so how can we calculate the speed of a satellite that is moving in a circular orbit at a height of thirty eight hundred kilometers above the surface of the earth so let's draw a picture so we're gonna say this is the earth and here is a satellite that orbits the earth maybe I could draw a better picture what can we do to find the speed of that satellite well first we need to derive an equation this satellite has a centripetal force any object that moves with circular motion has a centripetal force and it turns out that gravity provides the centripetal force in this example this force is the gravitational force that keeps it moving in a circle so we're gonna set the force of gravity equal to the centripetal force and let's say that the satellite has a mass lowercase M and the earth has a mass Capital m so to calculate the gravitational force between two objects its G times the mass of the Earth times the mass of the satellite divided by the squared distance between the Centers of the earth and the satellite the centripetal force acting on the satellite because the satellite is moving in a circle it's gonna be the mass times the square of the speed divided by the radius so what we can do is cancel em and if we multiply both sides by R we can cancel an R so we're left with GM over R is equal to V squared so now we need to take the square root of both sides so the speed of a satellite is the square root of the universal gravitation constant times the mass of the Earth divided by the distance between the Centers now let's think about what we already know we know the gravitational constant is a fixed number we have the mass of the earth now what is our in its problem what should we replace art with R is not the radius of the earth and it's not the height above the surface of the earth it's actually the sum of those two values so let me draw a better picture to illustrate it so let's say this is the earth and this is the satellite so first we have the radius of the earth which is six point three eight times 10 to the 6 meters and we have the height of the satellite above the surface of the earth which is thirty-eight hundred kilometers to add these two we need to make sure that the unit matches so one kilometer is a thousand meters so 3800 kilometers that's 3.8 million meters you got to multiply this by a thousand so that's equivalent to three point eight times 10 to the 6 meters so R is the sum of these two numbers so if we add six point three eight times ten to the six plus three point eight times ten to the six that's equal to one point zero one eight times ten to the seven meters so R is the distance between the Sun of the earth and the center of the satellite now we can use the formula so we have G the gravitational constant six point six seven times ten to the minus eleven multiplied by the mass of the earth which is five point nine seven times ten to the 24 divided by R which is one point zero one eight times ten to the seven meters so the speed of the satellite let me erase a few things the speed of the satellite is six thousand two hundred fifty four point three meters per second so that's how fast it's going now that we have this speed what is the period of the satellite in hours so what do you think we need to do in order to calculate the period so let's say this is the earth and this is the satellite the satellite travels around the earth and it's moving at constant speed whenever you have an object moving at constant speed you can use this equation D is equal to V T so V is d over T the distance around a circle is the circumference which is 2 pi R and the time it takes to complete one revolution or one cycle around the circle is the period now we need to solve for T so I'm going to multiply both sides by T so VT is equal to 2 PI R and dividing both sides by V the period is 2 PI R over V and we have the R value the R value is this number so the period is going to be the circumference 2 pi times 1 point 0 1 8 times 10 to the 7 divided by the speed of the satellite which is 62 54.3 meters per second so you should get 10,000 and 227 seconds now let's convert that into hours so the 60 seconds in a minute and there's 60 minutes in one hour so we could cancel the unit seconds and minutes so it's ten thousand two hundred twenty seven divided by sixty and divided by sixty again so the period is about two point eight four hours so that's how long it's going to take for this satellite to orbit the Earth at this speed and at this orbital radius now let's move on to our second question a geosynchronous satellite is one that stays above the same point on the equator of the earth what is the period of such a satellite in seconds so let's say if this satellite stays at the same point so it's gonna rotate at the same rate as the Earth rotates it's gonna have the same angular speed now it takes the earth 24 hours to rotate on its internal axis so the satellite is gonna take 24 hours for the satellite to revolve around the earth make sure you understand the difference between rotation and revolution when an object rotates it rotates about its internal axis so like the earth it has its own internal access that it rotates around now the satellite revolves around an external axis which is basically the axis of the earth but that's outside of the satellite so it's external so the satellite traveling around the earth its revolving the earth spinning on its own axis it's rotating so what is the period of that satellite so as we mentioned it takes 24 hours for the earth to rotate on its own axis so for geosynchronous satellite it's going to be 24 hours for it to revolve around the earth now we need to convert that into seconds so one hour is equal to 60 minutes and one minute is equivalent to 60 seconds so it's 24 times 60 times another 60 so it's 86,400 seconds so that's the period now let's move on to Part B what is the satellites height above the surface of the earth in kilometers so how can we use the period to calculate the height of the satellite so let's draw a picture so let's say this is the earth and this is the satellite so let's say that lowercase R is the radius of the earth and H is the height of the satellite capital R is the distance between the centre of the earth and the set of the satellite our goal is to calculate H the height above the surface of the earth so let's write this r is equal to h plus lowercase R now somehow we need an equation that relates the period with R so let's see if we can come up with such an equation now recall that we said the speed of the satellite it's a square root of GM over R and we're gonna use that equation for Part C but we don't know what R is so we got to find that first now V squared it's gonna be GM over R and without the square root and we call on the last problem that V is 2 PI R over T so I'm gonna replace V with 2 PI over T so I'm gonna have 2 PI R over the period and that's squared and that's equal to GM over R so 2 squared is 4 so we're gonna have 4 PI squared R squared over T squared is equal to GM over R and now I'm running out of space so let's get rid of some stuff so what I'm gonna do now is I'm gonna cross multiply so I'm gonna have 4pi squared times R cubed that's R squared times R and on the right side GM times T squared so I can leave the equation like this or I could solve for R so I'm gonna do that so RQ who's going to be GM T squared over 4pi squared and then taking the cube root of both sides r is going to be the cube root of GM t squared over 4pi squared so this equation allows me to calculate the radius of the orbit if i know the period so this is gonna be the cube root of g which is six point six seven times ten to the minus eleven times the mass of the earth which you can find a base on the last problem that's five point nine seven times ten to the twenty four and then multiplied by the period which is 86,400 and don't forget to square it and all of this will be divided by four five squared you may want to put this in parentheses taking the cube root is the same as raising it to the 1/3 power so you should have an R value of 4.2 2 times 10 to the 7 meters but we're not quite finished yet we need to find the height so the height is going to be the radius of the orbit minus the radius of the earth so that's going to be four point two two times ten to the seven and then minus the radius of the earth which is in the last problem six point three eight times ten to the six meters so this is equal to three point five eight times ten to the seven meters now let's convert that to kilometers so there's a thousand meters per kilometer so we got to divide that by a thousand and so it's thirty-five thousand eight hundred kilometers so that's how high this satellite is above the surface of the earth now the last thing that we need to do is we need to calculate the speed of the satellite so let's go back to this equation so we have G we have the mass of the earth five point nine seven times ten to the twenty four and the orbital radius is four point 22 times 10 to the seven that's the distance between the center of the earth and the center of the satellite so the speed of the satellite is 3072 meters per second and so that's it for this video so now you know how to calculate the speed of the satellite the satellites height above the surface of the earth and also the period of the satellite if you need to thanks for watching