Force from angle and magnitude

Data

• Angle with x axis  α = 120º
• Magnitude of the force F = 5N

Previous considerations

Observe that exercise can be considered the complementary to this. If on the previous one we were asked to calculate the angle from the analytic expression of the force, here we are asked for the analytic expression of the force from the angle and the magnitude of the force.

On the other hand, we know that the analytic expression of a force in two dimensions is given by the general expression:

$$\overrightarrow{F}=F_x·\overrightarrow{i}+F_y·\overrightarrow{j}$$

So, in this exercise we must calculate F_x and F_y.

Resolution

Ideally we should do a general idea through a graphic:

Mathematically, we can determine the expression F_x according to the definition of cosine:

$$F_x=F·\cos \left(\alpha \right)=5·\cos \left(120º\right)=-2.5 N$$

For the expression F_y we will use the definition of cosine:

$$F_y=F·\sin \left(\alpha \right)=5·\sin \left(120º\right)=4.33 N$$

So, the analytic expression we are looking for is:

$$\overrightarrow{F}=-2.5·\overrightarrow{i}+4.33·\overrightarrow{j}$$

Observe that, instead of considering α, occassionally is better to take into account the smaller angle formed by $\overrightarrow{F}$ with the axis x. On that case, it would be β=180º-120º=60º and the relations with the sine and cosine would be given by:

$$\begin{gathered} F_x=-F·\cos \left(\beta \right)=-5·\cos \left(60º\right)=-2.5 N \\ {F}_y=F·\sin \left(\beta \right)=5·\sin \left(60º\right)=4.33 N \end{gathered}$$