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Current time:0:00Total duration:10:13

Acceleration due to gravity at the space station

Video transcript

most physics books will tell you that the acceleration due to gravity near the surface of the earth is 9.81 m/s^2 and this is an approximation and what I want to do in this video is figure out if this is the value we get when we actually use Newton's law of universal gravitation and that tells us that the force of gravity between two objects and let's just talk about the magnitude of the force of gravity between two objects is equal to the gravitational constant the universal gravitational constant times the mass of one of the bodies m1 times the mass of the second body divided by the distance between the center of masses of the body squared so the distance between the center of the masses of the body squared so let's use this the universal law of gravitation to figure out what the acceleration due to gravity should be at the surface of the earth and I have G right over here I have the mass of the earth which I've looked up over here and we also have the radius of the earth and for the sake of this we're going to assume that the distance between the body if we're at the center of if we're at the surface of the earth the distance between that and the center of the earth is just going to be is just going to be the radius of the earth and so this will give us the magnitude of the force if we want to figure out the magnitude of the acceleration which this really is I actually didn't write this as a vector so this is just the magnitude of the acceleration if you wanted the acceleration which is a vector you would have to say downwards or towards the center of towards the center of the earth in this case but if you want the acceleration we just have to remember we just have to remember that force force is equal to mass times acceleration and if you wanted to solve for acceleration you just divide both sides times mass so force divided by mass is equal to is equal to acceleration or if you want the magnitude if you take the magnitude of your force and you divide by mass you're going to get the magnitude of your acceleration this is a scalar quantity this is scalar quantity right over here so if you want the acceleration due to gravity you divide let's say let's let's write this in terms of the force of gravity on earth so the magnitude of the force of gravity on earth this one right over here so this will be in the case of earth are destroyed earth really really small so one of these masses is going to be earth it's going to be this mass right over here and so if you wanted the acceleration of at this acceleration due to gravity at the surface of the earth you would just have to divide you just have to divide by the mass that is being accelerated due to that force and in this case it is the other mass it is the mass that's sitting on the surface so let's divide both sides by that mass let's divide both sides by that mass and this will give us the magnitude of the magnitude of the acceleration on that mass due to gravity so this is equal to the magnitude of acceleration the magnitude of the acceleration due to gravity and the whole reason why this is actually a simplifying thing is that these two this m2 right over here and this m2 cancels out and so the magnitude of our acceleration the magnitude of our acceleration due to gravity using Newton's universal law of gravitation is just going to be this expression right over here it's going to be it's going to be the gravitational constant times the mass of the earth divided by the distance between the object the object's center of mass and the center of the mass of the earth and we're going to sue that the object is right at the surface then its center of mass is right at the surface so this is actually going to be the radius of the earth squared so divided by radius squared sometimes this is also viewed as the gravitational field at the center at the surface of the earth if because if you multiply it by a mass it tells you how much force is pulling on that mass but with that out of the way let's actually use a calculator to calculate what this value is and then what I want to do is figure out well one I want to compare it to the value that the textbooks give us and see maybe why it may or may not be different and then think about how it changes as we get further and further away from the surface of the earth and per tick if we get to an altitude that the Space Shuttle or the International Space Station might be at and this is an altitude of 400 kilometers is where it tends to hang out give or take a little bit depending on what it is up to so first let's just figure out what this value is when we use the universal law of gravitation so let's get my calculator out let's get the calculator out so we know what G is it is six point six seven three eight times 10 to the negative 11 this EE button means literally times 10 to the negative 11 so this is six point six seven three eight times 10 to the negative 11 and then I want to multiply that times the mass of Earth times the mass of Earth which is right over here it is five point nine seven two two times ten to the 24th so times 10 to the 24th power and we want to divide that by the radius of Earth squared so divided by the radius of Earth is so this is in kilometers and I just want to make sure that everything is in the same meet in the same units so 6,371 kilometers actually let me scroll over well you can't see the kilometers right over this is in kilometers is the same thing as six million three hundred seventy one thousand meters if you just multiply this by a thousand or you could even write this as six point three seven seven one six point three seven one times 10 to the sixth times 10 to the sixth meters and we're going to square this that's the radius of the earth the distance between the center of mass of Earth and the center of mass of this object which is sitting at the surface of the earth and so let's get our drumroll and we get nine point eight and if we round we actually get something a little bit higher than what the textbooks give us we get nine point eight two let's just round so we get nine point eight two nine point eight two meters per second squared and so you might say well what's going on here why do we have this discrepancy between what the universal law of gravitation gives us and with the average measured acceleration due to due to the force of gravity at the surface of the earth and the discrepancy here the discrepancy between these two numbers is really because earth is not a uniform sphere of uniform density earth is and that's what we had to assume right over here when we use the universal law of gravitation it's actually a little bit flatter then then you would then you would then a perfect sphere and it definitely does not have uniform density the different layers or earth have different densities you have all sorts of different interactions and then you also if you measure effective gravity there's also a little bit of a buoyancy effect from from the air very very very very negligible I don't know if it would have been enough to to change this but there's other minor minor effects irregularities earth is not a perfect sphere it is not of uniform density and that's what accounts for the bulk of this now that out of the way what I'm curious about is what is the acceleration due to gravity if we go up 400 kilometers so now the main difference here G will stay the same the mass of Earth will stay the same but the radius is now going to be different because now we're placing the center of mass of our object whether it's a Space Station or someone sitting in the space station they're going to be 400 kilometers higher and I'm going to exaggerate what 400 kilometers looks like this is not drawn to scale but now the radius is going to be the radius of the earth plus 400 kilometers so now for this for the case of the space station R is going to be not 6,371 kilometers it's going to be 6,000 we're going to add 400 to this 6771 kilometers which is the same thing as 6 million seven hundred seventy one thousand meters which is the same thing as six point seven seven one times 10 to the sixth meters this is one two three four five six ten to the sixth meters so let's go back to our calculator and we should just be able so second entry that's the last entry we had instead of six point three seven one times 10 to the sixth let's add 400 kilometers to that so then we get six point sorry six point seven so we're adding 400 kilometers so it was three seven one now it's seven seven one times ten the sixth and what do we get we get eight point six nine meters per second squared so now the acceleration here acceleration here is a eight point six nine meters per second squared and you can verify that the unit's work out because over here gravity is in meters cubed per kilogram second squared you multiply that times the mass of Earth which is in kilograms the kilograms cancel out with these kilograms and then you're dividing by meters squared so you divide this by meters squared you get you're left with meters per second squared so the units work out as well so there's an important thing to realize and this is a misconception we do a whole video on it earlier when we talk about the universal law of gravitation is that there is gravity when you are in orbit up here the only reason why it feels like there's not gravity or it looks like there's not gravity is that this space station is moving so fast that it's essentially in free fall but it's moving so fast that it keeps missing the earth it keeps missing the earth and in the next video we'll figure out how fast does it have to travel in order for it to stay in orbit in order for it to not plummet to earth due to this due to the force of gravity due to the acceleration that is occurring this centripetal this Center seeking acceleration