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...

You grind your way through a problem, get an answer, and want to know if it is correct.  How can you tell? In the previous post I talked about changing the angle in a problem (the inclined plane) to see how that affected other angles.  It can help you see how the angles relate to each other.  In this post I will use the same technique, but this time to test if my answer could be wrong. Let's return to the inclined plane.  Suppose a block of mass m = 2.5 kg slides down a frictionless plane at a slope of θ = 35 degrees to the horizontal.  You need to determine the block's acceleration.    You calculate an acceleration of a = 8.0 m/s/s down the slope.  Your friend, attempting the same problem, gets an answer of a = 5.6 m/s/s.  Who is right? After both checking that your calculators are set to degrees (not radians), you look at each other's work.  You have a = g cos θ while your friend has a = g sin θ where g is the usual accelerat...

How often have I walked into a lecture theatre planning to talk about some deep secret of physics only to end up arguing with students about which angle is which in a problem?  To be fair, I also often struggled to follow geometrical arguments when I was a student.  One trick I learned was to try to see the problem as dynamic rather than static, by which I mean imagine the angles changing so I can see how they relate to each other. For example, consider the inclined plane.  In these problems, some object is placed on a surface that is at an angle to horizontal: something sliding down a ramp, for example. We have the usual free-body diagram, indicating the relevant forces (weight of the object, normal force, friction), and the angle of the slope is labelled θ (theta).  To determine the acceleration (if any) of the object we need the net force acting on it.  And one of the things that will require is the angle between the normal force (N) and the vertical.  I...