This question deals with the acceleration due to gravity on the surface of the planet. The value of the acceleration due to gravity on the surface of two planets, that is Earth and on the surface of another planet are compared to find out the relation between the radii of both planets, provided their masses are the same.
The correct option for the radius of the new planet is "Option 3: Square Root of 3 R"
The value of the acceleration due to gravity on the surface of a planet is given by the following formula:
[tex]g = \frac{GM}{R^2}[/tex]----------- eqn(1)
where,
g = acceleration due to gravity on Earth
G = Universal Gravitational Constant
M = mass of the Earth
R = Radius of the Earth
Now, the same formula for the acceleration due to gravity on the surface of the other planet can be written as follows:
[tex]g' = \frac{GM'}{R'}\\\\[/tex]------------- eqn(2)
where,
g' = acceleration due to gravity on the new planet
G = Universal Gravitational Constant
M' = mass of the new planet
R' = Radius of the new planet
According to the given condition, the new planet has the same mass as the Earth's mass but the acceleration due to gravity on the surface of the new planet is one-third of the acceleration due to gravity on the surface of Earth.
[tex]M'=M\\\\g'=\frac{1}{3}g[/tex]
using eqn (1) and eqn (2):
[tex]\frac{GM}{R'^2}=\frac{1}{3}\frac{GM}{R^2}\\\\R'^2=3R^2\\\\R' = \sqrt{3}R[/tex]
Hence, the radius of the new planet is found to be equal to the square root of 3 times the radius of the Earth (Square Root of 3 R).
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