Question

The exact equation of motion of a simple pendulum is$d^2\theta /d{t}^2=-{\omega }^{2}sin\theta$where${\omega }^2=g/l$. By method (c) above, integrate this equation once to find$d\theta /dt$if$d\theta /dt=0$when$\theta =90°$. Write a formula for$t\left(\theta \right)$as an integral. See Problem 5.34.

Short answer

The solution of the given differential equation is$t\left(\theta \right)=\int \sqrt{\frac{l}{2g}}\cdot \sqrt{sec\theta }d\theta +c$

Step-by-step solution

Given information

The given equation.

$\frac{d^2\theta }{d{t}^2}=-{\omega }^{2}sin\theta$

Differential equation

Definition: A differential equation is an equation that relates one or more unknown functions and their derivatives.

Solve for differential equation

Equation is$\frac{d^2\theta }{d{t}^2}=-{\omega }^{2}sin\theta$

Multiply both side of the equation by $d\theta /dt$

Solve further,

It is given that $\frac{d\theta }{dt}=0$at$\theta =90°$.

So,

$$\begin{gathered} 0=2{\omega }^{2}cos{90}^{°}+2c \\ 0=0+2c \\ c=0 \end{gathered}$$

Thus,

Integrate above equation

Integrate both sides,

Therefore, the solution of the given differential equation is $t\left(\theta \right)=\int \sqrt{\frac{l}{g}}\cdot \sqrt{sec\theta }d\theta +c$