Problem 1. Mathematics of rotations

Transcription

1 Problem 1. Mathematics of rotations (a) Show by algebraic means (i.e. no pictures) that the relationship between ω and is: φ, ψ, θ Feel free to use computer algebra. ω X = φ sin θ sin ψ + θ cos ψ (1) ω Y = φ sin θ cos ψ θ sin ψ (2) ω Z = φ cos θ + ψ (3) (b) Determine the projection of ω on to the lab frame axes ẽ 1, ẽ 2, ẽ 3. (You may use either algebraic means, or use the appropriate picture form Goldstein/lecture/internet, or both.) 1

2 Problem 2. Nutation of a Heavy Symmetric Top Consider a heavy symmetric top with one end point fixed. (a) Write down the Lagrangian from class. Carry out Routh s procedure explicity by Legendre transforming with respect to the the conserved momenta p ψ and p φ. Show that θ obeys the equation of motion following from the Routhian: where Also show that I θ = U eff θ, (4) U eff = mgl cos θ + (p φ p ψ cos θ) 2 2I 1 sin 2 θ φ = p φ p ψ cos θ I 1 sin 2 (θ). (5). (6) (b) In class we analyzed the limit when gravitational torque is small to the rotational kinetic energy, mgl/(p 2 ψ /I 1) 1. Take p φ /p ψ = r with 0 < r < 1. Within this approximation (known as the fast top approximation), if the energy E is adjusted to the minimum of the effective potential, the tip of the top will slowly precess with θ = 0, and φ = mgl p ψ. (7) This is is shown in Fig. 1(d) which shows the trajectory of the tip of the top on the sphere. Now if the energy of the system is slightly larger than the minimum of U eff, describe qualitatively the motion in θ and φ. For what range in E do the first (a) and second (b) figures describe the top s motion? Explain. Work in the fast top approximation (a) (b) (c) (d) Figure 1: Motion of the tip of the heavy symmetric top (c) Using the fast top approximation outlined in (b), compute the period of oscilations for a given energy E, and determine the precession rate φ(t), and angle θ(t), as a function of time. What is the average precession rate? 2

3 θ Problem 3. OCW) Foucault Pendulum and the Coriolis Effect (MIT- Consider a pendulum consisting of a long massless rod of length l attached to a mass m. The pendulum is hung in a tower that is at latitude λ on the earths surface, so it is natural to describe its motion with coordinates xed to the rotating Earth. Let ω (i.e. once per day) be the Earths angular velocity. Use the spherical coordinates (r, θ, φ) shown in the gure to investigate the Coriolis force. Here z is perpendicular to the Earths surface and y is tangent to a circle of constant longitude that passes through the north pole. The radius of the earth is R e z r m φ N y x Image by MIT OpenCourseWare. (a) Determine the Lagrangian of the Pendulum using the coordinates θ and φ. From the start you may keep terms up to first order in ω, and of course you may neglect total time derivatives to simplify the analysis. Derive the equations of motion of the pendulum. Do not assume small oscillations. (b) Since l is large, consider the small angle approximation for θ and simplify your equations of mo tion from (a). Demonstrate that the pendulum undergoes precession with a rate φ = ω sin λ. 3

4 Problem 4. A point mass on a disk (MIT-OCW) A thin uniform disk of radius R and mass M lies in the x-y plane, and has a point mass m = 3M/8 attached on its edge. (There is no gravity in this problem.) y B m A x (a) Find the moment of inertia tensor of the disk about its center (ignoring the mass m). Then nd the moment of inertia tensor of the combined system of the disk and point mass about the point A in the gure. (b) Find the principal moments of intertia and the principal axes about A. (c) The disk is constrained to rotate about the y-axis with angular velocity ω by pivots at A and B. What is the angular momentum about A as a function of time, and what is the torque required to maintain the motion? 4

5 Problem 5. A Rolling Cone (Adapted from Goldstein Ch.5 #17) A uniform right circular cone of height h, half-angle α, and density ρ rolls on its side without slipping on a uniform horizontal plane. It returns to its original position in a time τ. (a) Find the CM of the cone. Find the moment of inertia tensor for the body (or principal) axes centered on the tip with the y -axis going through the CM. (b) What is the moment of inertia tensor if we move our axes in (a) so they are centered on the CM? (c) Now assume the cone rolls on a fixed plane. Pick a new set of body axes (x, y, z) such that the z-axis is perpendicular to the plane, the y-axis coincides with the instantaneous line of contact, and the origin is the tip of the cone. Find the moment of inertia tensor for these axes. (d) Find the kinetic energy of the rolling cone and the components of the angular momentum. [There are two ways you could answer this, one uses your results from (b) and one uses those from (c).] 5