Question 1. **Course Module: Advanced Mechanics Problems**---### Problem Set#### 1. Particle in a Potential FieldA particle of mass \( m \) in 3D is subject to a potential \( V(x, y, z) = \frac{k}{2} x^2 e^{\lambda z} \). - **Task:** Write the Lagrangian and derive the Lagrange equations. Briefly, and without any further calculation, determine the small oscillation frequencies. Sketch the corresponding motion for each mode, indicating which are zero, and discuss the physical reasons for these results. *(Limit your explanation to 5-6 lines and include relevant sketches.)*#### 2. Dynamics of a Thin Hoop with a BeadA thin hoop of radius \( R \) and mass \( M \) oscillates in its vertical plane with one point fixed. A point mass \( M \) is attached, constrained to move frictionlessly along the hoop.- **Tasks:** - (a) Write the transformation equations for the center of the hoop and bead position, including time derivatives. Derive the kinetic energy, potential energy, and complete the Lagrangian for the system. - (b) Expand the Lagrangian to the second order in terms of small angular deviations from equilibrium. Identify and express matrices \( T \) and \( V \). - (c) Determine the normal mode frequencies. - (d) Compute the normal mode eigenvectors. Sketch the motion corresponding to each one. Ignore normalization constants.#### 3. Thumbtack on an Inclined Plane- **Problem Context and Tasks not visible in the current text.***(Note: Please refer to course materials or mathematical software for detailed derivations and validations of the theoretical content in these problems.)* --- **Graphical and Diagrammatic Explanations***Section to describe any associated diagrams of the problems if presented. Please request instructor-provided diagrams for visual aids.*---

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. A particle of mass m in 3D is subject to a potential V(x, y, z) = ²x Write the Lagrangian and the Lagrange Eqns. Very briefly, and without any further calculation find the small oscillation frequencies, sketch the corresponding motion for each, indicate which ones of them are zero, and discuss physically why they are zero. (This problem should take no more than 5-6 lines, plus drawings.)

. A particle of mass m in 3D is subject to a potential V(x, y, z) = ²x Write the Lagrangian and the Lagrange Eqns. Very briefly, and without any further calculation find the small oscillation frequencies, sketch the corresponding motion for each, indicate which ones of them are zero, and discuss physically why they are zero. (This problem should take no more than 5-6 lines, plus drawings.)