WEBVTT
00:00:00.880 --> 00:00:05.070
In this video, our topic is mechanical energy conservation in orbits.
00:00:05.520 --> 00:00:14.910
A foreign object that’s in orbit around some other body, we’re going to learn how to calculate its mechanical energy and also see how that energy changes over time.
00:00:15.560 --> 00:00:25.010
The first thing we can consider here is that an object’s mechanical energy, we’ll refer to it as capital ME, is equal to the sum of kinetic and potential energy.
00:00:25.530 --> 00:00:28.560
We know that kinetic energy is energy due to motion.
00:00:28.880 --> 00:00:34.730
And while there are many different kinds of potential energy, in this case gravity will always be its source.
00:00:35.190 --> 00:00:42.430
That is, for an object in orbit, its mechanical energy is equal to the sum of kinetic and gravitational potential energy.
00:00:42.910 --> 00:00:51.310
If we consider what energy due to motion is, we can recall that mathematically, it’s equal to one-half an object’s mass times its speed squared.
00:00:51.770 --> 00:01:01.610
So we can imagine that our orbiting body has some mass, we’ll call it 𝑚, as well as some speed 𝑣 so that there is kinetic energy for this body in motion.
00:01:02.100 --> 00:01:09.580
And what’s more, we can see that this body in orbit is specifically moving in a circle around the center of the mass that it orbits.
00:01:09.970 --> 00:01:21.700
Whenever a mass is moving in a circular path, we know it’s subject to a force called the centripetal force and that this force is always directed towards the center of the circle that the mass moves around.
00:01:22.120 --> 00:01:27.420
Now, a center-seeking or centripetal force always has some physical mechanism causing it.
00:01:27.770 --> 00:01:34.630
In this case, that force is the force of gravity, the attractional force between the mass being orbited and the mass in orbit.
00:01:35.070 --> 00:01:47.470
Mathematically, then, we can write that the centripetal force, 𝐹 sub 𝑐, this center-seeking force acting on our mass in orbit, is equal to the gravitational force acting on that mass, we’ll call it 𝐹 sub 𝑔.
00:01:47.900 --> 00:01:53.290
In other words, the physical cause of a center-seeking force on this mass is the force of gravity.
00:01:53.840 --> 00:02:06.210
We can recall that a foreign object in circular motion around an arc that has a radius we’ll call lowercase 𝑟, the centripetal force on that object equals its mass times its speed squared divided by 𝑟.
00:02:06.800 --> 00:02:10.700
Looking back to our sketch, we can say that this distance here is 𝑟.
00:02:11.200 --> 00:02:17.710
Once we know 𝑟 and 𝑚 and 𝑣, then we’re able to calculate the center-seeking force acting on our mass.
00:02:18.140 --> 00:02:22.370
As we’ve said, though, this is equal to the gravitational force acting on the mass.
00:02:22.740 --> 00:02:37.850
And that, written as an equation, is equal to the universal gravitational constant multiplied by the larger mass being orbited, we’ll call that capital 𝑀, and the smaller mass, lowercase 𝑚, all divided by the distance between the centers of mass squared.
00:02:38.310 --> 00:02:45.360
And in the case of our circular orbit, as we’ve seen, we can think of this distance as the radius of the circle that the smaller mass moves in.
00:02:45.950 --> 00:02:51.030
So the centripetal force acting on our object is equal to the gravitational force on it.
00:02:51.410 --> 00:02:54.670
And looking at this equation, we see that a couple of terms cancel.
00:02:55.120 --> 00:02:59.620
First, and interestingly, all this is independent of the smaller mass 𝑚.
00:03:00.060 --> 00:03:02.530
That mass disappears completely from this equation.
00:03:02.900 --> 00:03:07.480
We also see that a factor of 𝑟 cancels out in the denominator of these two fractions.
00:03:07.890 --> 00:03:23.670
And so we find this: the speed of our smaller mass squared is equal to the universal gravitational constant times the mass of the object being orbited, this is the larger mass, all divided by the distance between the centers of mass of that larger and our smaller mass.
00:03:24.150 --> 00:03:31.880
Now that we have this expression for 𝑣 squared, let’s go back to our equation for kinetic energy, which we see involves a 𝑣-squared factor.
00:03:32.280 --> 00:03:50.530
This means that we can take this expression for 𝑣 squared and substitute it in for 𝑣 squared here in our equation for kinetic energy and see that when we do this, we arrive at an equation for the kinetic energy of a mass, lowercase 𝑚, that’s in circular orbit around a larger mass, capital 𝑀.
00:03:50.980 --> 00:03:55.780
It’s important to keep in mind this limit, that this equation applies only to circular orbits.
