- #1

Combinatorics

- 36

- 5

- Homework Statement
- Given the following system (see the attached image) of two balls connected by a thin rod with negligible mass. The upper ball has a radius ##R_1## and density ##\rho_1## and the lower ball has a radius ##R_2## and density ##\rho_2##. This system is submerged in a liquid with density ##\rho_3##, where it floats in a stable equilibrium as shown in the image I attach.

Show that ##\rho_2>\rho_1##, and ## \rho_3 = \frac{\rho_1 R_1^3 + \rho_2R_2^3 }{R_1^3 + R_2 ^3 }##.

- Relevant Equations
- Bernoulli equation ## p+\rho gH+ 0.5 \rho v^2 = \text{constant}##

Buoyancy force ## B = \rho_3 V g##

Performing force balance on the two balls, I obtain

##

T+\frac{4\pi}{3}g \rho_3 R_1 ^3 = \frac{4\pi}{3}g \rho_1 R_1^3

##

##

T+\frac{4\pi}{3}g \rho_2 R_2^3 = \frac{4\pi}{3}g \rho_3 R_2^3

##

from which I obtain ## \rho_3 = \frac{\rho_1 R_1^3 + \rho_2R_2^3 }{R_1^3 + R_2 ^3 }## and ## 2T =\frac{4\pi}{3}g\rho_1R_1^3 - \frac{4\pi}{3}g \rho_2 R_2^3 ##.

Unfortunately, I have no idea how can I find any information regarding ## \rho_1,\rho_2##. I thought Bernoulli equation might help here, since the different pressures experienced by the two balls play a role here, but I do not know how to do it (as far as I know, Bernoulli equation may not be applicable in such a case, where there is more than just one liquid involved).

Will be happy to receive some help here. Thanks!

##

T+\frac{4\pi}{3}g \rho_3 R_1 ^3 = \frac{4\pi}{3}g \rho_1 R_1^3

##

##

T+\frac{4\pi}{3}g \rho_2 R_2^3 = \frac{4\pi}{3}g \rho_3 R_2^3

##

from which I obtain ## \rho_3 = \frac{\rho_1 R_1^3 + \rho_2R_2^3 }{R_1^3 + R_2 ^3 }## and ## 2T =\frac{4\pi}{3}g\rho_1R_1^3 - \frac{4\pi}{3}g \rho_2 R_2^3 ##.

Unfortunately, I have no idea how can I find any information regarding ## \rho_1,\rho_2##. I thought Bernoulli equation might help here, since the different pressures experienced by the two balls play a role here, but I do not know how to do it (as far as I know, Bernoulli equation may not be applicable in such a case, where there is more than just one liquid involved).

Will be happy to receive some help here. Thanks!