Question

A man weighs 200 pounds on earth.
How much would he weigh on a planet four times as massive as the earth with a radius five times as great.

Answer 1

Hint:The acceleration due to gravity depends upon the mass of the earth and the mass of the object on which we measure the acceleration due to gravity and the distance between the centre of the earth and the object.

Formula used:The formula of the acceleration due to gravity is given by,
$ \Rightarrow g = G\dfrac{{{M_e} \cdot m}}{{{r^2}}}$
Where the universal gravitational constant is G the mass of the earth is ${M_e}$ the mass of the object is m and the distance between the centre of the earth and the object is r.

Complete step by step solution:
It is given in the problem that a man weighs 200pounds on earth and we need to tell what will be the weight on the planet with four times as massive as the earth and has a radius five times as great.
The formula of the acceleration due to gravity is given by,
$ \Rightarrow g = G\dfrac{{{M_e} \cdot m}}{{{r^2}}}$
Where the universal gravitational constant is G the mass of the earth is ${M_e}$ the mass of the object is m and the distance between the centre of the earth and the object is r.
The acceleration due to gravity is g for normal condition is,
$ \Rightarrow g = G\dfrac{{{M_e} \cdot m}}{{{r^2}}}$………eq. (1)
The new planet has mass four times as earth and the radius is five times as that of the earth.
$ \Rightarrow g' = G\dfrac{{4{M_e} \cdot m}}{{{{\left( {5r} \right)}^2}}}$
$ \Rightarrow g' = G\dfrac{{4{M_e} \cdot m}}{{25{r^2}}}$………eq. (2)
Taking ratio of equation (1) and equation (2).
$ \Rightarrow \dfrac{{g'}}{g} = \dfrac{{G\dfrac{{4{M_e} \cdot m}}{{25{r^2}}}}}{{G\dfrac{{{M_e} \cdot m}}{{{r^2}}}}}$
$ \Rightarrow \dfrac{{g'}}{g} = \dfrac{4}{{25}}$
$ \Rightarrow g' = \dfrac{4}{{25}}g$
$ \Rightarrow g' = 0 \cdot 16g$.
The weight of the man on new planet is equal to,
$ \Rightarrow mg' = 0 \cdot 16mg$
The weight on the planet earth is 200 pounds.
$ \Rightarrow mg' = 0 \cdot 16mg$
$ \Rightarrow mg' = 0 \cdot 16 \times 200$
$ \Rightarrow mg' = 32{\text{ pounds}}$.

The weight of the man on the new planet is equal to 32 pounds.

Note:It is advisable for the students to always understand and remember the formula of the acceleration due to gravity. The acceleration due to gravity changes with change of the mass of the object and the distance between the centre of the earth and the object.