Patrick Teaches Physics

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 imm...

What are "cos" and "sin"?  They're buttons on a calculator -- right?  You enter an angle, push the button, and some number appears.  Somehow -- if you remember your trig rules (SOH, CAH, TOA) -- you can use this number get the side lengths of triangles. I think the above pretty much sums up the typical student's understanding, and it's adequate to complete most of the tasks a physics student has to with trig functions.  But it doesn't have to be so mysterious, and it can be very helpful if it isn't.  So in this post I'll explain what these functions are and how they are related. Consider trying to specify a point on the unit circle: We could use the Cartesian coordinates x and y, or we could use the angle θ. Both have their merits, and in practice we might want to go back and forth between them.  So, given an angle θ, what are x and y?  We call the functions that answer that question sine and cosine (usually written as sin and cos for sho...

When solving physics problems we always have the benefit of choosing our frame of reference.  Not only can we chose where the origin of our coordinate is, but we can also chose which part of our system is stationary.  Consider the following simple kinematic problem: A car is travelling along, and we want to know how long it takes to travel some distance.  We can plug numbers into a kinematic equation and get the answer.  If the car is accelerating then the problem is mathematically a bit more fiddly (a quadratic equation, or do it in two steps with two equations) but it's not conceptually harder. But now consider a similar problem with two moving bodies: Now there's two cars at different speeds, and the question is: how long will it take the red car to overtake the blue car?  This looks rather harder because the distance the red car has to travel is not immediately clear.  But that's only because of our choice of reference frame, which up till now has been ...