Introduction to Newton's law of gravitation

we're now going to learn a little bit about gravity and just so you know gravity is something that especially in introductory physics or even reasonably advanced physics we can learn how to calculate it we can learn how to realize what are the important variables in it but it's something that's that's really not well understood even once you learn general relativity if you if you do get there I have to say you know you can kind of say oh well it's you know the warping of space-time and all this but it's hard to get an intuition of why two objects just by a you know just because they have this thing called mass they are attracted to each other it's it's really at least to me a little bit mystical but with that said let's let's learn to detect deal with gravity and we'll do that learning Newton's law of gravity gravity and this works for most purposes so Newton's law of gravity says that the force between two masses and that's the gravitational force is equal to the gravitational constant G times the mass of the first object times the mass of the second object divided by the distance between the two objects squared so that's simple enough so let's see let's play around with this and and see if we can get some results that look reasonably familiar to us so let's use this formula to figure out what the acceleration at the surface the gravitational acceleration is at the surface of the earth so let's draw the earth just so we we know what we're talking about so that's my earth and let's say we want to figure out the gravitational acceleration on Sal that's me that's Sal and so how do we how do we apply this equation to figure out how how much I am accelerating down towards these the center of Earth or the Earth's center of mass let's say the force is equal to so what's this big G thing the G is the gravitational universal gravitational constant although I'm not as far as I know and I'm not an expert on this I actually think it it can its measurement can change it's not truly truly a constant or I guess when you you know on different scales it can be a little bit different but for our purposes it is a constant and the constant in most physical classes is this six point six seven times 10 to the negative 11th meters cubed per kilogram second squared I know these units are crazy but all you have to realize is these are just the units needed that when you multiply it times a mass and a mass divided by a distance squared you get Newtons or kilogram meters per second squared so we won't worry so much about the unit's right now just you know realize that you have to work with meters and kilograms and seconds so let's just write that number down so let's say I'll change colors to keep it interesting six point six seven times ten to the negative 11 and we want to know the acceleration on Sal so it would be so M 1 is the mass of Sal and I don't feel like revealing my mass in this video so I'll just leave it as a variable and then what's the mass - it's the mass of Earth and I wrote that here I looked it up on Wikipedia this is the mass of Earth so I multiplied times the mass of Earth times 5.9 seven times 10 to the 24 kilograms weighs a little bit not weighs is a little bit more massive than Sal divided by the distance squared now you might say well what's the distance between someone standing on the earth and the earth well it's zero because they're touching the Earth but it's important to realize that the distance between the two objects especially when we're talking about the universal law of gravitation is the distance between their center of masses for all general purposes my center of mass you know maybe it's like three feet above the ground because I'm not that tall it's probably a little bit lower than that actually anyway my center of mass might be three feet above the ground and whereas Earth's center of mass what's at the center of Earth so we have to know the radius of Earth right so the radius of Earth is I've also looked it up on Wikipedia 6,371 kilometers so how many meters is that so it's six million meters right and then you know the extra meter to get to my center of mass we could ignore from now because it would be zero zero one so we'll ignore that for now so it's six and I'll write it in scientific notation since everything else is in scientific notation six point three seven one times ten to the sixth meters right 6,000 kilometers is sick million meters so let's write that down so the distance is going to be six point three seven times ten to the sixth meters and we had a square that remember it's a distance squared so let's let's see if we can simplify this a little bit let's uh let's well just multiply those top numbers first so force is equal to let's bring the variable out mass of Sal times let's do this top part so we have six point six seven times five point nine seven times five point nine seven is equal to thirty nine point eight to thirty nine point eight two I just multiply this times this and after multiply the tens so ten to the negative 11 times 10 to the negative twenty-fourth we can just add the exponents to the same base so what's 24 minus eleven is ten to the thirteenth right and then what is the denominator look like it's going to be the six point three seven squared times ten to the six squared well 10 does it so it's going to be whatever this is going to be like 37 or something times what's ten to the sixth squared it's 10 to the twelfth right 10 to the twelfth so let's figure out what six point three seven squared is this little calculator I have doesn't have squared so I've to six point three seven so it's forty point five eight forty point five eight and so simplifying it that's the force is equal to the mass of Sal times let's divide thirty nine thirty nine point eight two divided by forty point five eight is equal to nine point eight one nine point eight one that's just this divided by this and then ten to the 30 / 10 to the 12 is what I'll actually know this is a 9.81 sorry it's 0.98 one 0.98 1 and then 10 to the 13 divided by 10 to the 12 is just 10 right 10 to the first times 10 so it's point 9 8 1 times 10 well so the force is equal to 9 point 8 1 times the mass of Sal and where does this guy how can we figure out the acceleration right now well acceleration force is just mass times acceleration right so that's also going to just be equal to the acceleration of gravity that's that's that's suppose to be a small G there times the mass of Sal right so we know the gravitational force is 9.81 times the mass of Sal and we also know that that's the same thing as the acceleration of gravity times the mass of Sal we can divide both sides by the mass of Sal we have the acceleration of gravity and if we use the unit's the whole way you would have seen that it is kilograms meters per second squared and we have just shown that at least based on the numbers of that they've given in Wikipedia the the the acceleration of gravity on the surface of the earth is almost exactly what we've been using and all the projectile motion problems it's 9.8 meters per second squared that's exciting let's do another quick problem with gravity because I got two minutes let's say let's say I have there's another planet to call the planet small earth and let's say you know the radius of small earth radius of small earth is equal to 1/2 the radius of Earth and the mass of small earth is equal to 1/2 the mass of Earth so what's what's the gravity what's what's the pull of gravity on any object say the same object on this how much smaller would it be on this planet well actually let me save that to the next video because I hate being rushed so I'll see you soon