Our Milky Way galaxy is very similar to our neighboring galaxy Andromeda. Consider that each galaxy contains approximately 4.0 x 10^11 stars each star having an average mass of 2.0 x 10^30 kg.
a. If the gravitational pull of Andromeda is 2.4 x 10^30 N, how far away is Andromeda?
b. How many light years away is the Andromeda Galaxy? (Note there are 365.25 days per year and light travels at 3.0 x 10^8 m/s).
To find the distance to the Andromeda galaxy, we can use Newton's law of universal gravitation, which states that the force of gravity between two objects is equal to the gravitational constant (represented by G) multiplied by the product of their masses, divided by the square of the distance between them.
a. Let's solve for the distance using the given gravitational pull of 2.4 x 10^30 N.
Step 1: Identify the known values:
- Mass of each galaxy (m1 = m2) = 4.0 x 10^11 stars * 2.0 x 10^30 kg/star = 8.0 x 10^41 kg
- Gravitational pull (F) = 2.4 x 10^30 N
- Gravitational constant (G) = 6.67430 x 10^-11 N m^2/kg^2
Step 2: Use Newton's law of universal gravitation to solve for the distance (r).
F = G * (m1 * m2) / r^2
Step 3: Rearrange the formula to solve for r.
r^2 = G * (m1 * m2) / F
r = √(G * (m1 * m2) / F)
Step 4: Substitute the known values and calculate the distance.
r = √[(6.67430 x 10^-11 N m^2/kg^2) * (8.0 x 10^41 kg * 8.0 x 10^41 kg) / (2.4 x 10^30 N)]
r ≈ 2.67 x 10^22 m
Therefore, Andromeda is approximately 2.67 x 10^22 meters away from us.
b. To convert this distance to light years, we need to divide it by the speed of light, which is approximately 3.0 x 10^8 m/s.
Step 1: Identify the known values:
- Distance (d) = 2.67 x 10^22 m
- Speed of light (c) ≈ 3.0 x 10^8 m/s
Step 2: Use the formula for light years:
1 light year = c * 365.25 * 24 * 60 * 60 seconds
Step 3: Substitute the known values and calculate the distance in light years.
d_ly = d / (c * 365.25 * 24 * 60 * 60)
d_ly = (2.67 x 10^22 m) / (3.0 x 10^8 m/s * 365.25 days * 24 hours * 60 minutes * 60 seconds)
d_ly ≈ 2.81 x 10^6 light years
Therefore, the Andromeda galaxy is approximately 2.81 x 10^6 light years away from us.