Question

(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.

Short answer

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

(b) The ratio of the force is obtained as: \({\rm{3}}{\rm{.9 \times 1}}{{\rm{0}}^{{\rm{ - 13}}}}\).

Step-by-step solution

Newton’s law of gravitation.

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.

Evaluating the force

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: \({\rm{2}}{\rm{.7 \times 1}}{{\rm{0}}^{{\rm{ - 10\;}}}}\,{\rm{N}}\).

Evaluating the ratio of the force

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: \({\rm{3}}{\rm{.9 \times 1}}{{\rm{0}}^{{\rm{ - 13}}}}\).