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A 112 kg astronaut is tethered to the International Space Station (ISS) and is 26 m from the center of mass of the ISS. The gravitational force between the astronaut and the ISS is [tex]\(4.64 \times 10^{-6} \, N\)[/tex].

Calculate the mass of the ISS. Write your answer using two significant figures.

[tex]\[ \boxed{\text{kg}} \][/tex]



Answer :

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].