Assignment:
Q1. The ground-state energy of a quantum particle of mass m in a pillbox (right-circular cylinder) is given by the following equation. Find the ratio of R to H that will minimize the energy for a fixed volume.
E = h2/2m ( 2.4048)2 /R2 + π2/H2
Q2. A particle, mass m, is on a frictionless horizontal surface. It is constrained to move so that Θ = ωt (rotating radial arm, no friction).
t = 0 , r = r0 , r= 0
(a) Find the radial positions as a function of time
(b) Find the force exerted on the particle by the constraint
Q3. A point mass m is moving over a flat, horizontal, frictional plane. The mass is constrained by a string to move radially inward at a constant rate. Using plane polar coordinates ( P, φ) P = P0 - kt:
(a) Set up the Lagrangian
(b) Obtain the constrained Lagrange equations
(c) Solve the φ-dependent Lagrange equation to obtain ω(t), the angular velocity. What is the physical significance of the constant of integration that you get from your `free`integration?
(d) Using the ω(t) from part (b), solve the P-dependent (constrained) Lagrange equation to obtain λ(t). In order words, explain what is happening to the force of constraint as P-->0.
Provide complete and step by step solution for the question and show calculations and use formulas.