69. To What Angle? A small metal ball of mass $m$ hangs from a pivot by a rigid, light metal rod of length $R .$ The ball is swinging back and forth with an amplitude that remains small throughout its motion, $\theta^{\max } \leq 5^{\circ} .$ Ignore all damping. (a) The equation of motion of this ideal pendulum can be derived in a variety of ways and is
$$
\frac{d^{2} \theta}{d t^{2}}=-\frac{g}{R} \sin \theta
$$
For small angles, show how this can be replaced by an approximate equation of motion that can be solved more easily than the one given. (b) Write a general solution for the approximate equation of motion you obtained in (a) that works for any starting angle and angular velocity (as long as the angles stay in the range where the approximation is OK). Demonstrate that what you have written is a solution and show that at a time $t=0$ your solution can have any given starting position and velocity. (c) If the length of the rod is $0.3 \mathrm{~m}$, the mass of the ball is $0.2 \mathrm{~kg}$, and the clock is started at a time when the ball is passing through the center $(\theta=0)$ and is moving with an angular speed of $0.1 \mathrm{rad} / \mathrm{s}$, find the maximum angle your solution says the ball will reach. Can you use the approximate equation of motion for this motion? If the starting angle is not small, you cannot easily solve the equation of motion without a computer. But there are still things you can do.
(d) Derive the energy conservation equation for the motion of the pendulum. (Do not use the small-amplitude approximation.) (e) If the pendulum is released from a starting angle of $\theta_{1}$, what will be the maximum speed it travels at any point on its swing?