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Sagot :
Sure! Let's start by analyzing the problem using the values provided and Newton's law of gravitation to calculate the theoretical gravitational force.
### Step 1: Setup Initial Values
Given:
- Mass of the first object, [tex]\( m_1 = 2 \times 10^9 \, \text{kg} \)[/tex]
- Distance between the objects, [tex]\( r = 5 \, \text{km} = 5 \times 10^3 \, \text{m} \)[/tex]
### Step 2: Mass of Object 2 and Gravitational Forces
We are provided a table of different [tex]\( m_2 \)[/tex] values in [tex]\( 10^9 \, \text{kg} \)[/tex] and corresponding recorded gravitational forces.
### Step 3: Record Table
Let's fill in the provided table with the given values.
[tex]\[ \begin{tabular}{|c|c|c|c|} \hline $m_1=2 \times 10^9 \, \text{kg}$ & $m_1 \times m_2$ & $r = 5 \, \text{km}$ & $r^2 \, (\text{m}^2)$ \\ \hline $m_2 \, (\times 10^9 \, \text{kg})$ & $(\text{kg}^2)$ & $F_G \, (\text{N})$ & $F_G \, (\text{N})$ \\ \hline 10 & $2 \times 10^{18}$ & 53.4 & 53.3944 \\ \hline 9 & $1.8 \times 10^{18}$ & 48.1 & 48.05496 \\ \hline 8 & $1.6 \times 10^{18}$ & 42.7 & 42.71552 \\ \hline 7 & $1.4 \times 10^{18}$ & 37.4 & 37.37608 \\ \hline 6 & $1.2 \times 10^{18}$ & 32 & 32.03664 \\ \hline 5 & $1 \times 10^{18}$ & 26.7 & 26.6972 \\ \hline 4 & $0.8 \times 10^{18}$ & 21.4 & 21.35776 \\ \hline 3 & $0.6 \times 10^{18}$ & 16 & 16.01832 \\ \hline \end{tabular} \][/tex]
### Explanation of Steps:
1. Calculate Mass Multiplication [tex]\( m_1 \times m_2 \)[/tex]:
- For each [tex]\( m_2 \)[/tex], compute [tex]\( m_1 \times m_2 \)[/tex] in [tex]\( \text{kg}^2 \)[/tex].
2. Calculate Gravitational Force:
- Use Newton's law of gravitation to calculate the theoretical gravitational force:
[tex]\[ F_G = \frac{G \cdot m_1 \cdot m_2}{r^2} \][/tex]
Where [tex]\( G = 6.67430 \times 10^{-11} \, \text{N} \cdot \left(\text{m}^2 / \text{kg}^2\right) \)[/tex].
### Summary of Results:
The final theoretical forces were calculated for each [tex]\( m_2 \)[/tex], and these values are, respectively:
- 10: Theoretical Force [tex]\( \approx 53.3944 \, \text{N} \)[/tex]
- 9: Theoretical Force [tex]\( \approx 48.05496 \, \text{N} \)[/tex]
- 8: Theoretical Force [tex]\( \approx 42.71552 \, \text{N} \)[/tex]
- 7: Theoretical Force [tex]\( \approx 37.37608 \, \text{N} \)[/tex]
- 6: Theoretical Force [tex]\( \approx 32.03664 \, \text{N} \)[/tex]
- 5: Theoretical Force [tex]\( \approx 26.6972 \, \text{N} \)[/tex]
- 4: Theoretical Force [tex]\( \approx 21.35776 \, \text{N} \)[/tex]
- 3: Theoretical Force [tex]\( \approx 16.01832 \, \text{N} \)[/tex]
### Conclusion:
This analysis confirms that the recorded gravitational forces correspond well with the theoretical values calculated using Newton's law of gravitation.
### Step 1: Setup Initial Values
Given:
- Mass of the first object, [tex]\( m_1 = 2 \times 10^9 \, \text{kg} \)[/tex]
- Distance between the objects, [tex]\( r = 5 \, \text{km} = 5 \times 10^3 \, \text{m} \)[/tex]
### Step 2: Mass of Object 2 and Gravitational Forces
We are provided a table of different [tex]\( m_2 \)[/tex] values in [tex]\( 10^9 \, \text{kg} \)[/tex] and corresponding recorded gravitational forces.
### Step 3: Record Table
Let's fill in the provided table with the given values.
[tex]\[ \begin{tabular}{|c|c|c|c|} \hline $m_1=2 \times 10^9 \, \text{kg}$ & $m_1 \times m_2$ & $r = 5 \, \text{km}$ & $r^2 \, (\text{m}^2)$ \\ \hline $m_2 \, (\times 10^9 \, \text{kg})$ & $(\text{kg}^2)$ & $F_G \, (\text{N})$ & $F_G \, (\text{N})$ \\ \hline 10 & $2 \times 10^{18}$ & 53.4 & 53.3944 \\ \hline 9 & $1.8 \times 10^{18}$ & 48.1 & 48.05496 \\ \hline 8 & $1.6 \times 10^{18}$ & 42.7 & 42.71552 \\ \hline 7 & $1.4 \times 10^{18}$ & 37.4 & 37.37608 \\ \hline 6 & $1.2 \times 10^{18}$ & 32 & 32.03664 \\ \hline 5 & $1 \times 10^{18}$ & 26.7 & 26.6972 \\ \hline 4 & $0.8 \times 10^{18}$ & 21.4 & 21.35776 \\ \hline 3 & $0.6 \times 10^{18}$ & 16 & 16.01832 \\ \hline \end{tabular} \][/tex]
### Explanation of Steps:
1. Calculate Mass Multiplication [tex]\( m_1 \times m_2 \)[/tex]:
- For each [tex]\( m_2 \)[/tex], compute [tex]\( m_1 \times m_2 \)[/tex] in [tex]\( \text{kg}^2 \)[/tex].
2. Calculate Gravitational Force:
- Use Newton's law of gravitation to calculate the theoretical gravitational force:
[tex]\[ F_G = \frac{G \cdot m_1 \cdot m_2}{r^2} \][/tex]
Where [tex]\( G = 6.67430 \times 10^{-11} \, \text{N} \cdot \left(\text{m}^2 / \text{kg}^2\right) \)[/tex].
### Summary of Results:
The final theoretical forces were calculated for each [tex]\( m_2 \)[/tex], and these values are, respectively:
- 10: Theoretical Force [tex]\( \approx 53.3944 \, \text{N} \)[/tex]
- 9: Theoretical Force [tex]\( \approx 48.05496 \, \text{N} \)[/tex]
- 8: Theoretical Force [tex]\( \approx 42.71552 \, \text{N} \)[/tex]
- 7: Theoretical Force [tex]\( \approx 37.37608 \, \text{N} \)[/tex]
- 6: Theoretical Force [tex]\( \approx 32.03664 \, \text{N} \)[/tex]
- 5: Theoretical Force [tex]\( \approx 26.6972 \, \text{N} \)[/tex]
- 4: Theoretical Force [tex]\( \approx 21.35776 \, \text{N} \)[/tex]
- 3: Theoretical Force [tex]\( \approx 16.01832 \, \text{N} \)[/tex]
### Conclusion:
This analysis confirms that the recorded gravitational forces correspond well with the theoretical values calculated using Newton's law of gravitation.
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