Vector Decomposition
Whenever any physicist is dealing with any type of motion of an object, it is vital that they construct a free-body diagram so that they can effectively analyze the forces that are acting on the object. Although free-body diagrams are helpful for forces acting on an object that lays perfectly horizontal on a table or any flat surface. But when an object lays of a diagonal surface, such a book on a ramp, you must decompose the vectors of the forces that are not aligned with the x or y axis into a diagram commonly known as a pseudo free-body diagram. For example, below is a free-body diagram of a book that is on a table:
As you can see, the two forces that are acting upon the object, in this case the book, both reside on the y-axis. Since the two forces do this, we are not required to decompose any vectors. But if we were to have the object placed on a ramp, we would have the following free-body diagram:
As you can see, the forces that are acting on the mass are not aligned with the two conventional axes, the x and the y. Because of this, we must decompose the forces so that it is aligned with a traditional grid. Since the normal force and the force of friction are perpendicular to each other, we can use them as our axes when constructing the pseudo free-body diagram. The reason that we are not using the already vertical force of gravity as the y axis is because then we would have to unnecessarily decompose two vectors instead of just one. Below is a pseudo free-body diagram constructed to show the force for the scenario stated above:
As you can see, we have decomposed the force of gravity so that it is perfectly aligned with our axes. The way that you get the expressions next to both of the decomposed vectors is by drawing and imaginary right triangle under the mass in a way such that the triangle is tilted just as much as the mass. The figure below shows the imaginary triangle under the mass:
For the decomposed vector that is perpendicular to the motion of the mass, we have adjacent. The trigonometric function that we use when we have an adjacent value is cosine which results in mgcosƟ. For the decomposed vector parallel to the motion of the mass, we have opposite. The trigonometric function that we use when dealing with opposite is sine. This gives us a resultant vector of mgsinƟ. The angle of Ɵ on the hypothetical triangle shown above is the same as the inclination of the ramp. If we give the object a mass of 10 kg, then the force of gravity in the x direction would be (10)(9.8)(sin(30°)). The force of gravity in the y direction would be (10)(9.8)(cos(30°)). These come out to be 49 m/s/s in the x direction and 84.9 m/s/s.