The motion of the simple pendulum is described by the following differential equation
$$\frac{d^2 \theta}{dt^2}+\frac{g \sin \theta }{l}=0$$
Multiply through $$\frac{2d \theta}{dt}$$
and integrate and apply initial condition $\theta=\theta_0$ and $\dfrac{d \theta}{dt}=0$
Then, separate the resulting equation in the variable to obtain
$$ \frac{d \theta}{\sqrt{\cos\theta- \cos \theta_0}}=\pm \sqrt{\frac{2g}{l}}dt $$
My work
$$\frac{2 d \theta}{d t} \frac{d^2 \theta}{d t^2}+\frac{2 d \theta}{dt}\frac{g \theta }{l}=0$$
I got stuck here.I don't think I can proceed any further. Can someone show me some hint
$\endgroup$ 12 Answers
$\begingroup$$$\ddot\theta+k^2\sin\theta=0$$
can be rewritten
$$\dot\theta\ddot\theta+k^2\dot\theta\sin\theta=0$$
and after integration from $0$ to $t$,
$$\dot\theta^2-2k^2(\cos\theta-\cos\theta_0)=0.$$
Then we have a separable equation
$$\frac{\dot\theta}{\sqrt{\cos\theta-\cos\theta_0}}=\sqrt2k.$$ and by a second integration
$$\int_{\theta_0}^\theta\frac{d\theta}{\sqrt{\cos\theta-\cos\theta_0}}=\sqrt2kt.$$
This integral is a difficult one, requiring an Elliptic Integral of the First Kind:
$$\int_{\theta_0}^\theta\frac{d\theta}{\sqrt{\cos\theta-\cos\theta_0}}=\frac{2F\left(\frac\theta2,\csc^2\frac{\theta_0}2\right)}{\sqrt{1-\cos\theta_0}}.$$
For more tractability, one often assumes small angles so that $\cos\theta\approx1-\frac{\theta^2}2$ and
$$\int_{\theta_0}^\theta\frac{\sqrt2d\theta}{\sqrt{\theta_0^2-\theta^2}}=\sqrt2\arccos\frac\theta{\theta_0}.$$
Finally,
$$\theta=\theta_0\cos kt.$$
(This result can be directly obtained from the linearized equation $\ddot\theta+k^2\theta=0$, a much easier one.)
$\endgroup$ 0 $\begingroup$I'm kind of confused about what the question is exactly. But anyway, starting from $$ \frac{d^2\theta}{dt^2}+\frac{g}{l}\sin \theta=0 $$ and multiplying by $2d\theta/dt$ you get $$ 2\frac{d\theta}{dt}\frac{d^2\theta}{dt^2}+\frac{2g}{l}\frac{d\theta}{dt}\sin \theta=0 $$ Note that $$\frac{d}{dt}\left[\frac{d\theta}{dt}\right]^2=2\frac{d\theta}{dt}\frac{d^2\theta}{dt^2}$$ So $$ \frac{d}{dt}\left[\frac{d\theta}{dt}\right]^2=-\frac{d\theta}{dt}\frac{2g}{l}\sin \theta $$ I leave the rest to you. (Hint integrate the above equation over $t$ given the initial conditions.)
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