Gravitational Fields - A Level Physics

hello today we're continuing in the a-level physics revision series and in the next two videos we're going to be looking at the subject of fields and in particular gravitational and electric fields this video is looking at gravitational fields by fields I do not mean the things in the country that cows graze in a field is something which has a value at all points in space for example in a loom you might take a thermometer and measure the temperature at every single point in the room now it's unlikely that the temperature will be the same at every point near the window it might be a little cooler near the fire it might or the radiator it might be a little warmer but there will be a value at every point in the room and that is the temperature field and what does a temperature field do well it just sits there it doesn't do anything until you put something in the field with which that field might be able to interact for example some ice in the room will melt and the temperature field will have an impact on that so a field is essentially the capability to do something but you don't do anything until you put something in the field with which it will interact now in respect of a gravitational field it is a gravitational field that will sit there until you put a mass in that field and if you put a mass in the field the field will cause the mass to accelerate similarly when we get to an electric field if we have an electric field you'll find that that will sit there unless you put a charge in the field and if you put a charge in the field that charge will be accelerated by the field so the field is essentially a capability to do something if something with which that field interacts is placed in the field now it was Newton who discovered that all masses attract one another and the force of attraction between two masses is equal to minus I'll tell you why it's minus moment G which is called the gravitational constant times the mass of the first object times the mass of the second object divided by R squared where R is the distance between the two objects and this makes it what's called an inverse square law in other words as the distance doubles the force goes down fourfold R squared why is it - because the force operates in the opposite distance in the opposite way - the distance so for example if you're standing on earth and they you're above the earth by a distance of R then R is measured upwards but the force will act downwards and that's the reason for this - term here it's important to note that if we're talking about the earth or the Sun or any large body R is the distance from the centre of the Sun not the surface the center of the earth not the surface of the earth don't make that mistake R is always from the center the value of g is six point six seven times ten to the minus 11 Newton meter squared per kilogram squared which might sound like a mouthful but when you think about it the constant has got to balance the two sides of the equation force is measured in Newtons mass is measured in kilograms and since there are two of them that will be the kilogram squared and R is measured in meters and since its squared there will be meters squared so for this to be dimensionally consistent G has to be in these units and this force as we shall see is the force that keeps the moon going around the earth and keeps the earth going around the Sun now let's consider what happens if you're standing on the surface of the earth the earth has a radius R and not to scale here are you standing on the surface of the earth you have a mass which we're going to call little in the earth has a man which we're going to call capital in and so it will be true from this formula that the force between you and the earth will equal minus G times your mass little m times the mass of the earth Big M divided by R squared because you remember I said you have to measure it from the center of the earth and that means the distance is the radius of the earth now on the surface of the earth G is a gravitational constant that doesn't change M is the mass of the earth that doesn't change and R squared the distance from the center of the earth doesn't change if you remain on the surface so the only thing that changes is if instead of you being on the surface of the earth we're talking about me so the value of minus GM over R squared is always a constant for anything or anyone on the surface of the earth the only thing that changes is their mass and this value if you work it out comes to 9.81 m/s^2 and that we call little g it's the value of gravity on the surface of the earth or not too far above it for the first hundred meters or so this is a very good approximation to the value of the gravitational attraction so using this formula and replacing minus GM over R squared by little G you get that F is equal to M times little G you can put a minus sign but now there's no radius term so actually we can just say the force which would of course act downwards is equal to the mass of the object on the earth times G and that has a very close similarity to Newton's second law F equals MA what we're really saying is that the acceleration that you will experience is equal to the value of G of course when you're on the surface of the earth you won't go through it because the earth is strong enough to stop you doing that but if you're a few meters above the earth then you will accelerate towards the surface with the acceleration G 9.81 m/s^2 now this is the position only when you are close to the earth if you are observing the earth from outer space then the field lines of gravity the gravitational field will move in they always of course go towards the earth radially from all directions but if you are standing on the surface of the earth then a very good approximation is that gravity just acts straight down now since the force of the gravitational field is acting towards the earth if you want to move something in the opposite direction you will have to do work and in doing that work you will give the object potential energy that means energy to fall back down again and the potential energy which we describe as EP the potential energy gravitational potential energy is equal to minus G the cap at the gravitational constant times the mass of the earth or one object multiplied by the mass of the object that you're moving away from the earth divided by R and in that respect it only differs from the formula for the force force has R squared the potential energy simply has R that's the potential energy there is another thing which is slightly different called the gravitational potential and that's what I was talking about the capability of the field and this is the capability of moving a unit mass so that little m there were a unit mass then the gravitational potential which we call V is equal to EP divided by M which is minus G over our that's the capability of moving a unit mass gravitational forces