 Hi, I'm Zor. Welcome to Unisor Education. We continue discussing gravitational field and in this lecture we will talk about potential and kinetic energy related to gravitational field and how it's all related to energy conservation. This lecture is part of the course called Physics for Teen, presented on Unisor.com. The course is, well, it has prerequisites and one of the prerequisites is the course called Mass for Teens on the same website. Also what's important is, presented on Unisor.com is a course. Now, if you found this lecture on YouTube by searching for whatever topic, that would be just a single lecture and I do recommend to take the whole course because obviously all lectures are interrelated and presented in certain logical sequence. Okay, now, by the way, the site is free and contains no ads, no problem with that. Okay, talking about gravitational field and its energy. Well, we have already started amount of work which gravitational field performs by attracting objects which are in that field. Let's just consider an ideal and simple case if this is the source of energy, a gravitational field and you have an object at certain location r1 from this point mass which is the source of gravitation and we would like to move it to location r2. Now, the amount of work which is needed to do this particular transformation which is performed either by the gravitational field if we are going towards the source when it attracts or by us if r2 is greater than r1 which means we are taking away from the gravitational field. It's calculated using the formula which we have derived in the previous lecture w from r1 to r2 is equal to mass of this object times gm divided by r2 minus gm divided by r1 or if you wish gm m1 over r2 minus 1 over r1. Now, let's just think about the sign of this, sign is very important. If r2 as on this picture is smaller then the difference will be positive. So, amount of work which is performed by the gravitational field is positive. Now, if r2 is greater it will be negative because it's not the gravitational field it's we who are performing the work against the gravitational field. So, we always have to take into consideration who does the job and from which perspective we are actually looking. If we are looking from our viewpoint the sign should be minus of this. If you are looking from the gravitational viewpoint it should be like it is. But anyway, we know what the sign actually is so let me just say that we will probably deal with absolute value of this in all cases to have the work positive and then we will think about which sign to apply. Now, so let's consider for a moment that we have placed our object at location r1 from the source of gravity and it's at rest which means something is holding this particular object at this particular place. But for instance if this is the source of gravity like surface of the earth and this is the direction which is actually vertical upwards then we have some kind of a support like a table or whatever it is where the object lies upon and the gravitational field force is equivalent to reaction of the support and that's why the object is at rest. Now, then we just take this support away and let the object basically do whatever it's supposed to do in this situation and what is this? Well, there is gravity which attracts this object it goes towards location r2. The field performs this work. Now, the work is supposed to be conserved so where the energy goes. Now, since the field performs some work it means that in this particular position our object has certain potential energy potential because there is some work which is spent which is done by the field to move it from here to here. Where the energy goes? Well, obviously when it reaches the point r2 its potential energy is less but it has certain speed. Why the speed? Well, because there is a force which is actually accelerates object as it moves from r1 to r2. Mind you the force is variable so we can't really just use the plane formulas of mechanics where the force usually is constant acceleration therefore it's constant and therefore the speed very easy to calculate. It's not easy to calculate because the force is variable and acceleration therefore is variable and you know the force is basically proportional to product of the mass divided by distance from the source of gravity so it's changing because r is changing. Alright, however the law of conservation of energy can be applied here and what I can say is the following. Now, potential energy at any point let's just think about this. Potential energy at any point should be measured by amount of work which is needed to place this particular object into this particular place so let's just assume that object is not there at all well or you can also say it's somewhere very very far away infinitely far away from the source not to have any influence of the gravitational field on the object and now we are bringing this object towards this particular position now we are doing so we are supposed to do some work but however it's not actually it's gravitational field which does the work so it will be a negative thing but in any case what's the amount of work which is necessary to bring the object into this particular position in absolute value. Well, again that's basically the same formula except you can put r1 equals to infinity and r2 equals to r1 so it will be basically gmm divided by r1 right so r1 is infinity r2 is equal to r1 so in this particular case this is amount of energy amount of work so to speak which is needed to bring object here so this is an amount of potential energy in this particular case so this is a difference between potential energy in this field and potential energy in this field which means this is a measure of increment of potential energy either increasing or decreasing well now obviously this same increment of potential energy should correspond to increment of the kinetic energy now kinetic energy in the beginning well since speed at point r1 is equal to zero and speed at point r2 is vr2 so but kinetic energy at the very end of this traveling is this one this is kinetic energy at point r2 so what I'm saying is that this sorry square so mv2 divided by 2 this is kinetic energy mechanical energy which object will have as it ends this trip from point r1 to point r2 and this is the kinetic energy which is supposed to be equal to the difference in potential energies so let's put it this way that the absolute value of kinetic energy at point r2 should be equal to absolute value of work from r1 to r2 so I don't want to deal with the signs so let's assume this is exactly the position which means this is positive and if this is positive then I can actually equate this with mv2 vr2 divided by 2 now obviously this thing goes out and this is assuming that the speed at this point is equal to zero so this is basically how we can calculate speed at the point at the end point of traveling within the gravitational field now again back to bringing an object from infinity to point r let me just do it this is one r only so if r1 is equal to infinity and r2 is equal to r our formula would be mvr2 divided by 2 is equal to gmm so 1 over r minus okay just divide it like this that's what it will be from which again it's independent of mass of the probe object so vr2 is equal to 2 gm divided by r and from this vr is equal to square root of 2 gm divided by r so this is the speed of the object if it was in the infinity and the gravitational force brings it into point at the distance r so let me just do another picture here so this is my object this is my force of gravity so whenever I'm bringing the object from infinity to this point well not I am bringing it's gravitational field actually which brings then the final speed would be this so it depends on the object itself which is the source of energy and it depends on the position well g is universal constant you know that so it depends on the mass