Gravity Calculation on a New Planet

The acceleration due to gravity (\(g\)) on the surface of a celestial body is determined by its mass (\(M\)) and radius (\(R\)) using the formula:

\(g = \frac{GM}{R^2}\)

where \(G\) is the universal gravitational constant.

Comparing Planet Gravity to Earth

Let \(g\) represent the acceleration due to gravity on Earth, with mass \(M_E\) and radius \(R_E\). The formula is:

\(g = \frac{GM_E}{R_E^2}\)

For the new planet, let its acceleration due to gravity be \(g^1\), its mass be \(M_p\), and its radius be \(R_p\).

We are given the following relationships:

• Mass of the planet: \(M_p = \frac{M_E}{2}\)
• Radius of the planet: \(R_p = \frac{R_E}{2}\)

Using the gravity formula for the new planet:

\(g^1 = \frac{GM_p}{R_p^2}\)

Substitute the given values for \(M_p\) and \(R_p\) into the equation:

\(g^1 = \frac{G \left( \frac{M_E}{2} \right)}{\left( \frac{R_E}{2} \right)^2}\)

Simplify the expression step-by-step:

\(g^1 = \frac{G \cdot M_E / 2}{R_E^2 / 4}\)

\(g^1 = \frac{G M_E}{2} \times \frac{4}{R_E^2}\)

\(g^1 = 2 \times \frac{GM_E}{R_E^2}\)

Recognize that \(\frac{GM_E}{R_E^2}\) is equal to Earth's gravity, \(g\). Substitute \(g\) back into the equation:

\(g^1 = 2g\)

Conclusion on Planet Gravity

The acceleration due to gravity on the surface of this new planet is twice the acceleration due to gravity on Earth.