Sure! Let's solve this step-by-step.
1. Identify the known quantities:
- Mass of the first object, [tex]\( m_1 = 92.0 \)[/tex] kg
- Mass of the second object, [tex]\( m_2 = 0.894 \)[/tex] kg
- Distance between the objects, [tex]\( r = 99.3 \)[/tex] m
- Gravitational constant, [tex]\( G = 6.67 \times 10^{-11} \, \text{N} \cdot \text{m}^2 / \text{kg}^2 \)[/tex]
2. Write down the gravitational force formula:
[tex]\[
F = G \frac{m_1 m_2}{r^2}
\][/tex]
3. Plug in the known values:
[tex]\[
F = 6.67 \times 10^{-11} \, \frac{\text{N} \cdot \text{m}^2}{\text{kg}^2} \cdot \frac{92.0 \, \text{kg} \times 0.894 \, \text{kg}}{(99.3 \, \text{m})^2}
\][/tex]
4. Calculate the product of the masses:
[tex]\[
92.0 \, \text{kg} \times 0.894 \, \text{kg} = 82.248 \, \text{kg}^2
\][/tex]
5. Calculate the square of the distance:
[tex]\[
(99.3 \, \text{m})^2 = 9850.49 \, \text{m}^2
\][/tex]
6. Substitute these values into the formula:
[tex]\[
F = 6.67 \times 10^{-11} \, \frac{\text{N} \cdot \text{m}^2}{\text{kg}^2} \cdot \frac{82.248 \, \text{kg}^2}{9850.49 \, \text{m}^2}
\][/tex]
7. Perform the division inside the force formula:
[tex]\[
\frac{82.248 \, \text{kg}^2}{9850.49 \, \text{m}^2} \approx 8.350201 \times 10^{-3} \, \frac{\text{kg}^2}{\text{m}^2}
\][/tex]
8. Calculate the force value:
[tex]\[
F = 6.67 \times 10^{-11} \times 8.350201 \times 10^{-3} \approx 5.563558808943572 \times 10^{-13} \, \text{N}
\][/tex]
9. Express the force in scientific notation:
By simplifying, we get:
[tex]\[
\vec{F}=0.005563558808943572 \times 10^{-13 + 12} \text{N} = 0.005563558808943572 \times 10^{-1} \text{N}
\][/tex]
So, the gravitational force [tex]\( \vec{F} \)[/tex] can be written as [tex]\( (0.005563558808943572 \times 10^{-1}) \, \text{N} \)[/tex].
10. Simplify the notation:
Finally, the gravitational force between the two masses is:
[tex]\[
\boxed{0.005563558808943572 \times 10^{-1} \, \text{N}}
\][/tex]
Thus:
[tex]\[
\vec{F}=[0.005563558808943572] \times 10^{[-1]} \, \text{N}
\][/tex]
