By S.Cahn, B.Nadgorny
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Extra info for A Guide to Physics Problems. part 12 Mechanics, Relativity, and Electrodynamics
18 Yukawa Force Orbit (Stony Brook) A particle of mass moves in a circle of radius R under the influence of a central attractive force a) Determine the conditions on the constant such that the circular motion will be stable. b) Compute the frequency of small radial oscillations about this circular motion. 19). Assume that when the particle is reflected, its speed is unchanged and its angle of incidence equals its angle of reflection. The particle is attracted to the origin by a potential where c is some constant.
28)? 29 17 Unwinding String (MIT, Maryland (a,b), Chicago (a,b)) A point mass is attached to a long, massless thin cord whose other end is attached to a fixed cylinder of radius R. Initially, the cord is snugly and completely wound up around a circular cross section of the cylinder, so that the mass touches the cylinder. 29a). The point P is the initial position of the mass, and Q denotes the instantaneous contact point between the cord and the cylinder. a) Find the Lagrangian and equation of motion in terms of the generalized coordinate as a function of time, satisfying the initial conditions.
26). The motion is frictionless. 16 PROBLEMS a) Find the Lagrangian of the system and derive the equations of motion. b) Show that the orbit is stable with respect to small changes in the radius, and find the frequency of small oscillations. 27 Falling Chimney (Boston, Chicago) A tall, slender, cylindrical brick chimney of height L is slightly perturbed from its vertical equilibrium position so that it topples over, rotating rigidly around its base B until it breaks at a point P. Show that the most likely value for the distance of P from B is L/3.
A Guide to Physics Problems. part 12 Mechanics, Relativity, and Electrodynamics by S.Cahn, B.Nadgorny