00:03:56.070 --> 00:04:02.730
Remember that we derived it using an assumption that our object was moving in a circle and, therefore, experienced a centripetal force.
00:04:03.140 --> 00:04:06.850
For these types of orbits, though, we now have an expression for kinetic energy.
00:04:07.130 --> 00:04:11.610
And now, let’s consider the potential energy, specifically gravitational potential energy.
00:04:12.070 --> 00:04:24.400
We can refer to this energy using GPE and recall that it’s equal to negative the universal gravitational constant times the product of the two masses involved divided by the distance between their centers.
00:04:24.910 --> 00:04:41.110
So then if we want to write an expression for the mechanical energy, that is, kinetic plus potential, of a mass in circular orbit, then that’s equal to kinetic energy plus gravitational potential energy or this term here minus this term here.
00:04:41.590 --> 00:04:53.260
And we can see there’s a lot in common between these two terms, capital 𝐺, capital 𝑀, the larger mass that’s being orbited, lowercase 𝑚, the smaller mass that’s doing the orbiting, and one over 𝑟.
00:04:53.750 --> 00:05:03.470
We can factor all of these values out from both terms, which gives us 𝐺 times big 𝑀 times little 𝑚 divided by 𝑟 all multiplied by one-half minus one.
00:05:03.830 --> 00:05:06.590
But then one-half minus one is negative one-half.
00:05:06.970 --> 00:05:12.540
And now, we have a simplified expression for the overall mechanical energy of an object in circular orbit.
00:05:13.140 --> 00:05:16.830
And notice that this value doesn’t depend on the speed of the object.
00:05:16.980 --> 00:05:18.820
That doesn’t appear anywhere in the equation.
00:05:19.270 --> 00:05:26.670
The reason for that, as we saw a moment ago, is that the object’s speed squared can be expressed in terms of other parameters.
00:05:27.050 --> 00:05:35.500
We can say then that in the special case of a circular orbit, there’s a particular relation between an object’s kinetic and its gravitational potential energy.
00:05:36.000 --> 00:05:38.390
We can see that by dividing one by the other.
00:05:38.920 --> 00:05:46.040
When we do this, many of the factors cancel, big 𝐺, big 𝑀, little 𝑚, and 𝑟, the radius.
00:05:46.420 --> 00:05:48.910
And we end up simply with negative one-half.
00:05:49.440 --> 00:05:56.030
This relationship here is the specific one that KE and GPE will always share for a circular orbit.
00:05:56.490 --> 00:06:00.600
At this point, though, we might recall that circular orbits aren’t the only ones there are.
00:06:00.910 --> 00:06:05.680
It’s also possible for a mass to orbit another in the shape of what’s called an ellipse.
00:06:06.100 --> 00:06:08.790
This is like a circle that’s been smushed in one direction.
00:06:09.220 --> 00:06:14.280
We can see right away that not everything we’ve said so far will also apply to an elliptical orbit.
00:06:14.570 --> 00:06:21.300
Mostly, that’s because our object is no longer moving in a circular arc and, therefore, is not experiencing a centripetal force.
00:06:21.750 --> 00:06:28.410
So for an elliptical orbit, we can no longer say that the orbiting object’s kinetic energy is given by this relationship.
00:06:28.850 --> 00:06:32.100
But that doesn’t mean that everything is different for an elliptical orbit.
00:06:32.560 --> 00:06:39.950
For example, this equation here still applies for the gravitational potential energy between our orbiting mass and the mass being orbited.
00:06:40.430 --> 00:06:46.630
And it’s still true that the mechanical energy of our object is the sum of its kinetic and gravitational potential energies.
00:06:47.130 --> 00:06:54.940
So to write this object’s mechanical energy mathematically, once again, that’s kinetic energy plus gravitational potential energy.
00:06:55.350 --> 00:07:00.790
But now, we’ll write out kinetic energy as simply one-half the object’s mass times its speed squared.
00:07:01.110 --> 00:07:06.340
And expressed this way, we can see that there’s only one factor common to both of these terms.
00:07:06.590 --> 00:07:09.150
It’s the orbiting object’s mass, lowercase 𝑚.
00:07:09.650 --> 00:07:14.310
And so we can write the mechanical energy for an object in elliptical orbit this way.
00:07:14.810 --> 00:07:21.030
And here we see the object’s speed doesn’t cancel out of the expression like it did for our object in circular orbit.
00:07:21.370 --> 00:07:30.430
Despite the differences between these two equations, one thing that’s true for both is that the orbiting object’s mechanical energy is constant all throughout its orbit.
