8. Centripetal Forces & Gravitation

Acceleration Due to Gravity

# Find mass of planet in free fall

Patrick Ford

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Hey, guys, let's check out this problem here. So even astronaut who's dropping a rock from rest at the surface of unknown planets and we know what the radius of the Unknown Planet is, but we're being asked for the mass. So what is that variable We're looking for what we're looking for Big M. The mass of the planet. And how do we find that? Well, there's two basic approaches we can take so far, With all of our equations, we can take either a force approach or an acceleration due to gravity approach. Now, with forces, I need to know multiple masses and I need to be told from information about the force. But I don't have any of that. I don't have multiple masses and I don't have any reference to forces, so I can't go that route. Instead, I could use the acceleration due to gravity. So now my choices are which one of my working with, um I'm working with surface gravity or some gravity at a height. My told some heights and no, I'm not. But I am told that I'm on the surface of an unknown planet, so we're actually gonna be using the G surface equation. So G surface equals big g Big M over little r squared. Now, I'm gonna try to look for this big M right here. So we go ahead, just re arrange for that. So I want to just move this r squared over to the top, and then this g is gonna biggie is gonna come down, right? And so I'm gonna get that G surface. Times are squared, divided by big G is going to be equal to the mass of the planet. Now I have the radius and I have the capital G. I'm looking for mass. So now I just need to figure out what the what the surface gravity is. How do I do that? What's what other parts of this problem can I use? Well, I have that. The rock takes 0.6 seconds to fall 1.5 m, So let's just start. Diagram will quickly. What's going on? So we've got a surface here, a rock, and we're told that that rock is dropped from rest and it falls 1.5 m in a time of 0.6 seconds. We've seen these problems before. These air actually just kinda Matics problems. So sometimes your kingdom attics problems can actually pop up on gravitation problems. We need to remember those equations, right? So how do we figure out what G surface is? Remember, we need three out of the five variables and cinematics, so let's list them. Right. So I've got v knots. I need the knots. I need t. I need Delta y and I need the final. So let's take a look. I'm told that the rock is dropped from rest, which means that the initial velocity is equal to zero. So I have that and I'm also told the time, So I t and I'm told with the delta y is, remember, that's just the distance. That's just this delta y right here. So we actually have three out of five, which means we can go ahead and ignore the VF. So I'm gonna pick the equation that doesn't involve that one. Which is the delta t one or sorry, Delta y. So I've got Delta y equals V naughty plus one half g t squared. So now what happens is if I can find out what this little G is, I can plug this thing back into this equation and solve for EMS. That's the That's what I'm doing here. So now I know that the V not term is gonna cancel because zero and then I'm just gonna start plugging everything in. Now, I got 1.5 over here, and this 1.5 is if I just choose this downward direction to be positive, right? We don't have to deal with the negatives and positives and stuff like that. So that's just equal to one half G. And then t is just the time 0.6 squared. So now we go ahead and just figure this out. I've got G equals two times 1.5. This, too, comes from the fact you move in the half over the other side and then divided by 0. squared, and you should get 8.33 meters per second squared. So now we have this thing. We're gonna plug this back into this equation and solve. So you've got 8.33 on that. We've got the radius, which is four times 10 to the six. They were gonna square that radius and then divided by 6.67 times 10 to the minus 11. That equals the mass of the planet. And if you do that, you should get two times 10 to the 24th kilograms. And that is our final answer. Let me know if you guys have any questions with this.

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