David lands on planet X and observes objects free-falling with acceleration 4.9 m/s^{2 s2}. If the planet is 3 times larger than the Earth, what is the planet MASS expressed in Earth masses? A) 1/2 B) 3/2 C) 2/3 D) 9/2 E) 9/4

Question

Quizplus · Accepted Answer

The acceleration due to gravity (g) on the surface of a planet is given by the formula $g = \frac{G \cdot M}{R^2}$, where $G$ is the gravitational constant, $M$ is the mass of the planet, and $R$ is the radius of the planet. On Earth, $g = 9.8 \, \text{m/s}^2$, and on planet X, $g = 4.9 \, \text{m/s}^2$. Since planet X is 3 times larger than Earth, its radius is 3 times that of Earth's. Let $M_{\text{Earth}}$ and $M_{\text{X}}$ be the masses of Earth and planet X, respectively, and $R_{\text{Earth}}$ be the radius of Earth. The acceleration due to gravity on planet X can be expressed as:$$4.9 = \frac{G \cdot M_{\text{X}}}{(3R_{\text{Earth}})^2}$$And the acceleration due to gravity on Earth is:$$9.8 = \frac{G \cdot M_{\text{Earth}}}{R_{\text{Earth}}^2}$$Dividing the first equation by the second gives:$$\frac{4.9}{9.8} = \frac{M_{\text{X}}}{9M_{\text{Earth}}}$$Solving for $M_{\text{X}}$ gives:$$M_{\text{X}} = \frac{9}{2} M_{\text{Earth}}$$Therefore, the mass of planet X is $9/2$ times the mass of Earth.

David Lands on Planet X and Observes Objects Free-Falling with Acceleration