A Space Traveller determine the radius of a Planet to be half that of the earth after landing on its surface he find the acceleration due to gravity to be twice of the surface of the earth find the ratio of the mass of a planet to the earth
To calculate the ratio of the mass of the planet to that of the Earth, we need to use Newton's law of universal gravitation.
First, let's consider the given information:
- The radius of the planet is half that of the Earth.
- The acceleration due to gravity on the planet is twice that of the Earth.
Let's assume:
- The mass of the planet is "M".
- The mass of the Earth is "M_E".
- The radius of the planet is "r" (which is half of the Earth's radius, denoted as "r_E").
According to Newton's law of universal gravitation, the gravitational force between two objects is given by:
F = (G * M * M_E) / r^2
where:
- F is the gravitational force between the two objects.
- G is the gravitational constant (approximately equal to 6.674 × 10^-11 N m^2/kg^2).
On the surface of the planet, the acceleration due to gravity is given by:
g = G * M / r^2 -------- (Equation 1)
On the surface of the Earth, the acceleration due to gravity is given by:
g_E = G * M_E / r_E^2 -------- (Equation 2)
Given that g = 2 * g_E and r = r_E/2, we can substitute these values into equations 1 and 2:
2 * g_E = G * M / (r_E/2)^2
2 * (G * M_E / r_E^2) = G * M / (r_E/2)^2
Simplifying these equations, we get:
2 * M_E / r_E^2 = M / (r_E^2/4)
M_E = M / 4
Taking the ratio of the mass of the planet (M) to the mass of the Earth (M_E):
M / M_E = 4 / 1
Therefore, the ratio of the mass of the planet to that of the Earth is 4:1.