Question

22*. A point mass on the end of a light string rotates as a conical pendulum with angular velocity ω, the string being inclined at an angle θ to the vertical. Show that if the motion is slightly disturbed the result- ing small oscillations have an angular frequency o(1 +3 cos20)+.

Answer 1

For a conical pendulum Lagrangin l = K.E - P.E. = 1/2 mr^2 (theta^2 + phi^2 sin^2 theta) - m g l (1 - cos theta) theta = curst here rightdoublearrow theta = 0. [conical] phi is in plane of motion movement rightdoublearrow phi' = w partial differential L/partial differential phi = 0 rightdoublearrow d/dt partial differential L/partial differential phi = 0 partial differential L/partial differential phi = const rightdoublearrow ml^2 phi sin^2 theta = const as m & are cosnt rightdoublearrow phi sin^2 theta = const rightarrow partial differential L/partial differential theta rightdoublearrow ml^2 theta phi^2 ml^2 sin theta cos theta + m g l sin theta = 0 rightdoublearrow theta + g/l sin theta - theta^2 sin^4 theta cos theta/sin^3 theta = 0 - (1) rightarrow For theta = 0 rightdoublearrow phi^2 = g/l sec theta rightdoublearrow w = squareroot g/l sec theta T = 2 pi/w rightarrow Suppose that perturbation won't change ''T & ''phi'' sin theta = const'' From eqn (1) rightdoublearrow theta'' + g/l (sin theta - sin^4 theta_0/cos theta_0