The solar system is 25,000 light years from the center of our Milky Way galaxy. One light year is the distance light travels in one year at a speed of 3.0 x 10⁸ m/s . Astronomers have determined that the solar system is orbiting the center of the galaxy at a speed of 230 km/s . (c) The gravitational force on the solar system is the net force due to all the matter inside our orbit. Most of that matter is concentrated near the center of the galaxy. Assume that the matter has a spherical distribution, like a giant star. What is the approximate mass of the galactic center?

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Welcome back. Everyone in this problem. Imagine a planetary system located roughly 35,000 light years from the core of the Andromeda galaxy. A light year represents the distance light covers in a year traveling at three multiplied by 10 to the 8 m per second. This system orbits the galaxy center at a velocity of around 250 kilometers per second. What would be the estimated mass of the galaxy center if most of the mass is concentrated near the core and the mass within the orbit is spherical distributed similar to a giant star. For our answer choices. A says it's two multiplied by 10 to the 41st kilograms. B 3.1 multiplied by 10 to the 41st kilograms C 4.5 multiplied by 10 to the 42nd kilograms and the D 6.7 multiplied by 10 to the 42nd kilograms. Now, if we're going to figure out estimated mass, of course, let's start by making note of the information we already know. So for our system, we know that the planetary system is 35,000 light years from the core of the Andromeda galaxy. Now, if we represent that distance as R OK, then we'll need to convert it to si units. That is, we need to convert it to meters. So what is 35,000 light years in meters? How far away is that actually? Well, we have to think about how far one light year is now. One light year is the distance covered by light in one year. We know how fast light travels, we know how long one year is and we know that distance equals speed multiplied by time. So we can find one light year by multiplying the speed of light by the time for a year. And then multiply that by 35,000 to find a distance in meters, let's actually write what we mean. No, one light here is going to be equal to the speed of light, which is three multiplied by 10 to the 8 m per second, multiplied by the length of time for a year. Now we need to convert it to seconds. Ok. So how many seconds are in a year basically? Well, we know that a year has 365 days and one day has 24 hours. Ok. And one hour has 60 minutes. Ok? And one minute has 60 seconds. I feel like I need some more space here. Let me bring this back a bit. Ok. So we're talking about 60 minutes, 60 seconds per minute. Ok. So now when we multiply all of this, we get the value for one light here to be approximately 9.461 multiplied by 10 to the 15th meters. So since that's the length or how far away one light year is, then 35,000 light years is gonna be equal to 35,000 multiplied by that value, which is going to be approximately equal to 3.311 multiplied by 10 to the 20th meters. So that's how far 35,000 light years is in meters. Now, what else do we know? We also know that the system orbits the galaxy center at a velocity of around 250 kilometers per second. How fast is that in meters per second? Well, to figure that out, we can multiply 250 kilometers per second by 1000 m for every kilometer. And when we do that, then we get the speed to be equal to 2.5 multiplied by 10 to the fifth meters per second. So now that we have all the information from the problem, the velocity and the distance, how can we use that to find the estimated mass of the galaxy center? Well, if we draw a quick sketch, it can help us to better understand. So what are we really saying? Well, here's our planetary system and we say it's orbiting around the center of mass. OK. And let's just represent this as the orbit. OK? And for this center of mass, we can call it mass M. While for our planetary system, we can call it common M OK. Now, because it is in orbit here, we know that it has a central pal force. And according to Newton's second law of motion, the force is directly proportional to the acceleration by the formula that says the net force equals the mass multiplied by the acceleration. So in this case, by Newton's second law, OK, then we know that our centripetal force is going to be equal to the mass of our smallest of our planetary system. M multiplied by V squared divided by R. But when we look at this formula, it doesn't have anything to do with the mass of the center of the galaxy itself. So how can we relate? Well, we know that the gravitational force should be equal to the centripetal force and recall that the gravitational force FG is equal to the gravitational constant multiplied by the product of the masses divided by the distance between them squared. So this formula relates the mass of our center, the mass of our object and the distance between them are OK. So now since we know both of them are equal, that tells us then that our gravitational force equals our centripetal force. So GMM divided by R squared is going to be equal to MV squared divided by R. If we start to simplify, we can cancel M from both terms. OK. Uh We can cancel one of our R terms here R into R square lasers with R. So no notice that we're left with GM divided by R being equal to V squared. And now we can go ahead and solve for M if we multiply both sides by R and divide by G, that means M equals V squared R divided by G. And since we know all of those values, we can go ahead and solve, we know that the speed our masses or our planetary system is moving at is 2.5 multiplied by 10 to the fifth meters per second. So that's going to be squared, we know the distance between them is 3.311 multiplied by 10 to the 20th meters approximately. And we know that our gravitational constant G is 6.674 multiplied by 10 to the negative 11th Newton square meters per square kilogram. So now when we go ahead and calculate, OK, then we should get our mass M to two significant figures to be approximately 3.1 multiplied by 10 to the 41st kilograms. Thus, this is going to be the estimated mass of a center of our galaxy. When we go back to our answer choices that tells us then that B is the correct answer. Thanks a lot for watching everyone. I hope this video helped.

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Key Concepts

Gravitational Force

Orbital Motion

Mass Calculation Using Kepler's Third Law