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Answer :
To find the gravitational force between two masses, we can use the formula:
[tex]\[
\overrightarrow{F} = G \frac{m_1 \cdot m_2}{r^2}
\][/tex]
Where:
- [tex]\( G \)[/tex] is the gravitational constant, which is [tex]\( 6.67 \times 10^{-11} \, \text{N} \cdot \text{m}^2/\text{kg}^2 \)[/tex].
- [tex]\( m_1 \)[/tex] is the mass of the first object, given as [tex]\( 157 \, \text{kg} \)[/tex].
- [tex]\( m_2 \)[/tex] is the mass of the second object, given as [tex]\( 5.61 \, \text{kg} \)[/tex].
- [tex]\( r \)[/tex] is the distance between the centers of the two masses, given as [tex]\( 8.90 \, \text{m} \)[/tex].
Let's plug in the given values into the equation:
1. First, calculate [tex]\( m_1 \cdot m_2 \)[/tex]:
[tex]\[
157 \, \text{kg} \times 5.61 \, \text{kg} = 880.77 \, \text{kg}^2
\][/tex]
2. Calculate [tex]\( r^2 \)[/tex]:
[tex]\[
(8.90 \, \text{m})^2 = 79.21 \, \text{m}^2
\][/tex]
3. Now substitute these values into the gravitational force formula:
[tex]\[
\overrightarrow{F} = 6.67 \times 10^{-11} \frac{\text{N} \cdot \text{m}^2}{\text{kg}^2} \times \frac{880.77 \, \text{kg}^2}{79.21 \, \text{m}^2}
\][/tex]
4. Perform the division and multiplication:
[tex]\[
\overrightarrow{F} = 6.67 \times 10^{-11} \times 11.11943684 = 7.42 \times 10^{-10} \, \text{N}
\][/tex]
Thus, the gravitational force between the two masses is approximately [tex]\( 7.42 \times 10^{-10} \, \text{N} \)[/tex].
To express it in the given format [tex]\(\overrightarrow{F} = [7.42] \times 10^{[-10]} \, \text{N}\)[/tex], we have:
- [tex]\([7.42]\)[/tex] as the coefficient
- [tex]\([-10]\)[/tex] as the exponent to represent the force's magnitude in scientific notation.
Therefore, the final answer is:
[tex]\[
\vec{F} = 7.42 \times 10^{-10} \, \text{N}
\][/tex]
[tex]\[
\overrightarrow{F} = G \frac{m_1 \cdot m_2}{r^2}
\][/tex]
Where:
- [tex]\( G \)[/tex] is the gravitational constant, which is [tex]\( 6.67 \times 10^{-11} \, \text{N} \cdot \text{m}^2/\text{kg}^2 \)[/tex].
- [tex]\( m_1 \)[/tex] is the mass of the first object, given as [tex]\( 157 \, \text{kg} \)[/tex].
- [tex]\( m_2 \)[/tex] is the mass of the second object, given as [tex]\( 5.61 \, \text{kg} \)[/tex].
- [tex]\( r \)[/tex] is the distance between the centers of the two masses, given as [tex]\( 8.90 \, \text{m} \)[/tex].
Let's plug in the given values into the equation:
1. First, calculate [tex]\( m_1 \cdot m_2 \)[/tex]:
[tex]\[
157 \, \text{kg} \times 5.61 \, \text{kg} = 880.77 \, \text{kg}^2
\][/tex]
2. Calculate [tex]\( r^2 \)[/tex]:
[tex]\[
(8.90 \, \text{m})^2 = 79.21 \, \text{m}^2
\][/tex]
3. Now substitute these values into the gravitational force formula:
[tex]\[
\overrightarrow{F} = 6.67 \times 10^{-11} \frac{\text{N} \cdot \text{m}^2}{\text{kg}^2} \times \frac{880.77 \, \text{kg}^2}{79.21 \, \text{m}^2}
\][/tex]
4. Perform the division and multiplication:
[tex]\[
\overrightarrow{F} = 6.67 \times 10^{-11} \times 11.11943684 = 7.42 \times 10^{-10} \, \text{N}
\][/tex]
Thus, the gravitational force between the two masses is approximately [tex]\( 7.42 \times 10^{-10} \, \text{N} \)[/tex].
To express it in the given format [tex]\(\overrightarrow{F} = [7.42] \times 10^{[-10]} \, \text{N}\)[/tex], we have:
- [tex]\([7.42]\)[/tex] as the coefficient
- [tex]\([-10]\)[/tex] as the exponent to represent the force's magnitude in scientific notation.
Therefore, the final answer is:
[tex]\[
\vec{F} = 7.42 \times 10^{-10} \, \text{N}
\][/tex]
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