Final answer:
To find the minimum velocity for a dense sphere to complete a full loop in a liquid, principles of buoyancy and circular motion are used, but a specific velocity calculation requires more details, such as the size of the sphere and the loop.
Step-by-step explanation:
The question appears to involve finding the minimum velocity that must be given to a spherical object with a density double that of the surrounding liquid (2rho) to complete a full circle. The solution would likely involve the principles of buoyancy, circular motion, and possibly Bernoulli's theorem if fluid dynamics are considered.
Due to the lack of specific details in the question prompts, it's not possible to provide a fully accurate answer, but assuming a loop-de-loop scenario, it would involve balancing the forces of buoyancy, gravity, and the centrifugal force acting on the sphere in the circular path.
We can use the concept that the final velocity v is the sum of the initial kinetic energy and the work done by gravitational forces, expressed by the equation v² = v₁² +2g(h₁ − h₂). This equation is derived from the energy conservation principle, combining kinetic and potential energy changes when the fluid is incompressible and hence its density (rho) cancels out.
For a sphere to complete a full circle in liquid, the minimum velocity must be sufficient to overcome the gravitational pull at the top of the loop without falling out of the circular path.