The figure below shows a mass \(m\) connected to a spring of force-constant \(k\) along a wooden track. The mass is restricted to move along this track without friction. The entire system is mounted on a toy wagon of zero mass resting on a track along a second frictionless beam. The wagon is connected by a spring of force-constant \(k^{\prime}\) to an axle about which the whole apparatus is spinning with constant angular speed \(\omega\). The figure is a top down view, with gravity pointing into the page, and the rest length of each spring is zero.

(a) First, write the Lagrangian of the system in terms of the four variables \(r, \theta, R\), and \(\Theta\) shown on the Figure, without implementing any constraints.

(b) Identify two constraint equations. Implement the one keeping the two tracks perpendicular to one another into the result of part

(a) by eliminating \(R\). Do not implement the constaint causing everything to spin at constant angular speed \(\omega\).

(c) Introducing a Lagrange multiplier for the constraint having to do with the spin, write four differential equations describing the system.

(d) Identify the force on the mass \(m\) due to the spin of the system, and find all conditions for which this force vanishes.

Transcribed Image Text:

m k www R r k'