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A solid sphere of radius R has mass density rho(r) = rho0(1 - r²/R²) for 0 <= r <= R. Find the minimum density of a liquid in which this sphere will just float.
- rho0/5
- rho0/3
- 2rho0/3
- 2rho0/5
Correct answer: 2rho0/5
Solution
The sphere just floats when the liquid density equals the sphere's average density. Mass M = integral 0..R of rho0(1 - r²/R²) * 4 pi r² dr = 4 pi rho0 [R³/3 - R⁵/(5R²)] = 4 pi rho0 [R³/3 - R³/5] = 4 pi rho0 R³ (2/15) = 8 pi rho0 R³/15. Average density = M/((4/3) pi R³) = (8 pi rho0 R³/15)/((4/3) pi R³) = (8/15)*(3/4) rho0 = (24/60) rho0 = 2 rho0/5.
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