Buoyancy correction in a Kater's pendulum

In summary, Julien attempted to derive the buoyancy effect for a pendulum, but he was not able to solve for it. He is looking for help from his classmates.
  • #1
JulienB
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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 going to 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.
 
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  • #2
JulienB said:
##= (\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?

Can you expand ##\frac{\rho}{\rho - \rho_L}## to first order in a small quantity?
 
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  • #3
@TSny Hi and thanks for your answer. There is no other indication than what I have written. The problem with expansion is also that I get an extra ##+1##...

Julien.
 
  • #4
You should have something proportional to ##(1+\frac{ \varphi_0^2}{ 8})(\frac{\rho}{\rho - \rho_L})##. As you say, if you expand the second quantity to first order, the second quantity will be of the form 1 + ε, where ε is a small quantity. But, you can then multiply the whole thing out and keep only terms up to first order in small quantities.
 
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  • #5
@TSny I see. Thanks for your answer, our teacher explained something similar today.Julien.
 

1. What is buoyancy correction in a Kater's pendulum?

Buoyancy correction in a Kater's pendulum is a method used to account for the effects of air resistance and buoyancy on the oscillation period of the pendulum. This correction is necessary for accurate measurements in experiments.

2. How does buoyancy affect the oscillation period of a Kater's pendulum?

Buoyancy causes a reduction in the effective gravitational force acting on the pendulum, which leads to an increase in the oscillation period. This is because the buoyant force opposes the weight of the pendulum, making it swing slower.

3. How is buoyancy correction calculated for a Kater's pendulum?

The buoyancy correction is calculated by taking the difference between the actual density of the pendulum and the effective density (including the effects of air resistance and buoyancy). This difference is then multiplied by the oscillation period of the pendulum squared.

4. Why is buoyancy correction important in Kater's pendulum experiments?

Buoyancy correction is important because it allows for more accurate measurements of the oscillation period, which is a crucial factor in determining the value of gravitational acceleration. Without this correction, the results of the experiment may be significantly skewed.

5. Are there any other factors that may affect the accuracy of buoyancy correction in Kater's pendulum experiments?

Yes, there are other factors such as temperature and pressure that can also affect the accuracy of buoyancy correction. These factors can cause changes in the density of air, which in turn affects the oscillation period of the pendulum. It is important to control these variables in order to obtain precise results.

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