Question

An astronaut of mass m is working on a satellite orbiting the earth at a distance h from the earth’s surface. The radius of the earth is R, while its mass M. The gravitational pull ${F_G}$ on the astronaut is:
A. Zero since astronaut feels weightless
B. $0 < {F_G} < \dfrac{{GMm}}{{{R^2}}}$
C. $\dfrac{{GMm}}{{{{\left( {R + h} \right)}^2}}} < {F_G} < \dfrac{{GMm}}{{{R^2}}}$
D. ${F_G} = \dfrac{{GMm}}{{{{\left( {R + h} \right)}^2}}}$

Answer 1

The gravitational pull between any two objects depends only on the masses of the two objects and the distance between them. It does not depend on any other factors. To find the required solution of the given question we will apply the concept of Newton’s law of gravitation.
Formula Used: $F = \dfrac{{GMm}}{{{r^2}}}$

Complete answer:
Orbital velocity of the satellite is given by,
$v = \sqrt {\dfrac{{GM}}{r}}$
Here, we will substitute the value of ‘r’ as,
$\Rightarrow$r = R+h
Where ‘r’ is the position of the satellite from the earth’s centre.
Newton’s law of gravitation states that every particle in the universe attracts every other particle with a force. The magnitude of the force is given as,
It is directly proportional to the product of their masses, i.e. $F \propto Mm$
And is inversely proportional to the square of the distance between their centre, i.e. $F \propto \dfrac{1}{{{r^2}}}$
On combining the above relations, we get,
$\Rightarrow F \propto \dfrac{{Mm}}{{{r^2}}}$
$\Rightarrow F = \dfrac{{GMm}}{{{r^2}}}$
Now, according to Newton’s law of gravitation
$F = \dfrac{{GMm}}{{{r^2}}}$
Where ‘F’ is expressed as the force, ‘G’ is the gravitational constant, ‘m’ and ‘M’ are the masses and ‘r’ is the distance between the masses.
Thus, gravitational pull ‘${F_G}$’ on the astronaut is given as,
${F_G} = \dfrac{{GMm}}{{{{\left( {R + h} \right)}^2}}}$

Hence, option (D) is the correct answer.

Note:
Gravitation is defined as the force of attraction between any two objects or bodies. All of the objects in the universe attract each other with a certain amount of force. Weak forces are observed due to large separation between them. The force with which each body in the universe attracts other bodies towards itself is known as the Gravitational force.