A common problem presented to the introductory physics student is the “incline problem.” These problems usually involve some sort of box, skier, etc. positioned on an incline (or decline) and the student is asked to find the acceleration of the object and/or the velocity after a certain amount of time has passed.
We first attack this problem by rotating a standard Cartesian plane to the incline degrees stated in the problem. We rotate the x- and y-axis so that the new x-axis and the old x-axis make the angle of incline designated in the problem. For example, if the problem states that an object sits on an incline of 15 degrees, we can rotate the coordinate system so that a 15 degree angle is formed from the new (blue) x-axis and the old (black) x-axis. See below.
Keep in mind that when rotating the coordinate system, a 15 degree angle exists in each quadrant of the new plane. To help you keep straight your similar angles, go ahead and mark each one as soon as you rotate the axis. A handy trick is to draw arrows showing how you rotated the plane – see below.
Now that we have set up our new coordinate system, we can identify the forces that exist on the object lying on the incline. We have the force of gravity, which is directly down from the object. We also have the normal force which is upward from the new y-axis perspective.
Note that the normal force is opposite in magnitude as the y-component of the force of gravity.
Our problem is now set up and we have easily identified the major forces presented in a typical problem. From these forces, we can identify the magnitude and direction of the normal force, the acceleration of the object, the velocity of the object and even the time it takes for the object to travel a certain distance. I will post typical problems that focus on these questions in the near future.