Titania, the largest moon of the planet Uranus, has (1)/(8) the radius of the earth and (1)/(170

step 1 We're asked to find the gravitational acceleration on Titania. So we know that for any planets,

Titania, the largest moon of the planet Uranus, has (1)/(8)...

Titania, the largest moon of the planet Uranus, has $\frac{1}{8}$ the radius of the earth and $\frac{1}{1700}$ the mass of the earth. (a) What is the acceleration due to gravity at the surface of Titania? (b) What is the average density of Titania? (This is less than the density of rock, which is one piece of evidence that Titania is made primarily of ice.)

Key Concepts

Newton's Law of Universal Gravitation

This fundamental law states that every two masses attract each other with a force that is proportional to the product of their masses and inversely proportional to the square of the distance between their centers. It provides the basis for calculating gravitational forces in astronomical contexts and is essential for determining quantities like gravitational acceleration on a celestial body’s surface.

Surface Gravitational Acceleration

The acceleration due to gravity at the surface of a celestial body is derived from Newton's law and is given by the formula g = GM/R², where G is the gravitational constant, M is the mass, and R is the radius of the body. This concept is crucial for understanding how changes in a body's mass and size can affect the gravitational force experienced at its surface.

Average Density

Average density is a measure of how mass is distributed over a given volume, typically calculated for a spherical body using the formula density = M/(4/3 ?R³). This concept is important for determining the composition and internal structure of astronomical bodies, and it helps in distinguishing between objects made of denser materials like rock and less dense materials like ice.

Scaling Relationships in Astrophysics

Scaling relationships explore how variations in one property, such as radius, influence other properties, like mass and density. When given ratios of mass or radius relative to a standard (like Earth), these relationships allow astrophysicists to estimate other physical characteristics, demonstrating how relative comparisons can lead to insights about the structure and composition of celestial bodies.

Transcript

00:01 We're asked to find the gravitational acceleration on titania.

00:05 So we know that for any planets, the acceleration due to gravity is just going to be the gravitational constant times the mass of the planet and divided by the radius squared of that planet.

00:15 So we want to apply that to titania.

00:19 So we have g.

00:20 And then they don't directly give us the mass or the radius here.

00:24 They give it some relative to earth.

00:26 So we could calculate what the actual values would be for that, but it'll probably be a little bit simpler to just do this kind of proportional to what the value is on earth, which we know.

00:36 So we're told that its mass is just the 1 ,700th of what it is on earth.

00:41 So i'm going to write this as mass of the earth over 1 ,700.

00:47 So this whole term is just the m over here.

00:51 And then its radius is an eighth of earth, so divided by the radius of the earth, divided by 8, and that whole thing.

01:00 Squared.

01:03 So we can simplify this a little bit and we should get g and then i'm going to pull out the mass of the earth and the radius of the earth.

01:09 So we'll have g mass of the earth over radius of the earth squared.

01:15 And then the constants we're left with will be 64, 8 squared, which comes up to the numerator, and then 1 ,700.

01:25 So this whole term right here is just the acceleration due to gravity on earth.

01:31 So we know that that's 9 .8 meters per second squared.

01:35 So then we just multiply it by this constant 64 over 1 ,700.

01:41 And so we do that, and we get that it is 0 .377 times the gravitational acceleration on earth, so 9 .8.

01:57 And so then if we evaluate that further, we'll get that it's 0 .377 ,000 times the gravitational acceleration on earth, so 9 .8...

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