of the source of the gravity and the position now what's interesting is that let's just think about the symmetry of the situation if gravitational field spends this amount of energy to bring the object the probe object from the infinity to this point now then we if we want to bring it back to infinity to fly away from the source of gravity we have to spend exactly the same amount of energy because again energy must be conserved the same energy which comes in and gravitational field spends now we have to spend the same amount of energy to restore situation back to original when this object probe object is in infinity now let's go back to our space traveling interests what is this amount of energy which I have to spend to bring object from here to infinity this is amount of energy which is necessary for a spaceship let's say to completely leave the gravitational field of some object some planet or whatever whether it's an earth or a Jupiter or whatever all right now I have exactly the same amount of energy and therefore my speed if I just push very very strongly push this particular probe object towards this direction but the push should be should be very very very strong so it goes all the way to infinity because if it will be not as strong it will go and then return back and fall right so same thing if I'm on on the surface of the earth's so my radius is the radius of the earth's right this is basically the planet and I'm on the surface of the planet if I would like to throw a stone with certain speed such that this stone will fly away from the gravitational field of earth goes to Pluto Neptune whatever I have to have this speed so this is so-called escape speed now we can very easily calculate it because we know the mass of the earth we know the gravitational constant we know the radius of the of the planet so we just calculate and in the notes for this lecture I do have detailed calculations but basically the answer is my speed should be equal to eleven point two kilometers per second this is the speed of the well spaceship initial speed at least or the speed which it could it it should actually develop to escape the gravitational field of the earth well obviously in practice situation is slightly different because in the beginning the spaceship starts from the surface of the earth with a slow speed and then it speeds up during the first whatever number of minutes and by that time it's already farther away so my are as different so the maximum speed which it needs to leave the earth would be eventually it would be less than 11 points we don't really need the speed only if they're throwing in one shot the stone from the surface of the earth we need this speed to really leave the surface to the gravitational field of the earth if however slowly we reach a higher height above the surface of the earth then obviously this will be greater so the whole speed would be less but it still will be significant I mean if it's eleven point two and we are and well earth is a big planet right so we have to really move it's the radius of the earth is something like six hundred and forty six thousand four hundred kilometers that's the radius so you can imagine to substantially increase this number we have to really go up by hundreds of kilometers otherwise it will be an insignificant change so we can go slowly a few hundred kilometers a few hundred kilometers and only then we will start feeling that this speed is not really necessary we can have eleven point one for instance kilometers per second but it's still a lot I mean you still have to have very powerful engine to get away from from the plant so all these calculations obviously are made by people who are involved in in the space traveling now by the way the similar number to escape for instance the moon's gravitational field if moon is by itself without the earth I think it's two point four kilometers per second well obviously because the mass is smaller radius is also smaller but mass is significantly smaller alright so and and for the sun for instance to leave the gravitational field of the sun it would be much much bigger than that because the sun is huge obviously alright well basically that's all I wanted to talk about today just think about the whole purpose of this lecture I was trying to connect together potential energy which every probe object in the gravitational field exists and kinetic energy so if we are moving further for instance and we have to spend some energy then the potential energy will increase because we are spending our energy and that actually contributes that adds to the potential energy at this point if we move it to further from the planet moving against the gravitational field it increases potential energy then again if the object just left by itself and gravitational field moves it back to itself attracts it then potential energy is diminishing but its kinetic energy is increasing I would also like to warn you in this formula you see all these things were derived from the some kind of an assumption that we are dealing with a point mass source of gravitational field and if it's a point mass then obviously r can be however small and then the whole thing goes to infinity which obviously is not practical kind of situation because there are no point masses we have planets we have some other objects which are which which have certain size so this is not a point mass it's really like a planet which has certain radius which means that this thing has the lowest possible value which is the real radius of the object which is a source of gravity so there are no infinity here r cannot go to zero obviously only in this ideal situation then I would probably have to tell you another thing you see if we are dealing with a spherical source of energy like planet for instance ideal planet ideal sphere then for any point which is outside of the surface of its surface the planet acts as if the whole mass is concentrated in its center of the sphere now it can be proven by integrating it's a three-dimensional integral of this of this sphere because every point attracts at certain with a certain with a certain force right this is the planet and this is the object then as if the whole mass of this planet is concentrated in its center so the force which this thing this probe object is experiencing is exactly the same from the whole planet as if it would be a point object concentrated in the center of this sphere and again to prove it we have to do the three-dimensional integral because every piece is attracting something so we have three-dimensional integral which is this something like this and having this three-dimensional integral we can calculate attraction of each individual element three-dimensional element of the volume of this planet and after integration it will be obvious that the result will be the same as if the whole mass m is concentrated in one thing and the whole distance is the distance between the masses although in this particular case when we are integrating obviously every piece has a different direction different distance etc but this is a simple exercise in calculus which I don't think I want to do I'm just telling you that that's the case another interesting case also can be proven with integration that if this sphere has an empty space inside so it's actually like two spheres one inside of another but the matter is only in between these two spheres and the center is empty then what's interesting is that the gravitation inside would be zero will be basically weightless kind of a situation but that's another very simple exercise in calculus which I'll think about whether to go through this integration or not with you but that's kind of an interesting aspect of the gravitational field alright so that's it for today thank you very much I do suggest you to read the notes for this lecture they have all these detailed calculations about the moon and the earth and that's it thanks and good luck