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Homework 1 Solutions

# Homework 1 Solutions - Phys 200A(Theoretical Mechanics...

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Phys 200A (Theoretical Mechanics), Problem Set I Fetter & Walecka, problem #3.2. Done by Munirov V. R. 1) Lagrangian The Lagrangian of the system is straightforward: L = T - V, L = m 2 ˙ l 2 + m 2 2 l 2 sin 2 0 - mgl cos 0 . 2) Equilibrium orbit Euler-Lagrange equations of motion: d dt @ L @ ˙ l - @ L @ l = 0 , m ¨ l = m 2 sin 2 0 l - mg cos 0 . In equilibrium ¨ l = 0 , so we get the condition for an equilibrium circular orbit: l 0 = g cos 0 ( sin 0 ) 2 , QED. 3) Stability To consider the stability of this orbit against small displacements along the wire we write l = l 0 + 4 l ( Δ l ! 0 ) and put it into the equation of motion. By doing that we get: Δ ¨ l = 2 sin 2 0 4 l , which tells us that it is unstable equilibrium because coe cient before 4 l is positive. 4) Balance of force In non-inertial rotational reference frame there are three forces acting on the point mass: gravitational force m g acting downward, centrifugal force m 2 l sin 0 acting outward from rotational orbit and reaction force of the wire m 2 sin
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