Angular frequency - how is it derived?

[UrbanXrisis]

How is the formula for angular derived?

\omega=\sqrt{\frac{g}{L}}

my book has these equations:

F= -mg sin\theta = m \frac{d^2s}{dt^2}

\frac{d^2 \theta}{dt^2}=-\frac{g}{L}sin\theta

\frac{d^2 \theta}{dt^2}=-\frac{g}{L}\theta

\omega=\sqrt{\frac{g}{L}}

what exactly is m \frac{d^2s}{dt^2} and \frac{d^2 \theta}{dt^2}

 

[dextercioby]

\mbox{s} [/itex] is arc length...Because the movement is on a circle<br /> <br /> s=L\theta<br /> <br /> ,where \theta is the angle at the center of the circle...<br /> <br /> Daniel.

 

[Galileo]

s is the position of the particle, so the second derivative wtr time denotes the acceleration 'a' of the particle. (Hence F=ma in the first equation)

Solve the D.E. and check what that you get a periodic function. What is its angular frequency?

 

[dextercioby]

Instead of applying Newton's second law for the translation movement \frac{d\vec{\mbox{p}}}{dt}=\sum_{k} \vec{\mbox{F}}_{k},try to apply it for the rotation movement \frac{d\vec{\mbox{L}}}{dt}=\sum_{k} \vec{\mbox{M}}_{k}


Daniel.

 

[UrbanXrisis]

okay, I understand up to here:
\frac{d^2 \theta}{dt^2}=-\frac{g}{L}\theta


I think the next step is this...
\sqrt{\frac{d^2 \theta}{dt^2}}=\sqrt{-\frac{g}{L}\theta}
\sqrt{\frac{d^2 \theta}{dt^2}}=\frac{d \theta}{dt}= \omega
\omega = \sqrt{\frac{g}{L}

but I don't think I'm correct

 

[dextercioby]

Of course not.

It's a definition

\frac{g}{l}=:\omega_{0}^{2}

That's all to it.

Daniel.

 

[UrbanXrisis]

but how do I get \frac{d^2 \theta}{dt^2}=-\frac{g}{L}\theta to go to \omega=\sqrt{\frac{g}{L}}

 

[jtbell]

How much have you studied about solving differential equations? Here in the USA, very few people study differential equations at the K-12 level.

It turns out that a general solution of the differential equation

\frac {d^2 \theta} {d t^2} = - \frac {g}{l} \theta

is

\theta = A \sin \left(\sqrt {\frac {g}{l}} t + \theta_0\right)

where A and \theta_0 are arbitrary constants. You can verify this by working out the second derivative and plugging it back into the differential equation. A general form of a sinusoidal function is

x = A \sin (\omega t + \theta_0)[/itex]<br /> <br /> where \omega is the angular frequency. Matching up the preceding two equations gives you<br /> <br /> \omega = \sqrt {\frac {g}{l}}