Chapter 16, Problem 9PS

Pendulum Consider a pendulum of length L that swings by the force of gravity only.

For small values, the motion of the pendulum can be approximated by the differential equation

$\frac{d^2\theta }{d{t}^2}+\frac{g}{L}\theta =0$

where g is the acceleration due to gravity.

Find the general solution of the differential equation and show that it can be written in the form

$\theta (t)=A\cos \left[\sqrt{\frac{g}{L}}\left(t+\varphi \right)\right]$

Find the particular solution for a pendulum of length 0.25 meter when the initial conditions are $\theta (0)=0.1$ radian and ${\theta }^{\prime }(0)=0.5$ radian per second. (Use $g=9.8$ meters per second per second.)

Determine the period of the pendulum.

Determine the maximum value of $\theta$.

How much time from t = 0 does it take for $\theta$ to be 0 the first time? The second time?

What is the angular velocity ${\theta }^{\prime }$ when $\theta$⁼⁰ the first time? the second time?