Exercises on Analytical Mechanics Problem Set Number 4

Institut fur¨ Physik WS 2012/2013

Friederike Schmid

Exercises on Analytical Mechanics Problem set number 4

Questions on the lecture:

27. What do we mean by rigid body? 28. What are the Euler angles? 29. What is the characterizing property of a physical tensor? 30. Give the equation for the inertia tensor of a rigid body with mass distribution ρ(r). 31. What is the relation between the inertia tensor and the moment of inertia? 32. Give the general expressions for the angular momentum and the kinetic energy of a rigid body. Which is the translational contribution? Which the rotational contribution? 33. What are the principal moments of inertia? 34. What are the principal axes of inertia?

Problems (To be dropped until 12:00 am on November 11th, 2012 in the red box number 40 at Staudingerweg 7)

Problem 10) Jojo (6 Points) Consider a Jojo consisting of a cylinder of length L with radius R and mass M, which is attached to a string at its round side. The Jojo can roll up and down the string. It has the height h. a) Calculate the moment of inertia θ of the cylinder with respect to its long axis. b) Calculate the kinetic energy as a funciton of h˙ . (Don’t forget the rotational contribution!) c) Construct the Lagrange function L and determine the equations of motion. How does the solution look like?

Problem 11) Lagrange parameters (6 Points) Use Lagrange parameters to solve the following problems.

a) Determine the cuboid with largest volume V which can be fully contained in the ellipsoid x2 y2 z2 + + = 1. a2 b2 c2 Remember: Find ﬁrst the extremum of V − λf(x,y,z), where f(x,y,z) is the constraint. Then determine λ such that the constraint is fulﬁlled. b) Which is the smallest distance between the origin and the intersection curve between the two surfaces xy = 12 und x + 2z = 0? Note: Here you have two constraints, of course.

Problem 12) Variational calculus (6 Points) Calculate the curve of shortest length (the geodetic line) between two points (x,y,z) = (1, −1, cosh(1)) and (−1, 1, cosh(1)) on the surface z(x,y) = cosh y.

a) Consider ﬁrst a general parametrization of the curve, r(s) = (x(s),y(s), z(x(s),y(s)) with s ∈ [0, 1]. Give a general expression for the length of the curve I[{r(s)}] as a functional of the curve r(s). You will get an expression of the form 1 ′ ′ I = R0 dsL(x(s),x (s),y(s),y (s)). What is L? b) Give formally the Euler-Lagrange equations. c) There is a cyclic variable. Identify the corresponding ”conserved quantity”. Use it to solve the problem.