Pendulum period

1. Mar 18, 2007

Ja4Coltrane

I wanted to calculate the period of a pendulum withould the small angle approximation. I carried out the calculation and came out with a rather ugly integration which was the same as the one wikipedia had so I assume I did it right. The answer I got was:

T=4[root of(L/2g)]*integral of{1/[root of(cos@-cos@initial)] from 0 to @initial}

However when I performed a numerical integration with my calculator, the answer I got was completely absurd. Any help?

Last edited: Mar 18, 2007
2. Mar 18, 2007

tim_lou

the integral should be correct. However, if you use the small angle approximation, the $\cos\theta$ should be gone!

notice that for small angle,
$$\cos\theta \approx 1-\frac{\theta^2}{2}$$

you'll get a nice integral.

*notice that the integral is improper. that is, at theta=initial angle, the integrand goes to infinity. so some calculator will give you weird answers.

Last edited: Mar 18, 2007
3. Mar 18, 2007

Ja4Coltrane

but I just wanted to do it without the small angle.
I'm just trying to figure out why I'm not getting a reasonable answer for the numerical integration.

4. Mar 18, 2007

Ja4Coltrane

for a 5m string, I'm getting 258.3s

5. Mar 18, 2007

tim_lou

I edited my first post. the integral is improper, some calculators give you non-sense answer for improper integrals. if you want more info, look up elliptic integral, there are some nice infos on mathworld.com

http://mathworld.wolfram.com/EllipticIntegraloftheFirstKind.html

6. Mar 18, 2007

HallsofIvy

Staff Emeritus
Big, big, big problem! Without the "small" angle restriction, a pendulum may not even have a "period"!! It is theoretically possible to place a pendulum directly upward, where it is balanced, wait for some tiny ripple of air to "knock it over" and have it come right back up to a balance again.

1) What initial angle you used
2) What period you got
3) Why you think it is "absurd"

7. Mar 18, 2007

Ja4Coltrane

mmmmm
I see, I'm only in a first year of calculus so I dont know all this stuff. I know that there is some sort of a series to do this, but how would one normally carry out this calculation?

8. Mar 18, 2007

Ja4Coltrane

Well, I did the calculation with a 15 degree angle and 5 meters length and got a 258s period. By the way I only did a small angle to test it--to compare to 2pi*root of (L/g)

9. Mar 18, 2007

Saketh

One would normally use elliptic integrals. See the MathWorld page mentioned above for the mathematical intricacies. You can probably understand the page with first-year calculus, but it will take a lot of effort on your part.

10. Mar 18, 2007

Ja4Coltrane

Well uh-oh.

This is interesting--I felt like with my knowledge of calculus I could solve any summation problem like this, but it appears that I was quite wrong. Kind of frustrating really.