The "spherical pendulum" is just a simple pendulum that is...

The "spherical pendulum" is just a simple pendulum that is free to move in any sideways direction. (By contrast a "simple pendulum"- unqualified - is confined to a single vertical plane.) The bob of a spherical pendulum moves on a sphere, centered on the point of support with radius $r=R$ the length of the pendulum. A convenient choice of coordinates is spherical polars, $r, \theta, \phi,$ with the origin at the point of support and the polar axis pointing straight down. The two variables $\theta$ and $\phi$ make a good choice of generalized coordinates. (a) Find the Lagrangian and the two Lagrange equations. (b) Explain what the $\phi$ equation tells us about the $z$ component of angular momentum $\ell_{z^{*}}$ (c) For the special case that $\phi=$ const, describe what the $\theta$ equation tells us. (d) Use the $\phi$ equation to replace $\dot{\phi}$ by $\ell_{z}$ in the $\theta$ equation and discuss the existence of an angle $\theta_{\mathrm{o}}$ at which $\theta$ can remain constant. Why is this motion called a conical pendulum? (e) Show that if $\theta=\theta_{0}+\epsilon,$ with $\epsilon$ small, then $\theta$ oscillates about $\theta_{\mathrm{o}}$ in harmonic motion. Describe the motion of the pendulum's bob.

Key Concepts

Conical Pendulum

A conical pendulum is a special case in pendulum dynamics where the pendulum bob moves in a horizontal circular path while maintaining a constant angle with the vertical. This motion exhibits a balance between the gravitational force and the centripetal force required for circular motion, and it is analyzed by setting one of the coordinates constant, thereby reducing the complexity of the problem.

Conservation of Angular Momentum

The conservation of angular momentum is a fundamental principle stating that if a system's Lagrangian does not explicitly depend on an angular coordinate (i.e., there is rotational symmetry), then the corresponding component of angular momentum is conserved. This concept, which follows from Noether's theorem, is crucial in analyzing motions with rotational dynamics and simplifying the problem.

Small Oscillations and Harmonic Motion

The concept of small oscillations pertains to the approximation of a system's behavior around a stable equilibrium position. By assuming that deviations from equilibrium are small, the system's nonlinear equations can often be linearized, leading to simple harmonic motion. This approximation helps in understanding the system's oscillatory behavior and predicting its time evolution under minor perturbations.

Generalized Coordinates

Generalized coordinates are variables chosen to uniquely define the configuration of a system relative to its degrees of freedom. Unlike Cartesian coordinates, they can be adapted to the system's constraints and symmetry properties, often simplifying the problem. In many problems, they reflect natural parameters of the motion and facilitate the use of the Lagrangian formalism.

Lagrangian Mechanics

Lagrangian mechanics is a reformulation of classical mechanics based on the principle of least action. It uses the Lagrangian, defined as the difference between the kinetic and potential energies of a system, to derive the equations of motion through the Euler-Lagrange equations. This approach is particularly useful when dealing with complex systems and constraints, as it allows for a systematic derivation of motion equations in generalized coordinates.

Lagrange Equations

Lagrange equations are differential equations derived from the Lagrangian formalism using the Euler-Lagrange equation. They provide a systematic way to obtain the equations of motion for a system even when forces are nontrivial to compute directly. These equations are central in analyzing systems with multiple degrees of freedom and constraints.

Transcript

00:01 We're given the linearized equation for a pendulum.

00:05 So we assume that sine of theta is small, or theta is small, so that sine of theta is approximately theta.

00:12 So we get theta double dot plus g over l theta equals zero.

00:15 And we're given some initial conditions, so that this pendulum is at initial angle of 0 .2 radiance, and it has an initial velocity of one rating per second squared.

00:27 So you can say here that it's like here, and we give it some of velocity, so it's going to come back up and then it's going to swing.

00:34 And there's no dissipation in here, so it's going to, you know, in theory, swing forever.

00:43 So this is just a second order differential equation, and with a solution that looks like this, c1 and c2 we could find from initial conditions.

00:54 So c1 is 0 .2, and c2 is square root of l over g.

01:00 And i guess at some point, let's see here, i guess i didn't write that down here.

01:06 They told us that l was one meter.

01:10 I didn't use that until the very end or until, you know, getting solutions.

01:16 So our solution is given by this, right? and so if we plug in, you know, if we plug in l is one and we know g is 9 .8.

01:30 One meters per second squared.

01:34 Here's our solution.

01:37 Now, they ask us for the period, so how long it takes to go through one cycle.

01:45 And so that's 2 pi divided by the natural frequency.

01:48 And the natural frequency is the square root of this, square root of g over l.

01:54 And so that winds up being 2 pi times the square root of l over g.

01:58 And that turns out to be very roughly, very close to two seconds.

02:02 So it takes basically two seconds to go.

02:05 It's pretty slow...

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