act between all masses they even act between two people but they are so small because the masses are so small that you don't notice them you don't find yourself being pulled to somebody else but if at least the mass of one body is massive then there is a very strong pull and that can cause things to orbit so let's consider here the earth I let's suppose that we've got a body this could be a spacecraft could even be the moon that is orbiting the earth supposed to join up there and so here is the body and it is orbiting with a speed V of course the velocity will constantly change because it's constantly changing direction and it is at a distance R and R you remember has to be measured from the center of the earth now if the earth has a mass Capital m and the body whether it be a locket or spacecraft or the moon has a mass little m then we know that the force between those two bodies will equal minus G capital M little m over R squared that's the force that is acting between this body and this body but if that body is going in orbit around the earth then there is a centripetal force that is keeping that body in orbit and that centripetal force equals MV squared over R and that force must be been being provided by that force so those two forces are equal now before we proceed to look at the consequences of these two equations it might just be helpful if I demonstrate why the centripetal force is MV squared over R since that's something that P don't always know or understand let's take a body that is in orbit or indeed anything going round in a circle and that circle has a radius R and the body is traveling at velocity or speed V but it's velocity is constantly changing it's staying staying at V but it's just changing direction it's traveling and in a time DT and by DT it's got to bear and it travels through an angle that we will call D theta now what can we say let's call this distance around the arc of the circle DX then D theta if it is measured in radians is equal to DX divided by R because that is the definition of a Radian a Radian is the arc length divided by the radius but the distance DX that it travels round this arc is simply equal to the speed times the time distance is always speed times time speed is V time is delta T so DX is V DT now let's look to see how the velocity is changing we need to know what it's acceleration is because that although the speed isn't changing the direction is and so there is an acceleration well the velocity at this point is in that direction the velocity at this point is tangential to the radius which means it is in this direction and so we can do the usual trick with vectors we can say draw a diagram this is lengthy this is lengthy that distance there I'm going to call D s and that angle is d theta it's the same angle as that angle there you can do that just by geometry and so now we can say that D s almost is equal to V sorry I've got that wrong des divided by V equals D theta not quite what we're really doing is imagining that this is a circle of radius V and D s is not quite the arc length the arc length would be an arc so it's almost DS divided by V is D theta but not quite however if date D theta is very small then the arc length is pretty much equal to the length of the straight line and what's the significance of the length of the straight line well the length of the straight line is the change in velocity because if you want to know the difference between two velocities you put them tails to tail and then that line is the distance so D s is really DV it's the change in velocity so now we can say that DV over V is equal to D theta for small angles as long as the theta is small then DV is the straight line is virtually the same as the arc length so a brief recap because the velocity is the same we can imagine these two points as being on the surface of a circle of radius V if that were the case then the arc length between these two points divided by V would equal D theta what we actually want is the straight line distance between those two points but if D theta is very small then the arc length and the straight line are going to be broadly the same and so this is a reasonable approximation for small angles well now we can equate these data there to D theta there which means that we say that DV by V or DV divided so givi divided by V which is d theta is equal to VD tea over our which also equals D theta and this means if we bring V up here we get V squared divided by R bring the DT down here equals DV by DT but what is DV by DT velocity divided by time is acceleration and so the acceleration is equal to V squared over all and if you want the force well the force by Newton's second law is mass times acceleration and that is mass times V squared over R and that's why the centripetal force is MV squared over R okay after that little detour we can now go back and look at the significance of the gravitational force being supplied or being the cause of the centripetal force and if that's the case we can say that this force equals that force so let's write those down actually strictly of course this force should be minus as well since the force acts inwards whereas R is measured outputs so they're both minus which means you can lose the minuses when you equate them and you can say that G M M divided by R squared which is the gravitational force is equal to MV squared over R which is the centripetal force and that means that V squared equals G M little m over R squared and now you have to bring R over m the MS cancel one of the RS cancel and you get V squared is GM over R which means that V is equal to the square root of G capital M over R so the velocity of an orbiting body is equal to G the gravitational constant times mass mass of the earth leaves that's what we're thinking about times R which is the distance from the centre of the earth how long does it take that orbiting body complete one complete orbit well that time is called capital T and time will be the distance divided by the speed what is the distance of one orbit well it's the circumference of the circle 2 pi r where r is the distance from the center of the earth to the position where the orbiting body is divided by the speed which is V and that equals 2 PI R divided by V will V is square root of GM over R so if you divide by V that becomes the square root of R over GM and if you square 2 PI R you get 4 pi squared R squared that can now go inside the square root so we're squaring 2 PI R to get 4 pi squared R squared times R over GM that gives you that the time to complete the orbit is equal to the square root of 4 pi squared R cubed divided by G M that is the time to complete the orbit and it started to get sunny so I hope that it's not being too much of a problem seeing the words on the screen it is terrible to do so that I have blotted out the Sun there is something called a geosynchronous orbit if you take the earth you can have a satellite that takes precisely 24 hours to do