QuestionAs you will learn in Chapter 13, the angular acceleration of a simple pendulum is given by $\ddot{\theta}=-(g / L) \sin \theta$, where $g$ is the acceleration of gravity and $L$ is the length of the pendulum cord.
Problem 12.74. Derive the expression of the angular velocity $\ddot{\theta}$ as a function of the angular coordinate $\theta$. The initial conditions are $\theta(0)=\theta_{0}$ and $\dot{\theta}(0)=\dot{\theta}_{0}$
Answer Problem 12.75 Let the length of the pendulum cord be $L=1.5 \mathrm{~m}$. If $\dot{\theta}=3.7 \mathrm{rad} / \mathrm{s}$ when $\theta=14^{\circ}$, determine the maximum value of $\theta$ achieved by the pendulum.
Problem 12.76 The given angular acceleration remains valid even if the pendulum cord is replaced by a massless rigid bar. For this case, let $L=5.3 \mathrm{ft}$ and assume that the pendulum is placed in motion at $\theta=0^{\circ}$. What is the minimum angular velocity at this position for the pendulum to swing through a full circle?
Answer $$\dot{\theta}_{\min }=4.930 \mathrm{rad} / \mathrm{s}$$