A Simple Pendulum

1. Oct 25, 2005

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?

2. Oct 25, 2005

brentd49

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

3. Oct 26, 2005

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.

4. Oct 26, 2005

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.

5. Oct 26, 2005

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).

Last edited by a moderator: Apr 21, 2017
6. Oct 27, 2005

dextercioby

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

Daniel.

7. Oct 27, 2005

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.