Length of Pendulum with Variables Only

AI Thread Summary
To determine the length of the pendulum rod L for a grandfather clock pendulum with a given period T, the equations of motion and moments of inertia must be utilized. The relevant formulas include T=2*pi*sqrt(I/mgh) and I=(mr^2)/2+mh^2, where h is the distance from the pivot to the center of mass. The user attempted to isolate L from the derived equation but found it challenging, indicating a potential oversight in the simplification process. A suggestion was made to square the equation and multiply by the denominator to obtain a quadratic form, which could facilitate solving for L. The discussion emphasizes the importance of correctly manipulating the equations to find the desired variable.
efitzgerald21
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Homework Statement


A grandfather clock has a pendulum that consists of a thin brass disk of radius r and mass m that is attached to a long thin rod of negligible mass. The pendulum swings freely about an axis perpendicular to the rod and through the end of the rod opposite the disk, as shown in the figure below. If the pendulum is to have a period T for small oscillations, what must be the rod length L. (Use any variable or symbol stated above along with the following as necessary: g for the acceleration of gravity.)
Picture: http://www.webassign.net/hrw/15-54.gif

Homework Equations


T=2*pi*sqrt(I/mgh)
h=distance from pivot to com, meaning h=L+r
I=(mr^2)/2+mh^2

The Attempt at a Solution


I've tried to isolate L from the equation, but I can't isolate it. This is what I end up with:
T=2*pi*sqrt[((mr^2)/2+m(L+r)^2)/(mg(L+r))]
Cancelling m, I get T=2*pi*sqrt[((r^2)/2+(L+r)^2)/(g(L+r))]
After simplification, I get T=2*pi*sqrt[((r^2)/2+L^2+2Lr+r^2)/(gL+gr)]
There is no way to isolate L in this equation, I'm stuck.
I feel like there must be some piece of information I'm just not seeing, because otherwise this problem is impossible to solve.
Can you tell me what I'm missing here?
 
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This gets a simple quadratic equation if you square it (would be my first step to simplify it) and multiply it with the denominator.
 
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