To determine the magnitude of the force of gravity acting on a 70-kilogram mountaineer standing on the summit of Mt. Everest, we will use the gravitational force formula:
[tex]\[ F = G \frac{m_1 \cdot m_2}{r^2} \][/tex]
where:
- [tex]\( F \)[/tex] is the gravitational force,
- [tex]\( G \)[/tex] is the gravitational constant ([tex]\( 6.67 \times 10^{-11} \, \text{m}^3 \, \text{kg}^{-1} \, \text{s}^{-2} \)[/tex]),
- [tex]\( m_1 \)[/tex] is the mass of the mountaineer,
- [tex]\( m_2 \)[/tex] is the mass of the Earth,
- [tex]\( r \)[/tex] is the distance between the centers of mass of the mountaineer and the Earth.
Given data:
- Mass of the mountaineer ([tex]\( m_1 \)[/tex]) = 70 kg,
- Mass of the Earth ([tex]\( m_2 \)[/tex]) = [tex]\( 5.98 \times 10^{24} \)[/tex] kg,
- Distance from the center of the Earth to the summit of Mt. Everest [tex]\( r \)[/tex]:
- Earth's radius ≈ 6400 km,
- Height of Mt. Everest ≈ 8850 m (or 8.850 km).
First, we convert the Earth’s radius to meters since the height of Mt. Everest is given in meters:
[tex]\[ \text{Radius of Earth} = 6400 \, \text{km} \times 1000 \, \text{m/km} = 6,400,000 \, \text{m} \][/tex]
So, the total distance [tex]\( r \)[/tex] from the center of the Earth to the summit of Mt. Everest is:
[tex]\[ r = 6,400,000 \, \text{m} + 8,850 \, \text{m} = 6,408,850 \, \text{m} \][/tex]
Now we can plug these values into the gravitational force formula:
[tex]\[ F = 6.67 \times 10^{-11} \frac{(70 \, \text{kg}) (5.98 \times 10^{24} \, \text{kg})}{(6,408,850 \, \text{m})^2} \][/tex]
After calculating [tex]\( F \)[/tex], we find the gravitational force acting on the mountaineer to be:
[tex]\[ F \approx 679.77 \, \text{N} \][/tex]
Given the choices:
A. 90.5 newtons,
B. 179 newtons,
C. 684 newtons,
D. 781 newtons,
The closest value to 679.77 newtons is:
C. 684 newtons
Therefore, the correct answer is:
C. 684 newtons.
