To determine the gravitational force between the two masses, we will use Newton's law of universal gravitation, given by the formula:
[tex]\[
\vec{F} = G \frac{m_1 m_2}{r^2}
\][/tex]
where:
- [tex]\( \vec{F} \)[/tex] is the gravitational force,
- [tex]\( G \)[/tex] is the gravitational constant, [tex]\( G = 6.67 \times 10^{-11} \, \text{N} \cdot \text{m}^2 / \text{kg}^2 \)[/tex],
- [tex]\( m_1 \)[/tex] is the mass of the first object,
- [tex]\( m_2 \)[/tex] is the mass of the second object,
- [tex]\( r \)[/tex] is the distance between the centers of the two masses.
Given the values:
- [tex]\( m_1 = 92.0 \, \text{kg} \)[/tex],
- [tex]\( m_2 = 0.894 \, \text{kg} \)[/tex],
- [tex]\( r = 99.3 \, \text{m} \)[/tex],
we can substitute these values into the formula to find the gravitational force.
[tex]\[
\vec{F} = 6.67 \times 10^{-11} \, \text{N} \cdot \text{m}^2 / \text{kg}^2 \times \frac{92.0 \, \text{kg} \times 0.894 \, \text{kg}}{(99.3 \, \text{m})^2}
\][/tex]
First, compute the numerator:
[tex]\[
92.0 \, \text{kg} \times 0.894 \, \text{kg} = 82.248 \, \text{kg}^2
\][/tex]
Next, compute the denominator:
[tex]\[
(99.3 \, \text{m})^2 = 9860.49 \, \text{m}^2
\][/tex]
Now, divide the numerator by the denominator:
[tex]\[
\frac{82.248 \, \text{kg}^2}{9860.49 \, \text{m}^2} \approx 0.0083393935 \, \text{kg}^2 / \text{m}^2
\][/tex]
Finally, multiply by the gravitational constant [tex]\( G \)[/tex]:
[tex]\[
\vec{F} = 6.67 \times 10^{-11} \, \text{N} \cdot \text{m}^2 / \text{kg}^2 \times 0.0083393935 \, \text{kg}^2 / \text{m}^2 \approx 5.563558808943573 \times 10^{-13} \, \text{N}
\][/tex]
The result can be expressed in scientific notation by separating the mantissa and the exponent. Here, the mantissa is approximately [tex]\( 5.563558808943573 \)[/tex] and the exponent is [tex]\( -13 \)[/tex]:
[tex]\[
\vec{F} \approx 5.563558808943573 \times 10^{-13} \, \text{N}
\][/tex]
In conclusion, the gravitational force between the two masses is:
[tex]\[
\vec{F} \approx 5.563558808943573 \times 10^{-13} \, \text{N}
\][/tex]
Written in scientific notation, this is:
[tex]\[
\vec{F} = 5.563558808943573 \times 10^{-13} \, \text{N}
\][/tex]
