Derivation of Simple Harmonic Motion for a Spherical Bowl

[merryjman]

Problem: Prove that it is valid to model a particle in a smooth hemispherical bowl as a simple pendulum, provided its maximum displacement is small.

Answer: Usually, a FBD is used along with Newton's 2nd, and then the small angle approximation is used twice (once for siintheta in the weight component and once for tantheta to convert linear to rotational). Eventually you get the usual second-order Diffyq for angular displacement.

Question: Some of my students instead used the centripetal force requirement, setting the normal force (which is equal to the perp. component of the particle's weight) equal to the usual centripetal force equation. Using small angles, the costheta term in the normal force becomes 1, and the problem reduces to the familiar form.

Do you all feel that this is also a valid way to solve the problem? It seems OK to me; it involves Newton's 2nd, as well as the small angle approx., and I can't find anything wrong with it. But I thought I'd see what you all thought.

 

[mezarashi]

I think that this question is nearly the same as: Derive the "simple pendulum equation" from Newton's 2nd law. The only difference between the bowl and pendulum is the centripetal force is the normal force instead of the tension.

So to me, if the unusual approach you mention is acceptable for the question I state above, I would consider it acceptable for your question as well =)

 

[Astronuc]

Eventually you get the usual second-order Diffyq for angular displacement.

As long as one obtains the usual second-order differential equation (for simple harmonic motion) then it shows that the system is equivalent to a simple pendulum.

http://hyperphysics.phy-astr.gsu.edu/hbase/pend.html#c5

http://hyperphysics.phy-astr.gsu.edu/hbase/shm2.html#c2