# If only ball a is set at φ φ 0 initially obtain the

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4. If only ball A is set at φ = φ (0) initially, obtain the speed of each ball at the bottom after the n -th collision. What is the speed of each ball as n → ∞ ? Problem 45. 1992-Fall-CM-U-2. ID:CM-U-498 A toy consists of two equal masses ( m ) which hang from straight massless arms (length l ) from an essentially massless pin. The pin (length L ) and the arms are in plane. Consider only motion in this plane. 1. Find the potential and kinetic energies of the masses as a function of θ , the angle between the vertical and the pin, and the time derivatives of θ . (Assume the toy is rocking back and forth about the pivot point.) 2. Find the condition in terms of L , l , and α such that this device is stable. 3. Find the period of oscillation if θ is restricted to very small values. Classical Mechanics QEID#15664097 February, 2018
Qualification Exam QEID#15664097 20 Problem 46. 1992-Fall-CM-U-3. ID:CM-U-510 A homogeneous disk of radius R and mass M rolls without slipping on a horizontal surface and is attracted to a point Q which lies a distance d below the plane (see figure). If the force of attraction is proportional to the distance from the center of mass of the disk to the force center Q , find the frequency of oscillations about the position of equilibrium using the Lagrangian formulation. Problem 47. 1992-Spring-CM-U-1 ID:CM-U-517 For a particle of mass m subject to a central force F = ˆ r F r ( r ), where F r ( r ) is an arbitrary function of the coordinate distance r from a fixed center: 1. Show that the energy E is s constant. What property of the force is used? 2. Show that the angular momentum L is s constant. What property of the force is used? 3. Show that, as s consequence of the previous part, the motion of the particle is in a plane. 4. Show that, as a consequence, the trajectory of motion in polar coordinate can be solved by quadrature. ( i.e. , the time-dependence of the coordinates can be expressed as integrals, which you should express, but which you cannot evaluate until the function F r ( r ) is specified.) For this part it will be useful to introduce an effective potential incorporating the angular momentum conservation. 5. Suppose F r ( r ) is attractive and proportional to r n , where n is an integer. For what values of n are stable circular orbits possible? [ Hint: Use the effective potential defined before, make a rough drawing of the different possible situations, and argue qualitatively using this drawing.] Classical Mechanics QEID#15664097 February, 2018
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