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 ? ​

 

[PJC01]

I find this question intriguing and would like to provide some insights on the motion of a spherical pendulum. First, it is important to note that this type of motion is also known as conical pendulum motion, as the path traced by the point mass forms a cone shape.

To answer the question, yes, there is a function that describes the motion of the point mass in a spherical pendulum. It is known as the equation of motion and it can be derived using Newton's laws of motion. The equation is as follows:

θ'' + (g/l)sinθ = 0

Where θ is the angle made by the rod with the vertical, g is the acceleration due to gravity, and l is the length of the rod. This equation is a second-order differential equation, which means that it describes the acceleration of the point mass as a function of time.

Solving this equation can be quite challenging, but it can be done using techniques such as the small-angle approximation or numerical methods. The solution to this equation will give us the function that describes the motion of the point mass.

Furthermore, it is worth mentioning that the motion of a spherical pendulum is periodic, meaning that the point mass will repeat its motion after a certain period of time. This period is dependent on the length of the rod and the acceleration due to gravity.

In conclusion, the motion of a spherical pendulum can be described by a mathematical function known as the equation of motion. Solving this equation will give us insights into the behavior of the point mass and how it moves in a conical path. I hope this helps in solving the mystery you encountered and further sparks your curiosity in the fascinating world of physics.