Chapter D2.4, Problem 47E

(a)

To determine

To explain: The reason for the equation $ml{{\theta }^{\prime }}^{\prime }\left(t\right)=-mg\sin \theta \left(t\right)$ is true where g is the gravitational pull.

(b)

To determine

To write: The equation obtained in part (a) as ${{\theta }^{\prime }}^{\prime }+{\omega }_0{}^2\sin \theta =0$ where ${\omega }_0{}^2=\frac{g}{l}$.

(c)

To determine

To show: That the linear pendulum equation is ${{\theta }^{\prime }}^{\prime }+{\omega }_0{}^2\theta =0$.

(d)

To determine

To express: The period of the pendulum in terms of g and l and find the factor by which the period alters when the length is increased by a factor of 2.