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## Homework Statement

Hi everybody! While preparing my next experiment (Kater's pendulum), I was given for homework to derive an equation to correct the buoyancy when calculating ##g##. I am given the result:

##g_c = (\frac{2 \pi}{T(\varphi_0)})^2 l_r (1 + \frac{\varphi_0^2}{8} + \frac{\rho_L}{\rho})##

where ##\varphi_0## is the amplitude, ##l_r## is the length of the equivalent simple pendulum, ##\rho_L## is the density of the air, ##\rho## is the density of the pendulum.

**2. The attempt at a solution**

I manage to get the ##1 + \frac{\varphi_0^2}{8})## from the Taylor development of ##T(\varphi_0)## and taking the approximation ##\sin \varphi \approx \varphi##.

Then I thought I was gonna derive the buoyancy effect pretty easily from the torque, but that's what I get:

##\tau = m g l_r \sin \varphi - l_r \rho_L V g \sin \varphi = I \ddot{\varphi}##

##\implies \ddot{\varphi} - \frac{l_r g (m - \rho_L V)}{I} \varphi = 0##

##\implies T^2 = (2 \pi)^2 \frac{I}{l_r g (m - \rho_L V)}##

##\implies g = (\frac{2 \pi}{T})^2 \frac{I}{l_r (m - \rho_L V)}##

##= (\frac{2 \pi}{T})^2 \frac{m l_r^2}{l_r (m - \rho_L V)}##

##= (\frac{2 \pi}{T})^2 l_r \frac{\rho}{\rho - \rho_L}##

Mmm... That's not too far but not quite it! I've been thinking a lot about it now, and I don't get why it's not working. Does anyone have an idea? (Note that the Taylor development is added by putting ##T(\varphi_0)## inside the equation)

Thanks a lot in advance for your answers!

Julien.