# Solid pivoted pendulum attached to a spring – oscillation period?

1. The problem statement, all variables and given/known data
The figure shows a 200 g uniform rod pivoted at one end. The other end is attached to a horizontal spring. The spring is neither stretched nor compressed when the rod hangs straight down. What is the rods oscillation period? You can assume that the rods angle from vertical is always small.

2. Relevant equations

τ=2∏/ω
ω=√(k/M) for spring
ω=√(Mgl/I) for solid pendulum

3. The attempt at a solution

If not for the spring, this problem would simply be a solid pendulum, and the solution would simply be:
τ = $\frac{2∏}{√(Mgl/I)}$

And if this were just a particle mass attached to the spring, the solution would be:
τ = $\frac{2∏}{√(k/M)}$

I’m having trouble figuring out what happens to the equation in this case, though. One guess is as follows:

τ = $\frac{2∏}{√(k/M) + √(Mgl/I)}$

But I’m clueless as to whether this is right or how to derive a correct equation. It seems like I might have to figure out the restoring force and/or torque in this case, but isn’t that just -kx and -MglΘ? If anyone could help me toward the right direction, I’d appreciate it.

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