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.)
2. Relevant equations
h=distance from pivot to com, meaning h=L+r
3. 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:
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?