Spherical Pendulum Motion: Solve the Mystery

[praeclarum]

I thought of this question the other day, and I was unable to solve it. A Google search has not helped, so I thought I might post it here.

A point mass hangs from a rod of length "l" from the center of a pendulum. The only forces acting upon the point mass are the force of gravity and the force of constraint (keeping it distance "l" from the center). Is there a function that describes the motion of the point mass?

 

[tiny-tim]

hi praeclarum!

A point mass hangs from a rod of length "l" from the center of a pendulum. The only forces acting upon the point mass are the force of gravity and the force of constraint (keeping it distance "l" from the center). Is there a function that describes the motion of the point mass?

do you mean two pendulums hinged together?

show us what you've tried, and where you're stuck, and then we'll know how to help!

 

[praeclarum]

OK. It's not as complicated as a double pendulum. It's just a single pendulum where the mass is constrained to a sphere (rather than the 2-dimensional case where you have a circle).

Well, one thought I had was to solve for the potential energy of the system, since that's just

mgh+1/2mv^2 = C

The mass is just a constant, and we can get rid of it.

From this point, I am stuck, however, and I don't know where to go from here. I was thinking the initial velocity must be perpendicular to the force of constraint and was wondering if you could split up the motion into just x and y components to solve it, but that seemed fruitless upon inspection.

I am looking for a general function that describes the motion of the point around the sphere. Your help is appreciated greatly.

 

[tiny-tim]

so it's basically a mass moving on the inside of a sphere?

hmm … in linear problems we usually use conservation of energy and conservation of momentum, sooo …

have you tried conservation of angular momentum ? ​