Gravitational Acceleration on a Planet with Twice the Mass and Radius of Earth

Introduction

Understanding the gravitational force on the surface of a planet is a fundamental concept in physics and astronomy. In this article, we will explore the value of the gravitational acceleration on a planet that has twice the mass and twice the radius of Earth. We will derive the formula using basic principles and ensure our calculations align with the principles of gravitational force and volume scaling.

Gravitational Acceleration on Earth

The gravitational acceleration on the surface of Earth is approximately 9.81 m/s2. This value is derived using Newton's law of universal gravitation. The formula for gravitational acceleration is:

where:

g is the acceleration due to gravity, G is the gravitational constant, M_E is the mass of Earth, R_E is the radius of Earth.

The value of gravitational acceleration on Earth is:

Calculating Gravitational Acceleration on a Planet with Twice the Mass and Radius

Let's consider a planet with twice the mass of Earth and twice the radius of Earth. We will use the same formula for gravitational acceleration but with the updated values:

The new formula will be:

[ g_{new} frac{G cdot 2M_E}{(2R_E)^2} frac{2G cdot M_E}{4R_E^2} frac{1}{2} cdot frac{G cdot M_E}{R_E^2} ]

Since ( frac{G cdot M_E}{R_E^2} ) is the acceleration due to gravity on Earth, we can write:

[ g_{new} frac{1}{2} g_E ]

The average gravitational acceleration on the surface of Earth is approximately 9.81 m/s2. Therefore:

[ g_{new} frac{1}{2} cdot 9.81 , m/s^2 approx 4.905 , m/s^2 ]

Thus, the acceleration due to gravity on the surface of this new planet is approximately 4.91 m/s2.

Correcting the Misconception

A common mistake is to assume that doubling the mass and radius would result in four times the gravitational acceleration. However, the relationship is not linear. The formula for gravitational force includes both the mass and the radius squared in the denominator. Therefore, doubling both the mass and the radius results in a halving of the gravitational acceleration.

The original formula for gravitational force is:

If volume increases linearly with the radius, doubling the radius would mean doubling the mass to maintain the same density. Hence, doubling both the mass and the radius would result in the acceleration being halved.

Understanding the Volume and Mass Relationship

The volume of a sphere is given by:

where:

V is the volume, R is the radius of the sphere.

Doubling the radius increases the volume by a factor of 8, but since the mass also needs to be doubled to keep the density the same, the increase in mass is by a factor of 16. Therefore, the new acceleration would be:

[ g_{new} frac{16G cdot M_E}{(2R_E)^3} frac{G cdot M_E}{R_E^2} g_E ]

Thus, doubling the radius and the mass would actually result in the same gravitational acceleration as Earth, not half.

Conclusion

The gravitational acceleration on the surface of a planet with twice the mass and twice the radius of Earth is approximately 4.91 m/s2. This value is derived using the correct formula and understanding the relationship between mass, radius, and gravitational acceleration. The key takeaway is that doubling both the mass and the radius results in the acceleration being halved, not quadrupled.