Make it dynamic: Relating angles

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.  It turns out to be the same as the slope, θ.  Can you see why?

To some people this will be obvious, and the proof is not too hard (see below), but I find it really helpful to imagine the slope changing, and to see how the other angles change with it.


In the animation, the angle with the vertical is labelled φ (phi), but as you might now be able to see, θ = φ.  That's most obvious at the two extremes, where θ = 0 (the slope is horizontal) and θ = 90 degrees (the slope is vertical), but it's still true everywhere in between.

It's useful to remember that in most physics problems there is nothing special about the numbers presented.  Maybe the slope is 7 degrees, maybe it's 40 degrees, maybe its 79 degrees: working out the acceleration is the same process in all cases.  But it's easier to see what's happening in some cases than others, so try moving the problem around and see if it helps.

Addendum: Proof of the angles

In the above diagram I have kept the normal force, but removed the others to avoid clutter.  I have added two lines: one normal to the plane, the other vertical.  And I have labelled the two new angles α and β.  The key fact to remember is that the sum of the angles in a triangle is 180 degrees.  Thus

α+β = 90

and

θ+β = 90

so

θ = α

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