Answer :
To solve this problem, we will use Newton's law of universal gravitation, which 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]\[ F = G \frac{m_1 \cdot m_2}{r^2} \][/tex]
where:
- [tex]\( G \)[/tex] is the gravitational constant, which is approximately [tex]\( 6.67430 \times 10^{-11} \, \text{m}^3 \text{kg}^{-1} \text{s}^{-2} \)[/tex].
- [tex]\( m_1 \)[/tex] is the mass of the Earth, [tex]\( 6.0 \times 10^{24} \)[/tex] kg.
- [tex]\( m_2 \)[/tex] is the mass of Jupiter, [tex]\( 1.901 \times 10^{27} \)[/tex] kg.
- [tex]\( r \)[/tex] is the distance between the two planets, [tex]\( 7.5 \times 10^{11} \)[/tex] meters.
Plugging in the values into the formula, we get:
[tex]\[ F = (6.67430 \times 10^{-11}) \frac{(6.0 \times 10^{24}) \cdot (1.901 \times 10^{27})}{(7.5 \times 10^{11})^2} \][/tex]
Let's break this down step by step:
1. Multiply the masses:
[tex]\[ m_1 \cdot m_2 = (6.0 \times 10^{24}) \cdot (1.901 \times 10^{27}) \][/tex]
[tex]\[ = 1.1406 \times 10^{52} \, \text{kg}^2 \][/tex]
2. Square the distance:
[tex]\[ r^2 = (7.5 \times 10^{11})^2 \][/tex]
[tex]\[ = 5.625 \times 10^{23} \, \text{m}^2 \][/tex]
3. Divide the product of the masses by the square of the distance:
[tex]\[ \frac{m_1 \cdot m_2}{r^2} = \frac{1.1406 \times 10^{52}}{5.625 \times 10^{23}} \][/tex]
[tex]\[ = 2.0272 \times 10^{28} \, \text{kg} \cdot \text{m}^{-2} \][/tex]
4. Multiply by the gravitational constant [tex]\(G\)[/tex]:
[tex]\[ F = 6.67430 \times 10^{-11} \times 2.0272 \times 10^{28} \][/tex]
[tex]\[ = 1.3533700586666665 \times 10^{18} \, \text{newtons} \][/tex]
Thus, the force of gravity between Earth and Jupiter is approximately:
[tex]\[ 1.353 \times 10^{18} \, \text{newtons} \][/tex]
Therefore, the correct answer is:
C. [tex]\( 1.352 \times 10^{18} \)[/tex] newtons.
[tex]\[ F = G \frac{m_1 \cdot m_2}{r^2} \][/tex]
where:
- [tex]\( G \)[/tex] is the gravitational constant, which is approximately [tex]\( 6.67430 \times 10^{-11} \, \text{m}^3 \text{kg}^{-1} \text{s}^{-2} \)[/tex].
- [tex]\( m_1 \)[/tex] is the mass of the Earth, [tex]\( 6.0 \times 10^{24} \)[/tex] kg.
- [tex]\( m_2 \)[/tex] is the mass of Jupiter, [tex]\( 1.901 \times 10^{27} \)[/tex] kg.
- [tex]\( r \)[/tex] is the distance between the two planets, [tex]\( 7.5 \times 10^{11} \)[/tex] meters.
Plugging in the values into the formula, we get:
[tex]\[ F = (6.67430 \times 10^{-11}) \frac{(6.0 \times 10^{24}) \cdot (1.901 \times 10^{27})}{(7.5 \times 10^{11})^2} \][/tex]
Let's break this down step by step:
1. Multiply the masses:
[tex]\[ m_1 \cdot m_2 = (6.0 \times 10^{24}) \cdot (1.901 \times 10^{27}) \][/tex]
[tex]\[ = 1.1406 \times 10^{52} \, \text{kg}^2 \][/tex]
2. Square the distance:
[tex]\[ r^2 = (7.5 \times 10^{11})^2 \][/tex]
[tex]\[ = 5.625 \times 10^{23} \, \text{m}^2 \][/tex]
3. Divide the product of the masses by the square of the distance:
[tex]\[ \frac{m_1 \cdot m_2}{r^2} = \frac{1.1406 \times 10^{52}}{5.625 \times 10^{23}} \][/tex]
[tex]\[ = 2.0272 \times 10^{28} \, \text{kg} \cdot \text{m}^{-2} \][/tex]
4. Multiply by the gravitational constant [tex]\(G\)[/tex]:
[tex]\[ F = 6.67430 \times 10^{-11} \times 2.0272 \times 10^{28} \][/tex]
[tex]\[ = 1.3533700586666665 \times 10^{18} \, \text{newtons} \][/tex]
Thus, the force of gravity between Earth and Jupiter is approximately:
[tex]\[ 1.353 \times 10^{18} \, \text{newtons} \][/tex]
Therefore, the correct answer is:
C. [tex]\( 1.352 \times 10^{18} \)[/tex] newtons.
