Answer :
To find the force of gravity between Earth and an object [tex]\(10^{10}\)[/tex] meters away, we use the formula for gravitational force:
[tex]\[ F = G \frac{m_1 m_2}{r^2} \][/tex]
Where:
- [tex]\( F \)[/tex] is the gravitational force,
- [tex]\( G \)[/tex] is the gravitational constant [tex]\((6.673 \times 10^{-11} \, \text{N}\cdot \text{m}^2/\text{kg}^2)\)[/tex],
- [tex]\( m_1 \)[/tex] is the mass of the Earth [tex]\((6.0 \times 10^{24} \, \text{kg})\)[/tex],
- [tex]\( m_2 \)[/tex] is the mass of the other object (we assume it to be 1 kg for simplicity),
- [tex]\( r \)[/tex] is the distance between the centers of the two masses [tex]\((10^{10} \, \text{m})\)[/tex].
Substitute the values into the formula:
[tex]\[ F = 6.673 \times 10^{-11} \times \frac{(6.0 \times 10^{24}) \times 1}{(10^{10})^2} \][/tex]
Calculate the denominator and the numerator separately:
[tex]\[ (10^{10})^2 = 10^{20} \][/tex]
[tex]\[ 6.0 \times 10^{24} \times 1 = 6.0 \times 10^{24} \][/tex]
Now, divide the numerator by the denominator and multiply by the gravitational constant:
[tex]\[ F = 6.673 \times 10^{-11} \times \frac{6.0 \times 10^{24}}{10^{20}} \][/tex]
[tex]\[ F = 6.673 \times 10^{-11} \times 6.0 \times 10^{4} \][/tex]
[tex]\[ F = 6.673 \times 6.0 \times 10^{-11+4} \][/tex]
[tex]\[ F = 40.038 \times 10^{-7} \][/tex]
[tex]\[ F = 4.0038 \times 10^{-6} \][/tex]
Therefore, the calculated gravitational force is [tex]\(4.0038 \times 10^{-6} \, \text{newtons}\)[/tex].
After comparing the calculated force with the given options (13.52 newtons, 51.39 newtons, 13.52 × 10^{17} newtons, 51.39 × 10^{17} newtons), none of them matches the correct result in the problem.
Hence, the correct answer would be:
None of the above.
[tex]\[ F = G \frac{m_1 m_2}{r^2} \][/tex]
Where:
- [tex]\( F \)[/tex] is the gravitational force,
- [tex]\( G \)[/tex] is the gravitational constant [tex]\((6.673 \times 10^{-11} \, \text{N}\cdot \text{m}^2/\text{kg}^2)\)[/tex],
- [tex]\( m_1 \)[/tex] is the mass of the Earth [tex]\((6.0 \times 10^{24} \, \text{kg})\)[/tex],
- [tex]\( m_2 \)[/tex] is the mass of the other object (we assume it to be 1 kg for simplicity),
- [tex]\( r \)[/tex] is the distance between the centers of the two masses [tex]\((10^{10} \, \text{m})\)[/tex].
Substitute the values into the formula:
[tex]\[ F = 6.673 \times 10^{-11} \times \frac{(6.0 \times 10^{24}) \times 1}{(10^{10})^2} \][/tex]
Calculate the denominator and the numerator separately:
[tex]\[ (10^{10})^2 = 10^{20} \][/tex]
[tex]\[ 6.0 \times 10^{24} \times 1 = 6.0 \times 10^{24} \][/tex]
Now, divide the numerator by the denominator and multiply by the gravitational constant:
[tex]\[ F = 6.673 \times 10^{-11} \times \frac{6.0 \times 10^{24}}{10^{20}} \][/tex]
[tex]\[ F = 6.673 \times 10^{-11} \times 6.0 \times 10^{4} \][/tex]
[tex]\[ F = 6.673 \times 6.0 \times 10^{-11+4} \][/tex]
[tex]\[ F = 40.038 \times 10^{-7} \][/tex]
[tex]\[ F = 4.0038 \times 10^{-6} \][/tex]
Therefore, the calculated gravitational force is [tex]\(4.0038 \times 10^{-6} \, \text{newtons}\)[/tex].
After comparing the calculated force with the given options (13.52 newtons, 51.39 newtons, 13.52 × 10^{17} newtons, 51.39 × 10^{17} newtons), none of them matches the correct result in the problem.
Hence, the correct answer would be:
None of the above.
