By Sidney B. Cahn, Boris E. Nadgorny, C.N. Yang
Pt. 1. Mechanics, relativity, and electrodynamics -- pt. 2. Thermodynamics, statistical physics, and quantum mechanics
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Additional info for A Guide to Physics Problems. Mechanics, Relativity, and Electrodynamics
Use as variable the distance between the sleeves. 34 Double Pendulum (Stony Brook, Princeton, MIT) The double pendulum consists of a mass suspended by a massless string or rod of length from which is suspended another such rod and mass (see Figure P. 34). 21 MECHANICS a) Write the Lagrangian of the system for b) Derive the equations of motion. 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.
The gravitational acceleration is taken to be a constant in the negative direction. a) Write the expressions for the potential energy U and the length for a given curve b) Formulate the Euler-Lagrange equations for the curve with minimal potential energy, subject to the condition of fixed length. c) Show that the solution of the previous equation is given by A where A and B are constants. Calculate U and for this solution. 50 Rotating Hollow Hoop (Boston) A thin hollow cylindrical pipe is bent to form a hollow circular ring of mass and radius R.
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. Evaluate the integral in the limiting case and discuss the meaning of your answer.
A Guide to Physics Problems. Mechanics, Relativity, and Electrodynamics by Sidney B. Cahn, Boris E. Nadgorny, C.N. Yang