If the density of a small planet is the same as that of the earth while the radius of the planet is times that of the earth, the gravitational acceleration on the surface of the planet is _____.
Explanation for the correct option:
Step 1: State assumptions and known data
The density of Earth and the small planet is same.
Let the density of earth and the small planet be
Let be the radius of Earth. Then, is the radius of the small planet.
Step 2: Formulas used
Acceleration due to gravity is given as,
where is the gravitational constant, is the mass of the planet and is the radius of the planet.
Density of an object, , where is its volume
Step 3: Derive expression for acceleration due to gravity in term of density of the earth
From the equation relating density and mass,
The shape of a planet can be approximated to be that of a sphere (in fact, shape of the earth is geoid).
The volume of a sphere is , where is its radius.
Thus, the mass of the sphere is,
Substituting the above expression in ,
Step 4: Calculate the acceleration due to gravity on the small planet
The acceleration due to gravity on Earth is,
The acceleration due to gravity on the small planet is,
Therefore, if the density of a small planet is the same as that of the earth while the radius of the planet is times that of the earth, the gravitational acceleration on the surface of the planet is .
Hence, option A is correct.