Water droplets are hilarious

Classical Mechanics Level 3

When an astronaut drops a drop of water at the International Space Station, it appears as if the water droplet is oscillating, as shown in the video clip.

We cannot observe this phenomenon on the earth due to the earth's gravity. Regardless of this gravity effect, if an astronaut drops two types of fluids $(A$ and $B)$ of the same radius $r,$ the densities of which are $\rho_A$ and $\rho_B$ and the surface tensions of which are $\sigma_A$ and $\sigma_B,$ determine the ratio of oscillation frequencies between liquid $A$ and liquid $B.$

Details and Assumptions:

Fluid densities are $\rho_A=1 \text{ g/cm}^3$ and $\rho_B=12.1 \text{ g/cm}^3.$
Surface tensions are ${ \sigma }_{ A }=0.0405 \text{ N/m}$ and ${ \sigma }_{ B }=0.5 \text{ N/m}.$

The answer is 0.99.

1 solution

Laszlo Mihaly
Oct 10, 2017

We do this by "dimensional analysis", looking at the units of the parameters.

The surface tension \sigma is measured in N/m=kg/s^2. The radius R is meters, the density \rho is kg/m^3 and the frequency \omega is 1/s. In order to get the correct unit for the frequency we need to combine the other 3 quantities as \alha (\sigma/ \rho R^3)^0.5. The numerical factor \alpha depends on the mode of the oscillation. If the two droplets oscillate in the same mode and the radius is the same, the ratio of the frequencies will be (\sigma1/\sigma2 \rho2/\rho1)^0.5 = 0.99.

Water droplets are hilarious

The answer is 0.99.

1 solution

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