Question

The Large Magellanic Cloud is a small galaxy that orbits the Milky Way. It is currently orbiting the Milky Way at a distance of roughly 160000 light-years from the galactic center at a velocity of about 300 km/s.
Use these values in the orbital velocity law to get an estimate of the Milky Way's mass within 160000 light-years from the center. (The value you obtain is a fairly rough estimate because the orbit of the Large Magellanic Cloud is not circular.)

Answer 1

Answer: $2.0416(10)^{42}kg$

Explanation:

Approaching the orbit of the Large Magellanic Cloud around the Milky Way to a circular orbit, we can use the equation of velocity in the case of uniform circular motion:

$V=\sqrt{G\frac{M}{r}}$     (1)

Where:

$V=300km/s=3(10)^{5}m/s$ is the velocity of the Large Magellanic Cloud's orbit, which is assumed as constant.

$G$ is the Gravitational Constant and its value is $6.674(10)^{-11}\frac{m^{3}}{kgs^{2}}$

$M$ is the mass of the Milky Way

$r=160000ly=1.51376(10)^{21}m$  is the radius of the orbit, which is the distance from the center of the Milky Way to the Large Magellanic Cloud.

Now, if we want to know the estimated mass of the Milky Way, we have to find $M$ from (1):

$M=\frac{V^{2} r}{G}$     (2)

Substituting the known values:

$M=\frac{(3(10)^{5}m/s)^{2}(1.51376(10)^{21}m)}{6.674(10)^{-11}\frac{m^{3}}{kgs^{2}}}$     (3)

$M=\frac{1.362384(10)^{32}\frac{m^{3}}{s^{2}}}{6.674(10)^{-11}\frac{m^{3}}{kgs^{2}}}$    

Finally:

$M=2.0416(10)^{42}kg$ >>>This is the estimated mass of the Milky Way