To find the gravitational force between the two given masses, we'll use the gravitational force formula:
[tex]\[
\vec{F} = G \frac{m_1 m_2}{r^2}
\][/tex]
where:
- [tex]\( G = 6.67 \times 10^{-11} \, \text{N} \cdot \text{m}^2 / \text{kg}^2 \)[/tex] is the gravitational constant,
- [tex]\( m_1 = 5.00 \, \text{kg} \)[/tex] is the mass of the first object,
- [tex]\( m_2 = 7.50 \, \text{kg} \)[/tex] is the mass of the second object,
- [tex]\( r = 0.285 \, \text{m} \)[/tex] is the distance between the centers of the two masses.
Substitute the values into the formula:
[tex]\[
\vec{F} = \left( 6.67 \times 10^{-11} \right) \frac{(5.00) \times (7.50)}{(0.285)^2}
\][/tex]
First, calculate the numerator:
[tex]\[
5.00 \times 7.50 = 37.5 \, \text{kg}^2
\][/tex]
Next, calculate the denominator:
[tex]\[
(0.285)^2 = 0.081225 \, \text{m}^2
\][/tex]
Now, substitute these values into the force equation:
[tex]\[
\vec{F} = \left( 6.67 \times 10^{-11} \right) \frac{37.5}{0.081225}
\][/tex]
Calculate the fraction inside the parentheses:
[tex]\[
\frac{37.5}{0.081225} \approx 461.615
\][/tex]
Multiply this result by the gravitational constant [tex]\( G \)[/tex]:
[tex]\[
\vec{F} = 6.67 \times 10^{-11} \times 461.615 \approx 3.079409048938135 \times 10^{-8} \, \text{N}
\][/tex]
Thus, the gravitational force between the two masses is approximately:
[tex]\[
\vec{F} = 3.08 \times 10^{-8} \, \text{N}
\][/tex]
Therefore, the magnitude of the force is [tex]\( 3.08 \times 10^{-8} \, \text{N} \)[/tex].
Additionally, the force in terms of scientific notation can also be expressed as:
[tex]\[
\vec{F} = 3.079409048938135 \times 10^{-8} \, \text{N}
\][/tex]
So, the final force consists of:
[tex]\[
\vec{F} = (3.079409048938135 \times 10^{-8}) \, \text{N}
\][/tex]
The magnitude part [tex]\( A \)[/tex] is [tex]\( 3.079409048938135 \)[/tex] and the exponent part [tex]\( B \)[/tex] is [tex]\( -8 \)[/tex].
[tex]\[
\begin{array}{c}
\vec{F} = [(3.079409048938135)] \times 10^{[-8]} \, \text{N}
\end{array}
\][/tex]
