7.40 *** The "spherical pendulum" is just a simple pendulum that is free to move in any sidewaysdirection. (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, O, 4, with theorigin at the point of support and the polar axis pointing straight down. The two variables 6 and ø makea 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 thespecial case that o = const, describe what the 0 equation tells us. (d) Use the o equation to replace oby l, in the 0 equation and discuss the existence of an angle 0, at which 0 can remain constant. Whyis this motion called a conical pendulum? (e) Show that if 0 = 0, + €, with e small, then 0 oscillatesabout 0, in harmonic motion. Describe the motion of the pendulum's bob.

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Answered: 7.40 *** The "spherical pendulum" is…

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