Setup: Two masses (m_1, m_2), two massless rods length (L_1, L_2). Angles (\theta_1, \theta_2) from vertical. Find Lagrangian to second order in angles.
(Full solutions in main text; here only final results)
1.1 Shortest path between ( (0,0) ) and ( (1,1) ) is a straight line: ( y=x ).
2.1 Simple pendulum equation: ( \ddot\theta + (g/l)\sin\theta = 0 ). lagrangian mechanics problems and solutions pdf
2.3 Bead on parabolic wire ( y = ax^2 ): equation ( \ddot x + 2agx/(1+4a^2x^2) = 0 ).
3.3 Particle on sphere radius ( R ): conserved angular momentum about vertical; motion equivalent to a one‑dimensional problem in ( \theta ) with effective potential.
5.1 Noether current for translation: ( p = \sum m_i \dot x_i = ) const. Setup: Two masses (m_1, m_2), two massless rods
Problem: (Atwood Machine with a Massive Pulley)
A pulley of moment of inertia (I) and radius (R) has a massless string supporting masses (m_1) and (m_2) ((m_1 > m_2)). No slipping. Find acceleration.
Solution (as you would find in a PDF):
Setup: A mass (m) attached to a massless rod of length (L). The rod pivots without friction. Use the angle (\theta) from the vertical. Problem: (Atwood Machine with a Massive Pulley) A
(a) Find the Lagrangian.
(b) Derive the equation of motion.
(c) For small oscillations, find the period.
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Problem: A bead slides frictionlessly on a wire rotating at constant angular speed (\omega) in a horizontal plane. Find the radial equation. Solution Approach: Kinetic energy in polar coordinates: (T = \frac12 m (\dotr^2 + r^2 \omega^2)). No potential ((V=0)). The Euler-Lagrange gives (\ddotr - \omega^2 r = 0).