Sure! Let's solve the problem step-by-step.
We are given:
- Distance between the masses, [tex]\( r = 5.60 \)[/tex] meters
- Mass 1, [tex]\( m_1 = 4.17 \)[/tex] kilograms
- Mass 2, [tex]\( m_2 = 3.29 \)[/tex] kilograms
- Gravitational constant, [tex]\( G = 6.67 \times 10^{-11} \)[/tex] N [tex]\(\cdot\)[/tex] m[tex]\(^2\)[/tex] / kg[tex]\(^2\)[/tex]
We need to find the gravitational force [tex]\( \vec{F} \)[/tex] between the two masses using the formula:
[tex]\[
\vec{F} = G \cdot \frac{m_1 \cdot m_2}{r^2}
\][/tex]
Step-by-step solution:
1. Plug in the known values into the formula:
[tex]\[
\vec{F} = 6.67 \times 10^{-11} \cdot \frac{4.17 \cdot 3.29}{(5.60)^2}
\][/tex]
2. Calculate the product of the masses:
[tex]\[
4.17 \cdot 3.29 = 13.7193
\][/tex]
3. Calculate the square of the distance:
[tex]\[
(5.60)^2 = 31.36
\][/tex]
4. Divide the product of the masses by the square of the distance:
[tex]\[
\frac{13.7193}{31.36} = 0.4375710208530831
\][/tex]
5. Multiply by the gravitational constant [tex]\( G \)[/tex]:
[tex]\[
\vec{F} = 6.67 \times 10^{-11} \cdot 0.4375710208530831 = 2.917976116071429 \times 10^{-11} \text{ N}
\][/tex]
So the gravitational force is:
[tex]\[
\vec{F} = 2.917976116071429 \times 10^{-11} \text{ N}
\][/tex]
To express the force in the format [tex]\([a] \times 10^{[b]} \text{ N}\)[/tex]:
- [tex]\( a = 0.2917976116071429 \)[/tex]
- [tex]\( b = -10 \)[/tex]
Therefore, the gravitational force between the two masses is:
[tex]\[
\vec{F} = 0.2917976116071429 \times 10^{-10} \text{ N}
\][/tex]
