A spherical pendulum consists of a bob of mass \(m\) on the end of a light string of length \(R\) hung from a point on the ceiling, and with a uniform gravitational field \(g\) downward. The position of the bob can be specified by the polar angle \(\theta\) of the string (the angle of the string and bob from the vertical) and the azimuthal angle \(\varphi\) (the angle of the string and bob from, say, the north as projected down onto a horizontal base plane.)

(a) Show that the square of the velocity of the bob at any moment is \(v^{2}=R^{2}\left(\dot{\theta}^{2}+\sin ^{2} \theta \dot{\varphi}^{2}\right)\). Then in terms of any or all of \(m, R, g\), and the two coordinates \(\theta\) and \(\varphi\) and their first time derivatives

(b) find an expression for the energy \(E\) of the bob and explain why it is conserved;

(c) find an expression for the angular momentum \(\ell\) of the bob about the vertical axis passing through the point of support, and explain why it is conserved.