Answer :
1.58 m/s^2
For this problem, going to assume that the satellite is VERY close to the surface of a rather unusually smooth moon. In fact, I'm going to assume the satellite is almost skimming the surface, mere millimeters away from it.
The relationship between acceleration and velocity for a body in orbit is
a = v^2/r
Now we've been given r, and we still need v. Since velocity is defined as distance over time, we need to calculate the distance which will be 2 pi r. So
2 * 3.14159 * 1.74x10^6 m = 10932742 m
Therefore v will be
10932742 / (110 * 60) = 10932742/ 6600 = 1656 m/s
Substituting into the equation for a, gives us
a = (1656 m/s)^2/1.74x10^6 m
a = 2742336 m^2/s^2 / 1.74x10^6 m
a = 1.576055 m/s^2
Rounding to 3 significant figures gives 1.58 m/s^2
Note: If you wish a fun way of learning about orbital mechanics, you could do far worse than playing Kerbal Space Program. As the creator of XKCD, Randall Munroe, says. https://xkcd.com/1356/
The free-fall acceleration on the moon's surface is about 1.58 m/s²
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Further explanation
Newton's gravitational law states that the force of attraction between two objects can be formulated as follows:
[tex]\large {\boxed {F = G \frac{m_1 ~ m_2}{R^2}} }[/tex]
F = Gravitational Force ( Newton )
G = Gravitational Constant ( 6.67 × 10⁻¹¹ Nm² / kg² )
m = Object's Mass ( kg )
R = Distance Between Objects ( m )
Let us now tackle the problem !
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Given:
Period of Satellite = T = 110 min = 6600 s
Radius of Moon = R = 1.74 × 10⁶ m
Asked:
free-fall acceleration on the moon's surface = g = ?
Solution:
[tex]\Sigma F = ma[/tex]
[tex]G \frac{ M m} { R^2 } = m \omega^2 R[/tex]
[tex]G \frac { M } { R^2 } = \omega^2 R[/tex]
[tex]g = \omega^2 R[/tex]
[tex]g = (\frac{2 \pi }{ T} )^2 R[/tex]
[tex]g = ( \frac {2\pi} { 6600 } )^2 \times ( 1.74 \times 10^6 )[/tex]
[tex]g \approx 1.58 \texttt{ m/s}^2[/tex]
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Learn more
- Impacts of Gravity : https://brainly.com/question/5330244
- Effect of Earth’s Gravity on Objects : https://brainly.com/question/8844454
- The Acceleration Due To Gravity : https://brainly.com/question/4189441
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Answer details
Grade: High School
Subject: Physics
Chapter: Gravitational Fields
