Answer: The stability of circular orbits in a central potential is a fundamental concept in classical mechanics, crucial for understanding planetary motion and atomic structures. It is determined by analyzing the effective potential energy, which incorporates both the central potential and the centrifugal potential arising from angular momentum conservation. For a circular orbit to exist, the effective potential must have an extremum, and for it to be stable, this extremum must be a minimum.

Answer: In classical mechanics, particularly for central force fields, the Binet equation is a powerful tool to determine the force law from a given orbit. A central force implies that the angular momentum (L = mr²θ̇) remains conserved, which simplifies the equation of motion in polar coordinates. The Binet equation relates the force F(u) to the orbital shape, where u = 1/r, as F(u) = -mL²u²(d²u/dθ² + u). Given the orbit r = ke∝θ, we first express it in terms of u: u = (1/k)e^(-αθ). Next, we compute it...

Answer: To obtain the normal mode frequencies for a system of two equal masses connected by three springs, we begin by setting up the equations of motion for each mass. We assume a standard configuration where two equal masses, `m`, are connected by three springs, `k₁`, `k₂`, and `k₃`, in series. The outermost springs (`k₁` and `k₃`) are connected to fixed walls, and the middle spring (`k₂`) connects the two masses. Let `x₁` and `x₂` be the displacements of the first and second masses, respectively, fro...