Conical Pendulum: A Level Maths Required?

[dt19]

i know this isn't exactly homework, but i couldn't find anywhere else to put it!
we've been doing s.h.m. in physics, and when we considered a simple pendulum my teacher mentioned that you can also have a conical pendulum, but the maths for it is more complicated. i want to know more about this - is it something i could understand with A level maths? (differential equations, vectors, integration by parts, by substitution, inverse trig, functions, basic mechanics) I'm not adverse to learning new maths - i have a whole six weeks of holiday to come.
what does anyone think?

 

[neutrino]

http://en.wikipedia.org/wiki/Conical_pendulum

See if you can follow the article. If you find something difficult to grasp, you can always post here.

 

[DLinkage]

hah, this is quite and ugly problem if you are trying to determine the equations of motion. My best advice to you is to use the Largrangian approach as all the forces acting on the pendulum (assuming no drag or dampers) are conservative forces. This is to say that...

L = T - V

T = 1/2*m*(Vinertal dot Vinertial)
V = potential energy

V = -m*g*h
where h is the displacement from the pivot to the mass dotted with the vertical. If we measure the angle the pendulum makes with the vertical to be phi and the rotation angle in the horizontal plane to to be theta of a pendulum with length L then we simply find h = L*sin(phi)

I will leave T up to you, it is a measure of the kinetic energy. Since we have a particle there is no rotational energy as particles essentiall have no moment of inertia about their CM. The kinetic energy term is somewhat ugly in this equation.

Finally to find the EOMs apply

d/dt(dL/dthetadot) - dL/theta = 0 for the equation of motion in theta
d/dt(dL/dphidot) - dL/dphi = 0 for the EOM in phi

notice that theta and phi are implicit functions of time so the time derivatives will NOT vanish in the Euler-Lagrange equations (these are the two equations stated above)

On a side note, if non-conservative forces exist or if there are constraints, these terms appear on the right hand side of the equation replacing the zero

On a side note if you wish to learn of this method in a introductory way, either email me, I have several problems worked if you like learning by example or purcahse the following text. Its somewhat difficult but you will learn a lot - I used it in one of my graduate level courses.

Principles of Dynamics by Donald Greenwood (currently in its 2nd edition)

 

[maverick280857]

DLinkage's suggestion about using a Lagrangian is well taken but you'll have to learn at least how to apply the Lagrangian method first (see http://spinor.sitesled.com/#Classical%20Mechanics for some links).

If you are unfamiliar with the calculus of variations/functionals, you can either use the textbook DLinkage has suggested (I have not seen it so I can't say how it is organized) or you can take a leap and read this introductory article: http://arxiv.org/abs/physics/0004029 to get a hang of what this 'new' thing is about. I strongly advise reading this article...you need nothing more to read it than what you already know.

Finally, the conical pendulum does permit a Newtonian solution and if the wikipedia site given by neutrino isn't sufficiently detailed or explanatory, then please do read the section on conical pendulum in Classical Mechanics by Kleppener and Kolenkow (a friendly and easy-to-find book in the library/bookstores). Once you know how to solve the problem without using a Lagrangian, you can dive right into the Lagrangian method and appreciate its power. I always feel that knowing how Newtonian mechanics works first enables you to understand and appreciate the power of more advanced methods.