1.3 KiB
1.3 KiB
PROBLEMS PROBLEM 1. Determine the period of oscillations of a simple pendulum (a particle of mass m suspended by a string of length l in a gravitational field) as a function of the amplitude of the oscillations. SOLUTION. The energy of the pendulum is E = 1ml2j2-mgl cos = -mgl cos to, where o is the angle between the string and the vertical, and to the maximum value of . Calculating the period as the time required to go from = 0 to = Do, multiplied by four, we find -cos po) The substitution sin $ = sin 10/sin 100 converts this to T = /(l/g)K(sin 100), where 1/75
is the complete elliptic integral of the first kind. For sin 100 22 100 < 1 (small oscillations), an expansion of the function K gives T = §12 Determination of the potential energy 27 The first term corresponds to the familiar formula. PROBLEM 2. Determine the period of oscillation, as a function of the energy, when a particle of mass m moves in fields for which the potential energy is (a) U = Alx (b) U = Uo/cosh2ax, -U0 0, (c) U = Uotan2ax. SOLUTION. (a): T = By the substitution yn = u the integral is reduced to a beta function, which can be expressed in terms of gamma functions: The dependence of T on E is in accordance with the law of mechanical similarity (10.2), (10.3). (b) T = (7/a)V(2m/E). (c) T =(t/a)v[2m/(E+U0)]