By S.Cahn, B.Nadgorny
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Extra info for A Guide to Physics Problems. part 12 Mechanics, Relativity, and Electrodynamics
8 Car down Ramp up Loop (Stony Brook) A car slides without friction down a ramp described by a height function which is smooth and monotonically decreasing as increases from 0 to L. The ramp is followed by a loop of radius R. Gravitational acceleration is a constant in the negative direction (see Figure P. 8). a) If the velocity is zero when what is the minimum height such that the car goes around the loop, never leaving the track? b) Consider the motion in the interval before the loop. Assuming that the car always stays on the track, show that the velocity in the direction is related to the height as c) In the particular case that show that the time elapsed in going down the ramp from can be expressed as where and write as a definite integral.
C) Find the eigenfrequencies. 35). and attached to a sinrespectively from its point of a) Determine the value of such that one of the normal frequencies of this system will equal the frequency of a simple pendulum of length and mass You may assume the displacements of the masses from equilibrium are small. b) Find the mode corresponding to this frequency and sketch it. 36 Three Masses and Three Springs on Hoop (Columbia, Stony Brook, MIT) Three masses, each of mass are interconnected by identical massless springs of spring constant and are placed on a smooth circular hoop as PROBLEMS 22 shown in Figure P.
5 Relativistic Rocket (Rutgers) A rocket having initially a total mass ejects its fuel with constant velocity relative to its instantaneous rest frame. According to Newtonian mechanics, its velocity V , relative to the inertial frame in which it was originally at rest, is related to its mass M(V) by the formula a) Derive this result. b) Suppose the velocity of the ejecta is limited only by and derive the relativistic analogue of the above equation. Show that it reduces to the Newtonian result at the appropriate limit.
A Guide to Physics Problems. part 12 Mechanics, Relativity, and Electrodynamics by S.Cahn, B.Nadgorny