Respuesta :
Answer: M = 42.553·[tex]10^{36}[/tex] kg
Explanation: A orbit where a body remains traveling around a gravitating mass at constant radius is called Circular Orbit. Although in reality the orbit is more like an ellipse, the circular orbit is a good approximation to the real one.
In that system, it is possible to determine the velocity needed to maintain the orbit. The formula is: v = [tex]\sqrt{\frac{GM}{r} }[/tex] , where:
v is velocity;
G is the gravitational constant( = 6.67·[tex]10^{-11}[/tex][tex]\frac{N.m^{2} }{kg^{2} }[/tex])
M is the mass of the gravitating mass;
r is the distance between the center of the massive object and the orbiting object;
But, this question is asking for the mass M, so, rearraging:
[tex]v^{2} = \frac{GM}{r}[/tex]
[tex]M = \frac{v^{2}.r }{G}[/tex]
Transforming light-years in metres and dividing by 2 to find the radius:
r = (15.9.461 x 10¹⁵)·[tex]\frac{1}{2}[/tex] = 70.9575
M = [tex]\frac{(2.10^{5}) ^{2} .70.9575 }{6.67.10^{-11} }[/tex]
M = 42.553.[tex]10^{36}[/tex]kg
The mass of the massive object at the center of the ring is 42.553.[tex]10^{36}[/tex]kg.
The mass M of the massive object at the center of the Milky Way galaxy obtained is 4.26×10³⁷ Kg
How to determine the radius
- Diameter = 15 light-years
- Radius =?
Radius = diameter / 2
Radius = 15 / 2
Radius = 7.5 light-years
Radius = 7.5 × 9.461×10¹⁵
Radius = 70.9575×10¹⁵ m
How to determine the mass
- Velocity (v) = 200 km/s = 200 × 1000 = 200000 m/s
- Radius (r) = 70.9575×10¹⁵ m
- Gravitational constant (G) = 6.67×10¯¹¹ Nm²/Kg²
- Mass (M) =?
v = √(GM/r)
v² = GM / r
Cross multiply
GM = v²r
Divide both side by G
M = v²r / G
M = (200000² × 70.9575×10¹⁵) / 6.67×10¯¹¹
M = 4.26×10³⁷ Kg
Learn more about gravitational force:
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