Newton's law of gravitation:
\[
F \ = \ \frac{GM_{1}M_{2}}{r^{2}}
\]
\( M_{1}, M_{2} \) : masses in kg.
\( r \) : Distance between centers of mass in meters.
\( G \) : Netwon's gravitational constant, \( 6.67 \times 10^{-11}
\rm m^{3} \ kg^{-1} \ s^{-1} \).
\( F \): Force in Newtons.
Ratio of force between baby and Jupiter to that between mother and
baby:
\[
\begin{eqnarray}
\frac{F_{\rm baby-Jupiter}}{F_{\rm mother-baby}} & = &
\left( \frac{M_{\rm Jupiter} \ M_{\rm baby}}
{M_{\rm baby} \ M_{\rm mother}} \right)
\left( \frac{r_{\rm mother-baby}}{r_{\rm baby-Jupiter}} \right)^{2} \\
\frac{F_{\rm baby-Jupiter}}{F_{\rm mother-baby}} & = &
\left( \frac{M_{\rm Jupiter} }
{M_{\rm mother}} \right)
\left( \frac{r_{\rm mother-baby}}{r_{\rm baby-Jupiter}} \right)^{2}
\end{eqnarray}
\]
Mass of Jupiter:
$M_{\rm Jupiter} = 1.8981 \times 10^{27}$ kg.
Earth-Jupiter distance: Currently it is 5.94 AU, and varies between
about 6.44 AU and 4.21 AU. Let's use current value first.
1 AU (astronomical unit) $\equiv 1.49598 \times 10^{11}$ meters.
Hence $r_{\rm baby-Jupiter}({\rm now}) = 5.94 \times 1.49598 \times 10^{11}$ m
$= \ 8.89 \times 10^{11}$ m.
Mass of mother: Estimate 70 kg.
Mother-Baby separation: Close hug: $\sim 0.1$ m. Baby held at
arms length: $\sim 0.5$ m (these are estimated distances between centers of mass).
Hence, the close-hug ratio of the force between Jupiter and baby to the
that between mother and baby is:
\[
\begin{eqnarray}
\left[ \frac{F_{\rm baby-Jupiter}}{F_{\rm mother-baby}} \right] _{\rm close \ hug} & = &
\left(\frac{1.8981 \times 10^{27}}{70}\right)
\left(\frac{0.1}{8.89 \times 10^{11}}\right)^{2}
& \sim & 0.34
\end{eqnarray}
\]
If the mother holds the baby at arms length,
\[
\begin{eqnarray}
\left[ \frac{F_{\rm baby-Jupiter}}{F_{\rm mother-baby}} \right]_{\rm arms \
length} & = &
\left(\frac{1.8981 \times 10^{27}}{70}\right)
\left(\frac{0.5}{8.89 \times 10^{11}}\right)^{2}
& \sim & 8.58
\end{eqnarray}
\]
If Jupiter is at the closest distance to Earth,
\[
\begin{eqnarray}
\left[ \frac{F_{\rm baby-Jupiter}}{F_{\rm mother-baby}} \right]_{\rm close \ hug} & = & 0.68 \\
\end{eqnarray}
\]
\[
\begin{eqnarray}
\left[ \frac{F_{\rm baby-Jupiter}}{F_{\rm mother-baby}} \right]_{\rm arms \ length} & = & 17.1 \\
\end{eqnarray}
\]
Any mother with a mass greater than about 50 kg (110 lbs) will always dominate
the attraction in a close hug, even during Jupiter's closest approach
(students: prove this). Jupiter will always dominate in the arms-length
position for any mother (unless her mass is over 1,200 kg, or 2,640 lbs!).
Absolute force between Jupiter and a 7lb baby:
\[
\begin{eqnarray}
F_{\rm Jupiter-baby} & = & \frac{6.67 \times 10^{-11} \times
1.90 \times 10^{27} \times 3.2}{(8.89 \times 10^{11})^{2}} \\
& \sim & 5.1 \times 10^{-7} \ {\rm Newtons} \\
\end{eqnarray}
\]
or the equivalent of approximately one 10 millionth of a pound in weight.
Students: try repeating the same calculations for Mars and Venus.
Planetary mass data are available the from NASA solar system website and Earth-planet distances can be
calculated using the tool at this NASA JPL website.
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