To determine the force of gravity between Earth and Jupiter, we apply Newton's law of universal gravitation. This law states that the gravitational force [tex]\( F \)[/tex] between two masses [tex]\( m_1 \)[/tex] and [tex]\( m_2 \)[/tex] separated by a distance [tex]\( r \)[/tex] is given by:
[tex]\[ F = G \cdot \frac{m_1 \cdot m_2}{r^2} \][/tex]
where [tex]\( G \)[/tex] is the gravitational constant, approximately equal to [tex]\( 6.67430 \times 10^{-11} \, \text{m}^3 \text{kg}^{-1} \text{s}^{-2} \)[/tex].
Given:
- Mass of Earth, [tex]\( m_1 = 6.0 \times 10^{24} \, \text{kg} \)[/tex]
- Mass of Jupiter, [tex]\( m_2 = 1.901 \times 10^{27} \, \text{kg} \)[/tex]
- Distance between Earth and Jupiter, [tex]\( r = 7.5 \times 10^{11} \, \text{m} \)[/tex]
Plugging these values into the equation for the gravitational force, we have:
[tex]\[ F = 6.67430 \times 10^{-11} \cdot \frac{(6.0 \times 10^{24}) \cdot (1.901 \times 10^{27})}{(7.5 \times 10^{11})^2} \][/tex]
First, calculate the denominator [tex]\( r^2 \)[/tex]:
[tex]\[ (7.5 \times 10^{11})^2 = 56.25 \times 10^{22} \][/tex]
Next, multiply the masses:
[tex]\[ (6.0 \times 10^{24}) \cdot (1.901 \times 10^{27}) = 1.1406 \times 10^{52} \][/tex]
Now, calculate the gravitational force:
[tex]\[ F = 6.67430 \times 10^{-11} \cdot \frac{1.1406 \times 10^{52}}{56.25 \times 10^{22}} \][/tex]
Simplify the fraction inside the equation:
[tex]\[ \frac{1.1406 \times 10^{52}}{56.25 \times 10^{22}} = 2.027 \times 10^{29} \][/tex]
Finally, multiply by [tex]\( G \)[/tex]:
[tex]\[ F = 6.67430 \times 10^{-11} \cdot 2.027 \times 10^{29} = 1.3533700586666665 \times 10^{18} \, \text{N} \][/tex]
So, the gravitational force between Earth and Jupiter is approximately [tex]\( 1.353 \times 10^{18} \)[/tex] newtons.
Therefore, the correct answer is:
C. [tex]\( 1352 \times 10^{18} \)[/tex] newtons
