To find the gravitational force between the two masses, we use the formula for the gravitational force:
[tex]\[
F = G \frac{m_1 m_2}{r^2}
\][/tex]
where:
- [tex]\( G = 6.67 \times 10^{-11} \, \text{N} \cdot \text{m}^2 / \text{kg}^2 \)[/tex] is the gravitational constant,
- [tex]\( m_1 = 75.0 \, \text{kg} \)[/tex] is the first mass,
- [tex]\( m_2 = 68.4 \, \text{kg} \)[/tex] is the second mass, and
- [tex]\( r = 1.15 \, \text{m} \)[/tex] is the distance between the two masses.
Substituting these values into the formula:
[tex]\[
F = \left( 6.67 \times 10^{-11} \right) \frac{(75.0) (68.4)}{(1.15)^2}
\][/tex]
First, calculate the product of the masses:
[tex]\[
75.0 \times 68.4 = 5130.0 \, \text{kg}^2
\][/tex]
Next, calculate the square of the distance:
[tex]\[
(1.15)^2 = 1.3225 \, \text{m}^2
\][/tex]
Now, substitute these values back into the formula:
[tex]\[
F = \left( 6.67 \times 10^{-11} \right) \frac{5130.0}{1.3225}
\][/tex]
Compute the fraction inside the formula:
[tex]\[
\frac{5130.0}{1.3225} \approx 3879.1304347826085
\][/tex]
Finally, multiply by the gravitational constant:
[tex]\[
F = (6.67 \times 10^{-11}) \times 3879.1304347826085 \approx 2.587304347826087 \times 10^{-7} \, \text{N}
\][/tex]
Therefore, the gravitational force [tex]\( \vec{F} \)[/tex] between the two masses is [tex]\( 2.587304347826087 \times 10^{-7} \, \text{N} \)[/tex].
To write this result in scientific notation with proper coefficients and exponents:
[tex]\[
\vec{F} = 2.587304347826087 \times 10^{-7} \, \text{N}
\][/tex]
Thus,
[tex]\[
\vec{F} = a \times 10^b \, \text{N}
\][/tex]
where
[tex]\[
a = 2.587304347826087
\][/tex]
and
[tex]\[
b = -7.
\][/tex]
So, the gravitational force between the two masses is [tex]\( \mathbf{2.587304347826087 \times 10^{-7} \, \text{N}} \)[/tex]
