Nothing can escape the event horizon of a black hole, not even light. You can think of the event horizon as being the distance from a black hole at which the escape speed is the speed of light, 3.00 ✕ 10⁸ m/s, making all escape impossible. What is the radius of the event horizon for a black hole with a mass 5.0 times the mass of the sun? This distance is called the Schwarzschild radius.

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Accepted Answer

Welcome back everybody. We are looking at a black hole which is actually at the center of the Milky Way galaxy. And we are also looking at a star that is orbiting around this black hole. Now we are given information about the star. We are told that it is. Its orbital speed Is 2.1 times 10 - eight m/s. And we are told that the radius of its orbit is 1.12 times 10 to the eighth meters. And we are asked to find what the radius of its event horizon is. So we're actually giving a formula for the radius of an event horizon. This is given by two times Newton's gravitational constant times the mass of the black hole, all divided by the speed of light. Where now here's the thing, what is the mass of the black hole? Well, the mass of any unseen object and recalculated from kepler's third law given by this formula that the mass is equal to four, I squared times the radius of orbit cubed, divided by Newton's gravitational constant times the time in which it takes to make a full revolution. But we don't know this time. So what is that time? Well, we have another formula that we can use that. The time of revolution is just two times pi and the radius of revolution all over the orbital speed. We have these two values. So let's calculate T and then just work backwards. So the time it takes for the start to make one full revolution is two I times its radius of orbit of 1.12 times 10 to the eighth meters. All divided by its orbital speed of 2.1 times to the eighth meters per second. Great. So now we can calculate the mass of our black hole given by kepler's third law of four pi squared times the radius of orbit. 1.12 times 10 to 8. Cute. Divided by 6.67 times 10 to the negative 11th, which is just Newton's gravitational constant times our time. Oh, I'm so sorry. When you plug your time into your calculator, you get that. It is 3.34 seconds for one full revolution. So that's what we plug in here. 3.34 squared. And when you plug all this into your calculator, you get that. The mass of the black hole is 7.45 times 10 to the 34 kg. That's a big black hole. Alright, so now we can calculate the radius of the event horizon. Let's do that here. Radius of the event horizon is two times Newton's gravitational constant of 6.67 times 10 to the negative 11 Times the mass of the black hole. 7.45 Times 10 to the 34. All over. These constant for the speed of light. Three times 10 to the 8th squared Giving us our final answer of 1.1 times 10 to the eight m corresponding to answer choice C Thank you all so much for watching hope. This video helped. We will see you all in the next one.

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Key Concepts

Event Horizon

Schwarzschild Radius

Escape Velocity