an orbit that is to say that it spins or does one complete orbit in the same time as the Earth spins on its own axis and that means it appears to be stationary if you are on the earth and the satellite is overhead then as the Earth spins in 24 hours and the satellite spins in 24 hours the satellite will always be directly above you that's necessary if for example you want the satellite to beam you television pictures it's no use if the satellite moves out of position because then the dish aerial won't be able to pick it up so what we want to know is how far above the earth does a satellite have to be if it is going to orbit the earth in once every 24 hours well we know that by definition T must equal 24 hours and that will have to be converted into seconds we also know because we've just calculated it that the period is equal to the square root of 4 pi squared R cubed divided by G M this means that T squared which is squaring on both sides is 4pi squared r cubed over G M and now we want to rearrange that for R cubed we want to find out what R is because we want to know how far from the center of the earth remember from the center of the earth that satellite has to be but R cubed is going to be T squared G M divided by 4pi squared and that means that R is going to be the cube root now of T squared G M divided by 4pi squared now T we know has to be 24 hours but that has to be calculated in seconds so it's 24 times 60 times 60 seconds G is the gravitational constant we know that M is the mass of the earth if that's what we want to orbit and 4pi squared of course is well known and if you put all those numbers in you will find that R the distance from the center of the earth is 4 point 2 3 times 10 to the 4 kilometers but that's from the end of the earth how far above the earth does it have to be well the radius of the earth is very approximately 6 point 4 times 10 to the 3 kilometers about 6,400 kilometers so if you want to know how far above the surface of the earth then you have to take that distance - the radius of the earth and if you do that then you find that the height above the surface of the earth but the satellite has to be is thirty-five thousand nine hundred kilometres which is very roughly 22,000 miles so a geosynchronous satellite at his one that will stay above your head for the whole 24 hours has to orbit at about 22,000 miles above the earth man called Kepler lived just before Newton and he made three discoveries about the way planets orbit around the Sun most of us think planets orbit in a circle in fact Kepler found that they move in an ellipse an ellipse is a kind of squashed circle although in fact the circle of the planets around the Sun is almost a circle and a circle very often will nicely describe it and you get pretty accurate results but strictly it's just a slightly squashed circle and ellipse Kepler also found that the planet if this is the Sun and this is a planet say the earth the planet sweeps out equal areas in equal time so if the planet goes from there to there in a time let's say T seconds and then from there to there in a further time T seconds so it takes two seconds from better there and T seconds from there there and what Kepler found was that that area is equal to that area the planet sweeps out equal areas in equal time and the third thing that Kepler noticed was that if you take the period of the planet as T then T squared was proportional to the distance of that planet from the Sun cubed the time taken to do a complete revolution one orbit squared was proportional to the disk from the Sun cubed and we now know precisely why that is because that's what we worked up at worked out up here that an orbiting spacecraft rocket satellite whatever the x squared is proportional to the radius cubed because for any given object like the earth for pi squared over G M are all constants so when Kepler deduced that he was on the way to identifying what Newton subsequently formulated we can now think about escape velocities let's suppose we want to get away from the earth away from the Earth's gravitational field which is pushing us downwards and we want to escape so that we're no longer attracted by the gravitational field how fast do we have to go well let's call that escape velocity V and that means we will have an energy a kinetic energy of 1/2 MV squared now that has got to be at least as great as the potential energy in the gravitational field that would otherwise cause us to come back and the potential energy is minus G M where m is the mass of the Earth times M which is the mass of the rocket or whatever it is you're trying to get out divided by R where R is the distance from the surface of the earth because that's where we're starting and the center of the earth those two things have to be at least equal and that means that the velocity is going to be or the velocity squared is equal to minus 2 G M M divided by M R and the masses cancel and now you get that the velocity squared is we can forget about the minus sign because we know which direction it's going it's got to go upwards so the velocity now is taking the masses out to GM over R and if you work that out substituting the gravitational constant the mass of the earth and the radius of the earth you find that for the earth the escape velocity is approximately 11 kilometers per second which is very roughly 7 miles per second now you might observe that when a rocket is launched from one of the main rocket launch places it doesn't seem to go upwards at a rate of seven miles a second and the reason for that is it doesn't have to because it's got an engine and as long as you've got an engine providing a constant force then you can go up much slower because that engine will continue to give you thrust to overcome whoops to overcome the potential energy that would tend to pull you back down again what this velocity says is that if you could give an object a velocity of 11 kilometers per second then without any further engines it would go into outer space and out of the Earth's gravitational pull so if you could get say a cricket ball or a tennis ball and whack it with a bat or a racquet and if you could give it a speed of 11 kilometers per second 7 miles per second then you could get it into outer space now 7 miles per second in terms of miles per hour you have to multiply 7 by 3600 which I think comes to something like 25,000 miles an hour so you would have to hit your tennis ball at 25,000 miles an hour in order for it to go into outer space and escape the Earth's gravitational field and since the best tennis players in the world serve an average of 160 miles an hour they've got some way to go before those tennis balls go out into orbit

Channel: DrPhysicsA

Views: 182,433

Rating: 4.9381642 out of 5

Keywords: fields, energy, gravitational, potential, velocity, periodorbit, geosynchronous, satellites, Kepler, escape

Id: zdQ54siEfvc

Length: 28min 16sec (1696 seconds)

Published: Mon Mar 19 2012