4. If only ballAis set atφ=φ(0)initially, obtain the speed of each ball at thebottom after then-th collision. What is the speed of each ball asn→ ∞?Problem 45.1992-Fall-CM-U-2.ID:CM-U-498A toy consists of two equal masses (m) which hang from straight massless arms (lengthl) from an essentially massless pin. The pin (lengthL) 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 ofθ, theangle between the vertical and the pin, and the time derivatives ofθ. (Assumethe toy is rocking back and forth about the pivot point.)2. Find the condition in terms ofL,l, andαsuch that this device is stable.3. Find the period of oscillation ifθis restricted to very small values.Classical MechanicsQEID#15664097February, 2018

Qualification ExamQEID#1566409720Problem 46.1992-Fall-CM-U-3.ID:CM-U-510A homogeneous disk of radiusRand massMrolls without slipping on a horizontalsurface and is attracted to a pointQwhich lies a distancedbelow the plane (seefigure). If the force of attraction is proportional to the distance from the center ofmass of the disk to the force centerQ, find the frequency of oscillations about theposition of equilibrium using the Lagrangian formulation.Problem 47.1992-Spring-CM-U-1ID:CM-U-517For a particle of massmsubject to a central forceF= ˆrFr(r), whereFr(r) is anarbitrary function of the coordinate distancerfrom a fixed center:1. Show that the energyEis s constant. What property of the force is used?2. Show that the angular momentumLis s constant. What property of the forceis used?3. Show that, as s consequence of the previous part, the motion of the particle isin a plane.4. Show that, as a consequence, the trajectory of motion in polar coordinate canbe solved by quadrature. (i.e., the time-dependence of the coordinates can beexpressed as integrals, which you should express, but which you cannot evaluateuntil the functionFr(r) is specified.) For this part it will be useful to introduceaneffective potentialincorporating the angular momentum conservation.5. SupposeFr(r) is attractive and proportional torn, wherenis an integer. Forwhat values ofnare stable circular orbits possible?[Hint:Use the effective potential defined before, make a rough drawing of thedifferent possible situations, and argue qualitatively using this drawing.]Classical MechanicsQEID#15664097February, 2018

Qualification ExamQEID#1566409721

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