Respuesta :
Answer:
We can use Newton's law of universal gravitation to solve this problem:
\[ F = \frac{{G \cdot m_1 \cdot m_2}}{{r^2}} \]
Where:
- \( F \) is the force of gravity (given as \( 5.95 \times 10^6 \) N)
- \( G \) is the gravitational constant (\( 6.674 \times 10^{-11} \, \text{m}^3/\text{kg}\cdot\text{s}^2 \))
- \( m_1 \) is the mass of planet A (\( 3.67 \times 10^{20} \) kg)
- \( m_2 \) is the mass of planet B (which we want to find)
- \( r \) is the distance between the centers of the planets (\( 2.23 \times 10^{24} \) m)
We can rearrange the formula to solve for \( m_2 \):
\[ m_2 = \frac{{F \cdot r^2}}{{G \cdot m_1}} \]
Substitute the given values:
\[ m_2 = \frac{{5.95 \times 10^6 \cdot (2.23 \times 10^{24})^2}}{{6.674 \times 10^{-11} \cdot 3.67 \times 10^{20}}} \]
Now, calculate \( m_2 \).
\[ m_2 = \frac{{5.95 \times 10^6 \cdot (2.23 \times 10^{24})^2}}{{6.674 \times 10^{-11} \cdot 3.67 \times 10^{20}}} \]
\[ m_2 = \frac{{5.95 \times 10^6 \cdot 4.9729 \times 10^{48}}}{{6.674 \times 10^{-11} \cdot 3.67 \times 10^{20}}} \]
\[ m_2 = \frac{{29.6525 \times 10^{54}}}{{24.4574 \times 10^{9}}} \]
\[ m_2 = \frac{{29.6525}}{{24.4574}} \times 10^{54-9} \]
\[ m_2 \approx 1.211 \times 10^{45} \, \text{kg} \]
So, the mass of planet B is approximately \( 1.211 \times 10^{45} \) kg.
