Classical Mechanics - Homework 4 Prof. Uwe R. Fischer from 2014-04-23 (Wed) to 2014-05-07 (Wed) 5:00 pm Where to submit: HW box at the 1st ﬂoor, TA : Seok-young Choi (sychoi912 at snu.ac.kr) 1. (5 points) Heavy Top Consider a symmetric top whose principal moments of inertia about the x and y axes are equal (Ixx = Iyy ). This symmetric top is tilted with respect to z axis and rotates under gravitation(−gˆ z ). Its mass is m and the distance of COM with respect to the support point is s so that the potential is V = mgs cos θ where θ is Euler angle. Use Eulerian angles coordinates that you learned in the class. (a) Write down the Lagrangian of the system and obtain the equation of motion. What are the conserved quantities? (b) What is the eﬀective potential in Lagrangian, Veff = Veff (θ), in terms of the conserved quantities? 2. (4 points) Solid Cone z Calculate the principal moments of inertia of a uniform solid cone of vertical height h, and base radius a about its vertex. For what value of the ratio h/a is every axis through the vertex a principal axis? For this case, ﬁnd the position of the center of mass and the principal moments of inertia about it. a h y x 3. (5 points) Centrifugal and Coriolis Force (a) Circular Lake The water in a circular lake of radius 1km in latitude 60◦ is at rest relative to the Earth. Find the depth by which the center is depressed relative to the shore by the centrifugal force. For comparison, ﬁnd the beight by which the center is raised by the curvature of the Earth’s surface. (Earth radius = 6400km.) (b) A Bird A bird of mass 2 kg is ﬂying at 10 ms−1 in latitude 60◦ N, heading due east. Find the horizontal and vertical components of the Coriolis force acting on it. 4. (4 points) Virial Theorem ∑ Suppose the pairwise interaction potential V (r1 , r2 , · · · , rN ) = 21 Vij where Vij = m aij rij , m ∈ Z. Here i, j denotes the index of given particles and aij is constant. Show ∑ that (a) ri · ∇i V = mV and that the Virial theorem reads (b) 2T¯ = mV¯ . 1 5. (10 points) Rotating Disk on a Line Consider a homogeneous disk of mass M and radius R with a point mass m = 12 M attached at the boundary of the disk. The disk rolls along a horizontal line without friction and without sliding. There is homogeneous gravitational ﬁeld g in −y direction. (a) Express the coordinates (xM , yM ) of the center of the disk in terms of φ assuming that xM = 0 when φ = 0. (b) Express the coordinates (xm , ym ) of the point mass m and the coordinates (xcom , ycom ) of the center of mass in terms of φ. (c) Calculate the kinetic energy T (φ, φ) ˙ and the potential energy V (φ) of the system. (d) Find the equation of motion for φ. And ﬁnd the frequency of small (φ ≪ 1, φ˙ ≈ 0) oscilation. (e) Find the constraint force Z exerted by horizontal ﬂoor to the center of mass. ˙ (f) Since ∂L ∂t = 0, the energy of the system is conserved. Find the expression for φ(t) in terms of φ(t) and the initial horizontal velocity v = Rx˙ M (t = 0) of the disk, by equating E(t = 0) and E(t). Find the expression for φ(t) ¨ in terms of φ(t) and v. (g) Find the condition for v under which the disk lifts oﬀ (loses contact with the ﬂoor) when φ = 3π 2 . (h) Find the rotational inertia around the center of mass. What is the cause(torque) for the rotational motion around the center of mass? Derive the same equation of motion obtained in (d) using Newtonian mechanics by investigating the rotational motion around the center of mass. 6. (4 points) Spaceship Maneuver A spaceship of mass 3t has the form of a hollow sphere, with inner radius 2.5 m and outer radius 3 m. Its orientation in space is controlled by a uniform circular ﬂywheel of mass 10 kg and radius 0.1 m. Given that the ﬂywheel is set spinning at 2000 r.p.m., ﬁnd how long it takes the spaceship to rotate through 1◦ . Find also the energy dissipated in this maneuver. 2