- #1

okalakacheekee

I substituted (w=omega) dw/dt for d2theta/dt2...and eventually got 1/2 w^2 = g/l cos theta + C...but that doesn't give me theta as a function of time?

Any help is appreciated...thanks

- Thread starter okalakacheekee
- Start date

- #1

okalakacheekee

I substituted (w=omega) dw/dt for d2theta/dt2...and eventually got 1/2 w^2 = g/l cos theta + C...but that doesn't give me theta as a function of time?

Any help is appreciated...thanks

- #2

ahrkron

Staff Emeritus

Gold Member

- 736

- 1

This problem usually asks for an approximation for small theta.

When that is the case, you can use the fact that

[tex]\sin \theta \rightarrow \theta[/tex],

[tex]\cos \theta \rightarrow 1 - \frac{\theta^2}{2}[/tex]

When that is the case, you can use the fact that

[tex]\sin \theta \rightarrow \theta[/tex],

[tex]\cos \theta \rightarrow 1 - \frac{\theta^2}{2}[/tex]

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- #3

okalakacheekee

but is there a way to solve it exactly?

- #4

okalakacheekee

so let's say O is theta, w is omega, for ease of writing

first i have

d2O/dt2 = -g/l sin O

I said w = dO/dt and then dw/dt=d2O/dt2

eventually i got to a point where i had

w dw = -g/l sin O dO

so i integrated and got

1/2 w^2 = g/l cos O + C

problem being, i had no time in there...

so then i put back in dO/dt for w and got

dO/dt = sqrt(2g/l * cosO)

so when you separate everything you get

dO/sqrt (cosO) = sqrt(2g/l) dO

then you integrate..here's where i ran into trouble yet again...how do you integrate the left hand side?

- #5

HallsofIvy

Science Advisor

Homework Helper

- 41,833

- 961

A standard attack is "linearization"- for small values of θ, replace sin(θ) by its linear approximation θ to get the linear equation d

Another method is "quadrature" which is basically what you are doing. Let ω= dθ/dt so that d

That can be integrated to get (1/2)ω

You could, of course, rewrite that as ω= dθ/dt= √((2g/l)cos(θ)+ C); but the resulting integral is an "elliptic integral" which cannot be integrated exactly.

- #6

krab

Science Advisor

- 896

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What you've done turns up the very useful relation of pendulum speed versus amplitude. Another way to derive it is to write KE + PE = [tex]{1\over 2}mv^2-mgl\cos\theta=\mbox{constant}[/tex].Originally posted by okalakacheekee

so let's say O is theta, w is omega, for ease of writing

first i have

d2O/dt2 = -g/l sin O

I said w = dO/dt and then dw/dt=d2O/dt2

eventually i got to a point where i had

w dw = -g/l sin O dO

so i integrated and got

1/2 w^2 = g/l cos O + C

problem being, i had no time in there...

so then i put back in dO/dt for w and got

dO/dt = sqrt(2g/l * cosO)

so when you separate everything you get

dO/sqrt (cosO) = sqrt(2g/l) dO

then you integrate..here's where i ran into trouble yet again...how do you integrate the left hand side?

You can plot curves in v vs. θ space (this space is called phase space); these will look like ellipses but become distorted into eye-shaped as the pendulum amplitude reaches large enough angles.

If you know enough "special functions", then yes, the problem is solvable in closed form. Look up info on "Jacobian Elliptic Functions". They're not as common as sines and cosines, but they are just as legitimate in the "finding a closed form solution" sense.

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