An astronaut is standing at the north pole of a newly...

An astronaut is standing at the north pole of a newly discovered, spherically symmetric planet of radius $R .$ In his hands he holds a container full of a liquid with mass $m$ and volume $V .$ At the surface of the liquid, the pressure is $p_{0} ;$ at a depth $d$ below the surface, the pressure has a greater value $p .$ From this information, determine the mass of the planet.

Key Concepts

Spherically Symmetric Gravitational Field

In the context of planetary bodies, a spherically symmetric mass distribution simplifies the analysis of gravitational effects because the gravitational field at the surface depends solely on the total mass and the radius. This symmetry allows the application of Newton's law in a straightforward manner to relate surface gravity to the planet's mass.

Fluid Density

Fluid density, defined as the mass per unit volume (? = m/V), is a pivotal quantity in hydrostatic calculations. It directly affects the pressure increase within a fluid under gravity, thus serving as a bridge between the fluid's intrinsic properties and the gravitational environment it experiences.

Hydrostatic Equilibrium

This concept refers to the balance in a fluid where the change in pressure with depth is governed by the weight of the fluid above. The relationship ?p = ? g d—where ? is the fluid density, g represents gravitational acceleration, and d denotes the depth—allows one to connect measurable pressure differences in the liquid to the gravitational force acting on it.

Newton's Law of Universal Gravitation

Newton's law states that a body with mass M exerts a gravitational force that results in an acceleration g = GM/R² at the surface of a spherical planet of radius R. This law is essential for linking the gravitational acceleration determined from the pressure gradient to the total mass of the planet.

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