7.40 *** The "spherical pendulum" is just a simple pendulum that is free to move in any sideways direction. (By contrast a “simple pendulum" – unqualified – is confined to a single vertical plane.) The bob of a spherical pendulum moves on a sphere, centered on the point of support with radius r = R, the length of the pendulum. A convenient choice of coordinates is spherical polars, r, 0, 6, with the origin at the point of support and the polar axis pointing straight down. The two variables 0 and o make a good choice of generalized coordinates. (a) Find the Lagrangian and the two Lagrange equations. (b) Explain what the o equation tells us about the z component of angular momentum €,. (c) For the special case that o = const, describe what the e equation tells us. (d) Use the o equation to replace o by l, in the e equation and discuss the existence of an angle 0, at which e can remain constant. Why is this motion called a conical pendulum? (e) Show that if 0 = 0, + €, with e small, then 0 oscillates about 0, in harmonic motion. Describe the motion of the pendulum's bob. %3D

7.40 *** The "spherical pendulum" is just a simple pendulum that is free to move in any sideways direction. (By contrast a “simple pendulum" – unqualified – is confined to a single vertical plane.) The bob of a spherical pendulum moves on a sphere, centered on the point of support with radius r = R, the length of the pendulum. A convenient choice of coordinates is spherical polars, r, 0, 6, with the origin at the point of support and the polar axis pointing straight down. The two variables 0 and o make a good choice of generalized coordinates. (a) Find the Lagrangian and the two Lagrange equations. (b) Explain what the o equation tells us about the z component of angular momentum €,. (c) For the special case that o = const, describe what the e equation tells us. (d) Use the o equation to replace o by l, in the e equation and discuss the existence of an angle 0, at which e can remain constant. Why is this motion called a conical pendulum? (e) Show that if 0 = 0, + €, with e small, then 0 oscillates about 0, in harmonic motion. Describe the motion of the pendulum's bob. %3D