Calculating Pendulum Period: A Challenge!

[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?

 

[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.

 

[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.

 

[Ja4Coltrane]

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

 

[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

 

[HallsofIvy]

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.

I don't know how to help you because I don't know:
1) What initial angle you used
2) What period you got
3) Why you think it is "absurd"

 

[Ja4Coltrane]

mmmmm
I see, I'm only in a first year of calculus so I don't 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?

 

[Ja4Coltrane]

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.

I don't know how to help you because I don't know:
1) What initial angle you used
2) What period you got
3) Why you think it is "absurd"

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)

 

[Saketh]

mmmmm
I see, I'm only in a first year of calculus so I don't 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?

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.

 

[Ja4Coltrane]

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.

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.