The asteroid belt circles the sun between the orbits of Mars and Jupiter. One asteroid has a period of 5.0 earth years. What are the asteroid's orbital radius and speed?

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Welcome back, everyone. We are making observations about an asteroid. Now, we are told that this asteroid is sharing an orbit with some planet X and it revolves around the sun. Now, the asteroid has an orbital period of four years or 1.26 times 10 to the eight seconds. And we are told that the gravitational constant in this case is 6.67 times 10 to the negative 11th newtons meters squared over kilograms squared. And we are tasked with finding two things here. One, what is the orbital radius and two, what is the velocity of our asteroid? Now, before getting started here, I do wish to acknowledge our multiple choice answers on the left hand side of our screen. Those are gonna be the values in which we strive for. So without further ado let us begin starting on part one here. Let me go ahead and change colors real quick. We can use the following equation here. This actually comes from Newton's second law applied to the centripetal direction we have that the gravitational force. So gravitational constant times the mass of the sun times the mass of our asteroid divided by the orbital radius squared is equal to mass times our centripetal acceleration, which is just V squared over R. Now, we don't know our velocity just yet. But what we can do is we know that velocity is equal to two pi R over T. So let's go ahead and sub that value to our equation. What we get is G big M little M over R squared is equal to M over R PS V squared. So that's going to give us four I squared R squared over T squared. As you can see, the R is going to cancel out with the exponent up top. But what we really want is we want to isolate our orbital radius. So what I'm gonna do is I'm going to multiply both sides of my equation by the following term P squared R squared divided by four I squared times the mass of our asteroid. All right, let me go ahead and copy that on the left hand side here. So T squared R squared over four pi squared M and you're gonna see a lot of things canceled out here on the right hand side, E squares cancel out. And we have, let's see what else the four pi squared cancel out as well as the mass of the asteroid cancel out. On the left hand side, the mass of the asteroids cancel out and the radii squared cancel out. What this gives us is that our cubed from the right hand side of our equation is equal to G big MT squared over four pi squared, taking the cube root of both sides allows us to eliminate the exponent on the left hand side of our equation. So we have found that R is equal to the cube root of this entire term. Let's go ahead and plug in all of our values here. In fact, I'm actually going to move this over just a little bit. All right, let's go ahead and calculate our orbital radius here. We have the cube root of G which is 6.67 times 10 to the negative 11th times the mass of our sun, which is 1.99 times 10 to the 30th times our orbital period, which is 1.26 times 10 to the eighth. And we are going to square that value all divided by or pi squared. And what we get from this is that our radius is equal to 3.77 times 10 to the 11th meters. Right. Let me go ahead and change colors here again and move on to part two. All right. So from part two, let me go ahead and select a wider brush here. From part two, we need to find or for part two, we need to find our velocity and we have this equation right here. So our velocity is equal to two pi times our oral radius over our period. We have all of these values. So let's go ahead and plug that in. This is going to be two pi times 3.77 times 10 to the 11th, divided by 1.26 times 10 to the eighth. What this gives us is 18.76 times 10 to the third meters per second. But we want it in kilometers per second. So we know that there are 1000 m in one kilometer meter are going to cancel out on top and bottom. And what we get is a final velocity of 18.76 kilometers per second. So now what we have found is both our orbital radius as well as our orbital velocity corresponding to our final answer. Choice of B Thank you all so much for watching. I hope this video helped. We will see you all in the next one.

The asteroid belt circles the sun between the orbits of Mars and Jupiter. One asteroid has a period of 5.0 earth years. What are the asteroid's orbital radius and speed?

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Key Concepts

Kepler's Third Law

Gravitational Force

Orbital Speed