Question

Calculate the acceleration due to gravity on the surface of a planet whose mass and radius are double than the mass and radius of earth.

Answer 1

Hint: As we know the formula of planet is \[g = \dfrac{{G \times M}}{{{R^2}}}\], where \[M\] is mass of any planet and \[R\] is radius of any planet and we can check from the formula that it is directly dependent on mass and inversely related to the radius of planet.

Complete step-by-step solution:
As in the given question we can assume mass of earth as, \[{M_e}\]
And radius of earth as, \[{R_e}\]
And in the given question we can assume the mass of planet as, \[{M_P}\]
And also we can assume the radius of planet as, \[{R_P}\]
As we know the formula for acceleration due to gravity is \[g = \dfrac{{G \times M}}{{{R^2}}}\]
And here by substituting the values of mass and radius of earth is, \[{g_e} = \dfrac{{G \times {M_e}}}{{{R_e}^2}}\]
Now we have calculated the acceleration due to gravity at earth now for that planet
As we know, \[{R_P} = 2{R_e}\]and \[{M_P} = 2{M_e}\]
We get:
\[{g_p} = \dfrac{{G \times 2{M_e}}}{{4{R_e}^2}}\]
\[{g_p} = \dfrac{{G \times {M_e}}}{{2{R_e}^2}}\]
Therefore relating acceleration due to gravity at planet and earth we get
\[{g_p} = \dfrac{{{g_e}}}{2}\]

Additional information: As in the formula above there is \[G\]present in there formula it is gravitational constant it is used in the equation to make a relation with geometry of space-time and energy momentum tensor.

Note:- The formula we used above to calculate the acceleration due gravity on any other planet is derived from newton’s second law of motion and newton’s laws of gravitation that every particle exerts force on any other particle and therefore these two will be equal which leads us to given formula.