Understanding the Relationship Between Gravitational Acceleration, Density, and Radius of a Planet

Have you ever wondered how the gravitational acceleration at the surface of a planet relates to its density and radius? In this article, we will explore the physical principles behind these relationships and provide a comprehensive solution for a specific scenario. We will use the example of a hypothetical planet with a density 1.5 times that of Earth and find its radius given that the gravitational acceleration at its surface is equal to that of Earth.

Gravitational Acceleration and Its Relation to Mass and Radius

Gravitational acceleration at the surface of a planet, ( g ), is given by the formula:

[ g frac{GM}{R^2} ]

where ( G ) is the gravitational constant, ( M ) is the mass of the planet, and ( R ) is the radius of the planet.

Expressing Mass in Terms of Density

The mass ( M ) of a planet can be expressed in terms of its density ( rho ) and volume ( V ) as:

[ M rho V ]

For a spherical planet, the volume ( V ) is given by:

[ V frac{4}{3} pi R^3 ]

Therefore, the mass ( M ) can be written as:

[ M rho left(frac{4}{3} pi R^3right) ]

Substituting Mass into the Gravitational Acceleration Formula

Substituting the expression for mass into the gravitational acceleration formula, we get:

[ g frac{G left(rho left(frac{4}{3} pi R^3right)right)}{R^2} frac{4}{3} pi G rho R ]

Setting Gravitational Acceleration Equal for Two Planets

Let ( g_E ) represent the acceleration due to gravity on Earth, and ( rho_E ) be the density of Earth with radius ( R_E R ). For the new planet, we have:

[ g g_E quad text{and} quad rho 1.5 rho_E ]

Thus, for the new planet, the gravitational acceleration ( g ) can be expressed as:

[ g frac{4}{3} pi G (1.5 rho_E) R_p ]

Equating the Two Expressions for Gravitational Acceleration

Setting the two expressions for gravitational acceleration equal gives:

[ frac{4}{3} pi G rho_E R frac{4}{3} pi G (1.5 rho_E) R_p ]

Canceling common terms, we have:

[ R 1.5 R_p ]

Solving for ( R_p ), we get:

[ R_p frac{R}{1.5} frac{2}{3} R ]

Conclusion

The radius of the planet is:

[ boxed{frac{2}{3} R} ]

Understanding this relationship helps us grasp fundamental concepts in physics and astronomy, particularly in planetary science. By applying the principles of gravitation, we can explore and predict the properties of distant planets without direct observation.

Keywords: gravitational acceleration, density, radius of a planet