How Resolving Forces Works

How forces can be resolved? How does multiplying sine or cosine angles to the force can get respective forces in that direction. How to believe that multiplying the trigonometric values gives the force equivalent in that particular direction, in reality?
- jagannath (age 19)
thanjavur,TN,India

Dear Jagannath,

That is a challenging question; it is something that most of us take for granted.

To see how it works intuitively, imagine a force acting on an object as a vector with some length and direction. Suppose the vector has length F and is pointed at and angle θ with respect to the horizontal x-axis. The magnitude of the force on the object is F, and it causes an acceleration (a) in the same direction, θ.

Now, if we shine light from "above" such that the shadows (of the force vector and the object) are projected onto the x-axis, the force vector looks shorter. In fact, its length is now Fcosθ instead of F. This is purely due to the definition of cosθ:

cosθ = Adjacent/Hypotenuse

When you plug a value of θ to determine Lcosθ in your calculator, it automatically gives the length of projection of a vector of length L pointing at an angle θ from the x-axis, onto the x-axis.

Therefore, by looking at the projection on the x-axis, we see that the x-component of the force is Fcosθ - which is the magnitude of the force in the x-direction. Then, by taking the projection of the acceleration vector onto the x-axis, the acceleration of the object in the x-direction is acosθ.

The same can be said for the y-direction. If we now shine light from the right to project shadows on the y-axis instead, the force would have a magnitude of Fsinθ. The acceleration in the y-direction is given by asinθ.

Thus,

Force (in x-direction) = mass x acceleration (in x-direction)

Force (in y-direction) = mass x acceleration (in y-direction)

So, multiplying by cosθ or sinθ gives you the components of the force in the x- and y-direction respectively.

*Extra note: You may be wondering why we can simply split the x- and y- directions up in the above. The argument above holds true because F=ma happens to be linear. That is, a increases in proportion to F. This is what made us able to resolve the force and acceleration together into components. Also, the vectors of the x- and y-components from a right angle - the net force can be found using Pythagoras's theorem, or alternatively, with (cosθ)^2 + (sinθ)^2 = 1.

Hope this helps!

-Mae

(published on 04/06/2011)