00:07:30.900 --> 00:07:37.550
That is, the mechanical energy of an object in orbit, whether that orbit is circular or elliptical, stays the same.
00:07:38.170 --> 00:07:47.900
Note that we’re not saying that all orbiting objects have the same mechanical energy, only that for a given object in orbit, that amount is maintained all through its orbit.
00:07:48.500 --> 00:07:53.270
This fact may seem more clearly true in the case of an object in circular orbit.
00:07:53.660 --> 00:07:58.740
In this case, everything on the right-hand side of the expression is either a constant or a fixed value.
00:07:59.150 --> 00:08:03.750
But what about for an elliptical orbit, where that mechanical energy is given by this equation?
00:08:04.210 --> 00:08:12.200
After all, we can see that 𝑟, the distance between the center of mass of our orbiting body and the body being orbited, clearly changes throughout the orbit.
00:08:12.630 --> 00:08:19.890
When our mass, for example, is out here, that value is relatively large, whereas when our mass is closer in, that value shrinks.
00:08:20.250 --> 00:08:24.900
Despite this, it is true that the mechanical energy of this orbiting body is constant.
00:08:25.180 --> 00:08:30.230
And that’s because as the distance 𝑟 varies, so does the speed of our orbiting object 𝑣.
00:08:30.670 --> 00:08:40.340
It turns out that the bigger 𝑟 is, that is, the farther our orbiting body is away from the body that it’s orbiting, the smaller that object’s speed 𝑣 gets.
00:08:40.880 --> 00:08:52.120
This means, for example, that when our orbiting mass is way out here, its speed 𝑣 is relatively small, whereas when 𝑟 is relatively small, say, when our mass is here, it has a greater speed.
00:08:52.560 --> 00:09:02.420
The speed of the object, 𝑣, and its distance from the center of mass of the object it’s orbiting, 𝑟, balance one another out so that the difference between these two terms is always the same.
00:09:02.880 --> 00:09:07.860
And this leads to the mechanical energy of our orbiting body being constant all throughout its motion.
00:09:08.370 --> 00:09:17.220
Knowing all this about how mechanical energy is conserved for objects in circular or elliptical orbit, let’s get some practice now with these ideas through an example.
00:09:17.920 --> 00:09:22.820
A spacecraft that has been launched into space is moving along a circular path around Earth.
00:09:23.360 --> 00:09:29.850
The radius of the orbit is 7,000 kilometers, and the spacecraft has a mass of 2,200 kilograms.
00:09:30.250 --> 00:09:32.510
What is the kinetic energy of the spacecraft?
00:09:32.880 --> 00:09:45.580
Use a value of 5.97 times 10 to the 24th kilograms for the mass of Earth and 6.67 times 10 to the negative 11th cubic meters per kilogram second squared for the universal gravitational constant.
00:09:46.100 --> 00:09:48.370
Give your answer to three significant figures.
00:09:48.980 --> 00:09:56.480
Okay, so here we have the Earth, and we’re told that there is a spacecraft, we’ll say that craft is right here, in circular orbit around it.
00:09:56.990 --> 00:10:04.500
We’re told that the radius of the orbit, which is the distance from the center of the spacecraft to the center of the Earth, is 7,000 kilometers.
00:10:04.600 --> 00:10:06.150
And we’ll call that distance 𝑟.
00:10:06.620 --> 00:10:11.210
We’re also told that this orbiting spacecraft has a mass of 2,200 kilograms.
00:10:11.240 --> 00:10:13.240
And we’ll call that mass lowercase 𝑚.
00:10:13.700 --> 00:10:16.820
We want to know what is the kinetic energy of the spacecraft.
00:10:17.120 --> 00:10:21.650
And we’re given values to use for the mass of the Earth and the universal gravitational constant.
00:10:22.110 --> 00:10:33.770
Now, if we recall that, in general, an object’s kinetic energy is given by one-half its mass times its speed squared, we might be confused about how to solve for the spacecraft’s kinetic energy because we don’t know its speed.
00:10:34.120 --> 00:10:44.820
However, we do know that this spacecraft is moving in a circular orbit, and any object moving in a circular arc is subject to what is called a centripetal or center-seeking force.
00:10:44.890 --> 00:10:46.190
We can call it 𝐹 sub 𝑐.
00:10:46.650 --> 00:10:55.460
This centripetal force acting on an object as it moves in a circle of radius 𝑟 is given by the object’s mass times its speed squared divided by that radius.
00:10:55.800 --> 00:11:00.540
Now, any center-seeking force needs to have some physical mechanism that causes it.
