Theoretical Mechanics: Fall-2021
HW 10: Due

1. Bead on a Spinning Wire Hoop

   (modified from Taylor Ex. 7.6)

   A small bead of mass \(m\) is threaded on a frictionless circular wire hoop of radius \(R\). The hoop lies in a vertical plane, which is forced to rotate about the hoop's vertical diameter with constant angular velocity \(\dot{\phi}=\omega\), as shown in Figure 7.9. The bead's position on the hoop is specified by the angle \(\theta\) measured up from vertical.

   1. Write down the Lagrangian for the system in terms of the generalized coordinate \(\theta\) and find \(\ddot{\theta}\). Discuss at least three strategies for making sense of your answer.

   2. Find the angles for which \(\ddot{\theta} = 0\) - these are equilibrium angles. Show these locations on a sketch of the hoop. Use at last three strategies for making sense of your answer. Include and discuss a plot the equilibrium angles vs. \(\omega\)

2. A Ball Confined to the Surface of a Sphere in Near-Earth Gravity A ball with mass \(m\) is confined to move on the surface of a sphere with radius \(r=R\). A convenient choice of coordinates is spherical, \(r\), \(\theta\), \(\phi\), with the polar axis pointing straight down. (Remember: in physics, \(\phi\) is the azimuthal angle in the \(xy\)-plane and \(\theta\) is the angle with the \(z\)-axis.)
   1. Find the equations of motion using a Lagrangian approach. Use at least three sense-making strategies to evaluate each equation.
   2. Explain what the \(\phi\) equation indicates about the \(z\)-component of angular momentum.
   3. Discuss the specific special case that \(\phi = \) constant. What does the equation of motion for \(\theta\) indicate?