier way to solve this is to use the method of the components:

1) Draw a diagram on an xy-plane.

2) Find the x and y components of the known force and of the resultant force.

3) Then, if you subtract the x component of the known force from the x component of the resultant force, you will get the x component of the unknown force.

4) Again, if you subtract the y component of the known force from the y component of the resultant force, you will get the y component of the unknown force.

5) Once you have the x and y components of the unknown force, it should be pretty simple to find the magnitude and direction.