(a) What is the approximate force of gravity on a\({\rm{70 - kg}}\)person due to the Andromeda galaxy, assuming its total mass is\({\rm{1}}{{\rm{0}}^{{\rm{13}}}}\)that of our Sun and acts like a single mass\({\rm{2 Mly}}\)away? (b) What is the ratio of this force to the person’s weight? Note that Andromeda is the closest large galaxy.

The force acting between two particles is given by Newton’s law of gravitation,

\(F = G\frac{{{m_1}{m_2}}}{{{r^2}}}\)

Here\(F\)is the force acting between two particles,\({m_1}\)and\({m_2}\)are the masses of the particle,\(G\)is the universal gravitational constant and\(r\)is the distance between the masses.

Using the Newton's gravitation force law as:

\(\begin{array}{}F & = G\frac{{Mm}}{{{r^2}}}\\ & = {\rm{6}}{\rm{.67 \times 1}}{{\rm{0}}^{{\rm{ - 11}}}}\frac{{{\rm{N}}{{\rm{m}}^{\rm{2}}}}}{{{\rm{\;k}}{{\rm{g}}^{\rm{2}}}}}\frac{{{\rm{1}}{{\rm{0}}^{{\rm{13}}}}{\rm{ \times 2 \times 1}}{{\rm{0}}^{{\rm{30}}}}{\rm{\;kg \times 70\;kg}}}}{{{{\left( {{\rm{2 \times 1}}{{\rm{0}}^{\rm{6}}}{\rm{ \times 3 \times 1}}{{\rm{0}}^{\rm{8}}}\frac{{\rm{m}}}{{\rm{s}}}{\rm{ \times 360 \times 24 \times 3600\;s}}} \right)}^{\rm{2}}}}}\\ & = {\rm{2}}{\rm{.7 \times 1}}{{\rm{0}}^{{\rm{ - 10\;}}}}\,{\rm{N}}\end{array}\)

Therefore, the force on a person due to Andromeda galaxy is: \({\rm{2}}{\rm{.7 \times 1}}{{\rm{0}}^{{\rm{ - 10\;}}}}\,{\rm{N}}\).

The ratio of the force is evaluated as:

\(\begin{array}{}ratio & = \frac{F}{{mg}}\\ & = \frac{{{\rm{2}}{\rm{.7 \times 1}}{{\rm{0}}^{{\rm{ - 10}}}}{\rm{\;N}}}}{{{\rm{70\;kg \times 9}}{\rm{.81}}\frac{{\rm{N}}}{{{\rm{kg}}}}}}\\ & = {\rm{3}}{\rm{.9 \times 1}}{{\rm{0}}^{{\rm{ - 13}}}}\end{array}\)

Therefore, the required ratio is: \({\rm{3}}{\rm{.9 \times 1}}{{\rm{0}}^{{\rm{ - 13}}}}\).