To determine the mass of the International Space Station (ISS), we can use the formula for gravitational force:
[tex]\[ F = G \frac{m_1 m_2}{r^2} \][/tex]
Given:
- [tex]\( F = 4.64 \times 10^{-6} \)[/tex] N (gravitational force)
- [tex]\( m_1 = 112 \)[/tex] kg (mass of the astronaut)
- [tex]\( r = 26 \)[/tex] m (distance between the astronaut and the ISS)
- [tex]\( G = 6.67430 \times 10^{-11} \)[/tex] m[tex]\(^3\)[/tex]kg[tex]\(^{-1}\)[/tex]s[tex]\(^{-1}\)[/tex] (gravitational constant)
We want to solve for [tex]\( m_2 \)[/tex], which is the mass of the ISS.
Rearrange the formula to solve for [tex]\( m_2 \)[/tex]:
[tex]\[ m_2 = \frac{F r^2}{G m_1} \][/tex]
Substitute the known values into the equation:
[tex]\[ m_2 = \frac{4.64 \times 10^{-6} \times 26^2}{6.67430 \times 10^{-11} \times 112} \][/tex]
Calculate the numerator:
[tex]\[ 4.64 \times 10^{-6} \times 26^2 = 4.64 \times 10^{-6} \times 676 = 3.13824 \times 10^{-3} \][/tex]
Calculate the denominator:
[tex]\[ 6.67430 \times 10^{-11} \times 112 = 7.475216 \times 10^{-9} \][/tex]
Now, divide the results to find [tex]\( m_2 \)[/tex]:
[tex]\[ m_2 = \frac{3.13824 \times 10^{-3}}{7.475216 \times 10^{-9}} = 419605.27 \, \text{kg} \][/tex]
Express the final result using two significant figures:
[tex]\[ m_2 \approx 4.2 \times 10^5 \, \text{kg} \][/tex]
Thus, the mass of the ISS is approximately [tex]\( \boxed{4.2 \times 10^5} \, \text{kg} \)[/tex].
