Certainly! Let's solve the problem step by step using the provided formula to calculate the gravitational force between two masses.
### Information Provided:
- Mass 1 ([tex]\( m_1 \)[/tex]) = 75.0 kg
- Mass 2 ([tex]\( m_2 \)[/tex]) = 68.4 kg
- Distance between the masses ([tex]\( r \)[/tex]) = 1.15 m
- Gravitational constant ([tex]\( G \)[/tex]) = [tex]\( 6.67 \times 10^{-11} \)[/tex] [tex]\( \text{N} \cdot \text{m}^2/\text{kg}^2 \)[/tex]
### Formula:
The gravitational force ([tex]\( F \)[/tex]) is given by:
[tex]\[ \vec{F} = G \frac{m_1 m_2}{r^2} \][/tex]
### Steps to Calculate:
1. Substitute the values into the formula:
[tex]\[
F = 6.67 \times 10^{-11} \times \frac{75.0 \times 68.4}{(1.15)^2}
\][/tex]
2. Calculate the value of [tex]\( m_1 \times m_2 \)[/tex]:
[tex]\[
m_1 \times m_2 = 75.0 \times 68.4 = 5130
\][/tex]
3. Calculate the value of [tex]\( r^2 \)[/tex]:
[tex]\[
r^2 = (1.15)^2 = 1.3225
\][/tex]
4. Apply these values in the formula:
[tex]\[
F = 6.67 \times 10^{-11} \times \frac{5130}{1.3225}
\][/tex]
Calculate the fractional division:
[tex]\[
\frac{5130}{1.3225} = 3877.56
\][/tex]
5. Then, multiply by [tex]\( 6.67 \times 10^{-11} \)[/tex]:
[tex]\[
F = 6.67 \times 10^{-11} \times 3877.56 \approx 2.587304347826087 \times 10^{-7}
\][/tex]
6. Express the force in scientific notation:
[tex]\[
F \approx 2.59 \times 10^{-7} \, \text{N}
\][/tex]
### Final Answer:
[tex]\[
\vec{F} \approx 2.59 \times 10^{-7} \, \text{N}
\][/tex]
So, the gravitational force between the two masses is approximately [tex]\( 2.59 \times 10^{-7} \)[/tex] Newtons.
