Question

Assume that the mass of the earth is hundred times larger than the mass of the moon and the radius of earth is 4 times that of moon show that the weight of the object is about one sixth of that on earth

Answer 1

Answer & Explanation:

The weight of an object can be expressed as the gravitational force at the surface.

The weight of an object of mass $m$ at the surface of the Earth can be written as

$W_E = \dfrac{ G M_E m }{ R_E^2 } \, \, ,$

where $M_E$ is the Earth's mass, $R_E$ is Earth's radius and $G$ is the gravitational constant.

Similarly, the weight of the same object on the surface of the Moon is given by

$W_M = \dfrac{ G M_M m }{ R_M^2 } \, \, ,$

where $M_M$ is the Moon's mass and $R_M$ is Moon's radius.

The ratio of the weight on Moon to the weight on Earth is

$\dfrac{ W_M }{ W_E } = \dfrac{ G M_M m / R_M^2 }{ G M_E m / R_E^2 }= \dfrac{ M_M }{ M_E } \dfrac{ R_E^2 }{ R_M^2 } \, \, .$

Now we have $M_E = 100 M_M$ and $R_E = 4 R_M$, so

$\dfrac{ W_M }{ W_E } = \dfrac{ 1 }{ 100 } \times 16 = \dfrac{4}{25} \approx \dfrac{1}{6} \, \, .$

Answer 2

(a) The acceleration due to gravity on the moon is about one-sixth of that on the earth. (b) In order to the force of gravitation between two bodies to become noticeable and cause motion, one of the bodies must have an extremely large mass.