Two equal masses, m1=m2=m, are joined by a massless string...

Two equal masses, $m_{1}=m_{2}=m,$ are joined by a massless string of length $L$ that passes through a hole in a frictionless horizontal table. The first mass slides on the table while the second hangs below the table and moves up and down in a vertical line. (a) Assuming the string remains taut, write down the Lagrangian for the system in terms of the polar coordinates $(r, \phi)$ of the mass on the table. (b) Find the two Lagrange equations and interpret the $\phi$ equation in terms of the angular momentum $\ell$ of the first mass. (c) Express $\dot{\phi}$ in terms of $\ell$ and eliminate $\dot{\phi}$ from the $r$ equation. Now use the $r$ equation to find the value $r=r_{0}$ at which the first mass can move in a circular path. Interpret your answer in Newtonian terms. (d) Suppose the first mass is moving in this circular path and is given a small radial nudge. Write $r(t)=r_{0}+\epsilon(t)$ and rewrite the $r$ equation in terms of $\epsilon(t)$ dropping all powers of $\epsilon(t)$ higher than linear. Show that the circular path is stable and that $r(t)$ oscillates sinusoidally about $r_{\mathrm{o}}$ What is the frequency of its oscillations?

Key Concepts

Lagrangian Mechanics

This concept involves formulating the dynamics of a system using the Lagrangian, which is defined as the difference between the kinetic and potential energies. The method allows one to derive the equations of motion (the Euler-Lagrange equations) by applying the principle of least action, making it central to analyzing systems with constraints and generalized coordinates.

Polar Coordinates

Polar coordinates are particularly useful in describing systems with rotational symmetry. In the context of the problem, the motion of a mass on a horizontal table is naturally expressed in terms of the radial coordinate r and the angular coordinate ?. This coordinate system simplifies the expression of the kinetic energy and naturally incorporates the geometric constraints of motion.

Euler-Lagrange Equations

The Euler-Lagrange equations are the fundamental equations derived from the Lagrangian formulation that provide the equations of motion for the system. They are obtained by setting the first variation of the action to zero and are used to determine how the generalized coordinates evolve over time, accounting for both kinetic and potential energy contributions.

Conservation of Angular Momentum

Angular momentum conservation arises when the Lagrangian is independent of the angular coordinate ? (i.e., ? is a cyclic coordinate). This is a direct consequence of Noether’s theorem, which links continuous symmetries of the action to conserved quantities. In the given system, this conservation law is used to simplify the problem by reducing the degrees of freedom in the equations of motion.

Circular Motion and Equilibrium Conditions

Finding the condition for circular motion involves determining the radius at which the forces (or effective potential) balance in such a way that the motion remains at a constant radial distance. This requires setting the radial acceleration to zero and leads to a relationship between the angular momentum, the centripetal force, and other system parameters, reflecting Newtonian mechanics in a rotating frame.

Stability Analysis and Small Oscillations

Stability analysis is performed by introducing small perturbations around the circular path and linearizing the equations of motion. The resulting linear differential equation for the perturbation reveals whether the system oscillates about the equilibrium configuration and, if so, with what frequency. This method is key in determining if a circular orbit is stable under small disturbances.

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