Radius Comparison Between Planet Zeta and Earth
How does the radius of Zeta compare with the radius of Earth?
The planet Zeta has 2 times the gravitational field strength and the same mass as the Earth. Given the information, how can we determine the relationship between the radius of Zeta and Earth?
Choose 1 answer:
a. r_zeta = 1/2r_earth
b. r_zeta = √2r_earth
c. r_zeta = r_earth
d. r_zeta = 1/√2 r_earth
Answer:
The radius of planet Zeta is 1/√2 times the radius of the Earth. Option D.
When comparing the radius of Zeta with Earth, we can use the information provided about the gravitational field strength and mass of both planets. The gravitational field strength (g) at the surface of a planet is determined by the equation:
g = G * (M / R^2),
where:
- g = gravitational field strength,
- G = gravitational constant,
- M = mass of the planet,
- R = radius of the planet.
Given that Zeta has 2 times the gravitational field strength of Earth and the same mass, we can set up a ratio of gravitational field strengths for Zeta and Earth:
g_zeta / g_earth = (G * M_zeta / R_zeta^2) / (G * M_earth / r_earth^2).
Simplifying the equation with M_zeta = M_earth, we get:
2 = r_earth^2 / R_zeta^2.
R_zeta^2 = r_earth^2 / 2.
R_zeta = √(r_earth^2 / 2).
R_zeta = 1/√2 * r_earth
Therefore, the radius of planet Zeta is 1/√2 times the radius of Earth, showcasing a clear comparison between the two planets.