Initialization

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Work

Pendulum Period Calculation

This worksheet is intended to get you STARTED, but it is not complete; there is work to be done yet.

The expression for the period of a pendulum oscillating to a maximum angle (Subscript[θ, max]) contains the integral


						(∫_0^(Subscript[θ, max]))(1/(Sqrt[cos(θ)-cos((Subscript[θ, max]))]))dθ.

Thetamaxdeg=90
A=Thetamaxdeg*Pi/180

Integrate[1/Sqrt[Cos[θ]-Cos[A]],{θ,0,A}]

Notice that the result is a special kind of function called "EllipticK", which you've probably never seen, but that's perfectly OK. Notice that Mathematica gives the functional form, not the numerical value.  Figure out how to find a numerical value, because you must use this number in your plot.

FunctionYouHaveToLookup[argumentyouhavetodecideupon]

You now have the tools to finish the exercise, but for those of you who want a bit more elegance from this exercise, read on.

You could use Mathematica to plot for you. What should you plot?  What are the limits?

Plot[]

How else could Mathematica help you understand and interpret your task?

More about the function "EllipticK":
This function is a special one - it's called an elliptical integral of the first kind and it is defined like this:

					EllipticK(k) = (∫_0^1)(1/((Sqrt[1-(t^2)])(Sqrt[1-(k^2)(t^2)])))dt.   
In our case, the argument k = sin(((Subscript[θ, max])/2)) and you should show analytically that out integral (above) is indeed Sqrt[2]*EllipticK[Sin[((Subscript[θ, max])/2)]].
					
In some versions of the course, you may be asked to show that the period of the pendulum is approximately  
						T = (Subscript[T, o])(1+(((Subscript[θ, max])^2)/16))				
and you should plot this approximation on the same plot as the exact result to see over what range this approximation is valid.  For extra practice, you can find the next term in the approximation, and plot this, too.