00:01
So for this problem, the first part of this problem, we are asked to calculate the force of gravity on a spacecraft that is at a distance, a separation distance, we're going to call the separation distance d, and that is 12 ,000 and 8 ,000 meters, and this above earth's surface.
00:32
If it's mass, and we are given the mass for this spacecraft, and that is equal to 1 ,489 kilometers.
00:49
Now, we need to take into account the radius and mass of the earth into your equations.
00:56
So to calculate the force of gravity, we need to consider that the earth has a radius, so the radius of the earth is equal to 6 .38 times 10 to the 6 meters.
01:19
We can also write this distance in scientific notation as the following as 1 .12 .8 times 10 to the 6 meters.
01:33
And we know that the mass of the earth, the mass of the earth is equal to 5 .98 times 10 to the 24 kilograms.
01:55
Okay, so now since the spacecraft is too earth ready about the surface of the earth, it is three earth ready from the center, which is the distance you, want to use in the formula for gravity.
02:11
So now we know that the center to the earth is the value that we are given plus this.
02:27
So we will find that this is, let's put in here the equation for the force of gravity.
02:36
We know that the force of gravity is defined as the gravitational constant times the mass of the earth times the mass of the spacecraft and this divided by the separation distance between these two objects.
02:52
Now that separation distance in this case is going to be the sum of these two values that we are given, the radius of the earth and the separation distance that we are given.
03:06
And then the mass is just going to be the product between these two.
03:12
So we just now substitute all of these values.
03:15
In here.
03:17
So we are going to have in the numerator we have the gravitational constant.
03:24
We know that that has a value of 6 .67 times 10 to the minus 11 neutrons times meters square and this divided by kilograms square.
03:42
And this times the mass of the earth that we know is 5 .98 times 10 to the 24 kilograms.
03:55
And this times the mass of the spacecraft that we are also given 1 ,489 kilograms, if i'm correct.
04:10
Yes.
04:12
And we divide this by the separation distance between the center of the earth and the and the spacecraft.
04:21
So that is the these two values, the sum of these two values.
04:26
So we will have 12 .8 times 10 to the 6 meters plus 6 .38 times 10 to the 6 meters and that to the square.
04:42
So using our calculator, we obtain a force of gravity of.
04:46
So the value that we obtain from this is equal to 1 ,6002 point, well, that's it.
05:02
And point, the problem states that answer to the one's place.
05:12
So we can conclude that the value of this is 1 ,6002 newton's.
05:21
So that's a solution for the first part of this problem.
05:26
Now, for the second part of this problem, we are told that an hypothetical planet has a radius of 2 .3 times that of the earth.
05:39
So we have the radius of that planet, and that is 2 .2 times the radius of the earth.
05:48
And then, but it has the same mass.
05:53
Mass of this planet is equal to the mass of earth.
05:58
And so for this problem, we are asked, what is the acceleration due to gravity on his surface? and answer to the thousand plays.
06:12
So with this, we know that the force is defined as the mass times the acceleration.
06:30
And so the acceleration of an object with a mass that is we can solve from this for the acceleration.
06:41
And then where f is the force of gravity, that we know the force of gravity is the force of gravity, is the one that we use from before, is equal to the gravitational constant times the product between the masses.
06:56
In this case the mass of the object, and this divided by the separation distance between these two objects.
07:15
And now, in this case, where m is the, well, mp is the planet's mass, and the other m is the mass of the object.
07:26
Now, we just plug the formula for the four, of gravity into newton's second law, and we get the acceleration of gravity at the planet's surface, because we just need to divide, as you can see, divide the force of gravity by the mass of the object.
07:44
So immediately, we obtained that the acceleration is the gravitational constant, the mass of the planet divided by the separation distance.
07:53
And the separation distance corresponds to the radius of the, the radius of the planet because we are assuming that that object is at the surface of the of the planet.
08:06
So we just need to simply substitute all of the values that we know for this case.
08:12
So we know that the gravitational constant, it has a value of 6 .67 times 10 to the minus 11 with units of newton, 6 meters, square per kilogram square.
08:26
And this times the mass of the planet.
08:29
Now, we are told that the mass of the planet is the same mass as the earth.
08:33
So we just plot in here the mass of the earth, which is 5 .98 times 10 to the 24 kilograms.
08:44
And this divided by the separation, well, in this case, the radius of the planet, but in this case, we are told that the radius of this planet is 2 .2 .2.
08:57
Times the radius of the earth.
08:59
So we need to just multiply that 2 .2 times the radius of the earth that we know from the previous item.
09:07
And that radius is 6 .38, 6 .38 times 10 to the 6 meters and all of that to the square.
09:16
So from this, we obtain an acceleration of.
09:21
So from this, we obtain a value of 2 .6.
09:30
025, if i'm correct, yes, 2 .025 meters per second square.
09:40
So that's a solution for part two of this problem.
09:44
Now for the other part of this problem, so now we are told that at the surface of a certain planet, the acceleration, we're going to call this part three.
10:03
And so we are told that at the surface of a certain planet, the gravitational acceleration g has a magnitude of 12 meters per second square.
10:14
So we are told that the acceleration due to gravity in this case is 12 meters per second square.
10:24
And then we are told that a 2 .10 kilograms brass ball is transported due to the apply...