Choose your own angle

The relativity principle says that we get to choose the coordinate system we use when describing physical problems, and that includes deciding which way is up.  To show how useful that can be I'm going to use the inclined plane.

We have an object sitting on a surface which is inclined at some angle: a box on a ramp or some such.  We want to know the object's acceleration.  Drawing the free-body diagram is straight-forward: there is weight, a normal force, and usually there is friction as well.

We can write down the equations of motion and solve them, but it won't be pretty!  The acceleration has horizontal and vertical components, and because the friction depends on the normal force, which depends on the weight and the angle, the trig functions are going to build up.  Also, it's not so easy to tell what the normal force should be.

And so on....

A much better approach is to rotate the coordinate system so that motion of the object is along the x-axis.

We can immediately write down the y-component of the acceleration: it's zero!  (This is actually a one-dimensional problem.) That fact will also tell us the size of the normal force.  The only disadvantage of this choice of coordinates is that gravity points at a weird angle.

Much better!

As always with problems including friction, remember that the friction model (f = μ N) is for the maximum friction.  If you get a positive value for the acceleration, then your block is sliding uphill!  What's actually happening is that the static friction is holding the block in place, a = 0.

I've discussed other aspects of the inclined plane in some previous posts.  Click on the "inclined plane" label below.