00:11:00.880 --> 00:11:07.310
In this case, our spacecraft is moving in a circle around the Earth thanks to the gravitational attraction between the two masses.
00:11:07.730 --> 00:11:20.300
In general, the force of gravity between two masses, we’ll call them uppercase and lowercase 𝑚, is equal to the product of those masses times the universal gravitational constant divided by the distance between their centers of mass squared.
00:11:20.760 --> 00:11:29.590
What we’re saying is that in the case of our satellite, the gravitational force experienced by it is equal to the center-seeking force that makes it move in a circle.
00:11:30.030 --> 00:11:44.330
In other words, the centripetal force 𝑚𝑣 squared over 𝑟 is equal to the gravitational force, where we’re saying that lowercase 𝑚 is the mass of our satellite and uppercase 𝑀 is the mass of the body being orbited, in this case, Earth.
00:11:44.860 --> 00:11:50.290
In this equation, the mass of our satellite cancels out, as does one factor of the radius 𝑟.
00:11:50.730 --> 00:11:58.270
And so we find this expression: 𝑣 squared, the speed of our satellite squared as it orbits the Earth, is equal to 𝐺 capital 𝑀 over 𝑟.
00:11:58.560 --> 00:12:04.290
And this now solves our problem of not knowing the speed of our satellite in order to calculate its kinetic energy.
00:12:04.730 --> 00:12:23.660
Now that we know its speed squared and can express that in terms of these factors, we can say that the kinetic energy of our orbiting spacecraft is equal to one-half its mass times 𝐺, the universal gravitational constant, multiplied by the mass of, in this case, the Earth, divided by the distance between our satellite and the center of the Earth.
00:12:24.090 --> 00:12:31.100
And we can now notice that we’re given the masses of these two bodies as well as the universal gravitational constant and this distance 𝑟.
00:12:31.450 --> 00:12:36.210
All that remains, then, is for us to substitute in these values and then calculate the kinetic energy.
00:12:36.600 --> 00:12:50.190
So here we have our satellite mass, 2,200 kilograms; our value for the universal gravitational constant; our mass of the Earth, 5.97 times 10 to the 24th kilograms; and 𝑟, 7,000 kilometers.
00:12:50.600 --> 00:12:54.180
There’s just one change we’ll want to make before calculating kinetic energy.
00:12:54.370 --> 00:12:57.520
And it comes down to what is right now a disagreement in units.
00:12:57.790 --> 00:13:06.340
Notice that in our value for the universal gravitational constant, we have units of distance given in meters, whereas our radius is currently in kilometers.
00:13:06.600 --> 00:13:16.570
If we recall, though, that 1,000 meters is equal to one kilometer, then that tells us that 7,000 kilometers is equal to 7,000 with three zeros added to the end meters.
00:13:16.970 --> 00:13:18.800
That is seven million meters.
00:13:19.250 --> 00:13:25.400
Now, the units all throughout our expression do agree, and we can go ahead and compute the kinetic energy of the spacecraft.
00:13:25.770 --> 00:13:31.140
To three significant figures, we find a result of 6.26 times 10 to the 10th joules.
00:13:31.490 --> 00:13:41.460
And if we recall that 10 to the ninth, or a billion joules, is equal to one gigajoule, then we can express our answer as 62.6 gigajoules.
00:13:41.880 --> 00:13:47.410
This is the spacecraft’s kinetic energy, which, notice, is constant all throughout its circular orbit.
00:13:48.870 --> 00:13:53.140
Let’s summarize now what we’ve learned about mechanical energy conservation in orbits.
00:13:53.490 --> 00:14:03.270
In this lesson, we first recall that an object’s mechanical energy, we refer to it using capital ME, equals the sum of that object’s kinetic and potential energies.
00:14:03.690 --> 00:14:19.260
We saw further that when an object is in circular orbit, we can express its kinetic energy this way, where all the factors in this equation are either constants or fixed values, whereas when an object is in elliptical orbit, we express its kinetic energy this way.
00:14:19.470 --> 00:14:21.340
And we note that it changes in time.
00:14:21.770 --> 00:14:34.820
Along with this, we recall that for all orbits, the gravitational potential energy shared between the two masses is equal to negative 𝐺, the universal gravitational constant, times the product of the two masses divided by the distance between their centers.
00:14:35.290 --> 00:14:46.960
All this led us to expressions for the mechanical energy of an object in both circular and elliptical orbit and, finally, the observation that these mechanical energies are constant in time.
00:14:47.450 --> 00:14:49.600
That is, these energies are conserved.
00:14:50.380 --> 00:14:53.760
This is a summary of mechanical energy conservation in orbits.