Calculating Time for a Simple Pendulum

[ursubaloo]

Consider the following problem: (this isn't homework, I thought this problem up myself and I'm wondering how to do it)

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You have a simple pendulum of mass M and a radius R, which is released from the horizontal. How much time does it take to reach the lowest point, as a function of R?
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It's easy to find the velocity as a function of the angle, but I couldn't figure out a way to factor time into it. There is also an approximation of the period of an osscilating pendulum which is equal to 2pi*root(L/g), but that holds only for small angle values.
So how do you do it?

 

[brentd49]

It's easy to find the velocity as a function of the angle, but I couldn't figure out a way to factor time into it.

v=dx/dt-> just separate and integrate.

 

[dextercioby]

Write the total mechanical energy balance and then use the definition of velocity.

Daniel.

P.S.It's an elliptic integral of the first kind at the end.

 

[HallsofIvy]

The equation of motion for a simple pendulum is
$$\frac{d^2\theta}{dt^2}= -(g/l) sin(\theta)$$.
Since t does not appear explicitely, if we let $\omega$ be the angular speed, we can convert this to
$$\omega\frac{d\omega}{d\theta}= -(g/l) sin(\theta)$$
$$\omega d\omega= -(g/l) sin(\theta)d\theta$$
which can be integrated to give
$$\omega^2= (2m/l) cos(\theta)+ C$$
Taking $\theta= \frac{\pi}{2}$ when $\omega= 0$
(releasing the pendulum from rest at the horizontal), we get
$$\omega^2= (2m/)(cos(\theta)- 1)$$
I presume that is what ursubaloo meant saying "It's easy to find the velocity as a function of the angle".

brentd49 is correct saying "just separate and integrate", except for the word "just". As dextercioby said, that's an elliptic integral and can only be done numerically.

 

[neutrino]

If anyone's interested...
An accurate formula for the period of a simple pendulum oscillating beyond the small-angle regime - http://arxiv.org/pdf/physics/0510206" (pdf).

 

[dextercioby]

It's unbelieveble that there are people nowadays which don't use latex when writing an article & post it on "arxiv"... :

Daniel.

 

[krab]

It's also sometimes unbelievable the feeble stuff some people publish. Elliptic integrals have been investigated for over a 100 years, and many approximations superior to the one given in this reference have been worked out. All they had to do was look.