How to find the mass (or volume) of Saturn's ring?

[Remon]

Homework Statement

Saturn has the largest ring system of any planet, with the inner edge about 7000 km above
Saturn’s surface and extending to its outer edge about 80,000 km above the surface. However,
the rings are only about 20 m thick. The rings are made of billions of small particles, most
of which are water ice, which has a mass density of 1000 kg/m3. Approximating the ring as
being made of solid ice, what is the mass of the ring compared to the mass of Earth’s Moon?
Show how your got your answer.

Homework Equations

Volume = L x W x H and Mass = Density x Volume

The Attempt at a Solution

I have the density (1 g/m³) but I don't know how to get the volume from the other information because I've never taken the volume of a ring before, could someone please help me find the volume? I know that once I have the volume I can easily get the mass and compare it with the moon's mass

 

[Dick]

Homework Statement

Saturn has the largest ring system of any planet, with the inner edge about 7000 km above
Saturn’s surface and extending to its outer edge about 80,000 km above the surface. However,
the rings are only about 20 m thick. The rings are made of billions of small particles, most
of which are water ice, which has a mass density of 1000 kg/m3. Approximating the ring as
being made of solid ice, what is the mass of the ring compared to the mass of Earth’s Moon?
Show how your got your answer.

Homework Equations

Volume = L x W x H and Mass = Density x Volume

The Attempt at a Solution

I have the density (1 g/m³) but I don't know how to get the volume from the other information because I've never taken the volume of a ring before, could someone please help me find the volume? I know that once I have the volume I can easily get the mass and compare it with the moon's mass

The volume is just the thickness times the area. And the area is the region between two circles. You should be able to find that just by subtracting the area of the inner circle from the area of the outer.

 

[Remon]

The volume is just the thickness times the area. And the area is the region between two circles. You should be able to find that just by subtracting the area of the inner circle from the area of the outer.

But how am I suppose to find the areas of both circles? I tried to use the equation of finding the area of a ring, which is: ∏(R² - r²)
I added both 80,000 km (which is R) and then 7,000 km (which is r) by the radius of Saturn which is 58,232 km (to extend their radiuses to Saturn's core/centre, not just its surface)
Then I got ≈ 4.6662 x 10¹⁰, I then multiplied it by 0.02 km (which is the thickness) to find the volume which I then plugged into the Mass = Density x Volume equation, and compared it to the moon's mass but still got an incredibly small number which is not one of the choices. Any more help would be greatly appreciated, thank you

 

[Dick]

But how am I suppose to find the areas of both circles? I tried to use the equation of finding the area of a ring, which is: ∏(R² - r²)
I added both 80,000 km (which is R) and then 7,000 km (which is r) by the radius of Saturn which is 58,232 km (to extend their radiuses to Saturn's core/centre, not just its surface)
Then I got ≈ 4.6662 x 10¹⁰, I then multiplied it by 0.02 km (which is the thickness) to find the volume which I then plugged into the Mass = Density x Volume equation, and compared it to the moon's mass but still got an incredibly small number which is not one of the choices. Any more help would be greatly appreciated, thank you

You are off to a good start. 4.6662 x 10¹⁰ is an ok numerical answer but the units on that are km^2. You need to keep track of the units, and show the rest of your work. I did it and I get that it's small compared to the mass of the moon. But not 'incredibly small'.

 

[Remon]

You are off to a good start. 4.6662 x 10¹⁰ is an ok numerical answer but the units on that are km^2. You need to keep track of the units, and show the rest of your work. I did it and I get that it's small compared to the mass of the moon. But not 'incredibly small'.

Thank you for replying back but I still don't think its right because the answer I got at the end was ≈1.27 x 10^-11, where all the choices are exactly the following:
0.0001 x Earth's moon, 0.001 x Earth's moon, 0.01 x Earth's moon, 0.1 x Earth's moon, and 1 x Earth's moon.
But the answer I have is 1.27 x 10^-11 x Earth's moon is is obviously much smaller than all of these answers, thank you again.
p.s I was aware of all the units but none of them needed to be changed

 

[Dick]

Thank you for replying back but I still don't think its right because the answer I got at the end was ≈1.27 x 10^-11, where all the choices are exactly the following:
0.0001 x Earth's moon, 0.001 x Earth's moon, 0.01 x Earth's moon, 0.1 x Earth's moon, and 1 x Earth's moon.
But the answer I have is 1.27 x 10^-11 x Earth's moon is is obviously much smaller than all of these answers, thank you again.
p.s I was aware of all the units but none of them needed to be changed

Yes, you do need to keep track of units. You've correctly got that the area of the rings is 4.6662*10^10*km^2. That's kilometers, not meters. The units on thickness (20m) and density (1000kg/m^3) are given in meters. The answer is not that small.

 

[Remon]

Yes, you do need to keep track of units. You've correctly got that the area of the rings is 4.6662*10^10*km^2. That's kilometers, not meters. The units on thickness (20m) and density (1000kg/m^3) are given in meters. The answer is not that small.

Actually I did say that I used 0.02 km for thickness in my previous comments, but for density I tried using both 1000 kg/m³ and 1 g/cm³ to see what answer they would give me. The 1000 kg/m³ gave me 1.27 x 10^-11 (which I stated before) and the 1 g/cm³ gave me 1.27 x 10^-14. So I really don't know what I'm doing here but I still get a much smaller answer than the smallest choice I have even after I tried using different units when I was unsure, thanks.

 

[Dick]

Actually I did say that I used 0.02 km for thickness in my previous comments, but for density I tried using both 1000 kg/m³ and 1 g/cm³ to see what answer they would give me. The 1000 kg/m³ gave me 1.27 x 10^-11 (which I stated before) and the 1 g/cm³ gave me 1.27 x 10^-14. So I really don't know what I'm doing here but I still get a much smaller answer than the smallest choice I have even after I tried using different units when I was unsure, thanks.

Can you please show your whole working for one of those? Your exponents are drastically off. You are NOT handling units correctly. 1000 kg/m^3 and 1 g/(cm^3) are exactly the same. How can using one or the other change the answer??

 

[Remon]

Can you please show your whole working for one of those? Your exponents are drastically off. You are NOT handling units correctly. 1000 kg/m^3 and 1 g/(cm^3) are exactly the same. How can using one or the other change the answer??

That's what I'm wondering too. But here's what I did:
I used ∏(R² - r²) to find the area of the ring
R being 138,232 and r being 65,232, the area I get is then = 4.6662 x 10¹⁰ km² (which is supposedly right), then I multiply it by the thickness (0.02 km) to get a volume of 933,240,000 km³.
Then I multiplied the volume by the density which is 1000 kg/m³, to get a mass of 933,240,000,000 (or 9.332 x 10¹¹)
Lastly, I divide the mass of saturn's rings by the moon's mass, therefore it looks like this: (9.332 x 10¹¹ kg)/(7.348 x 10²² kg) to give me the final answer of something ranging from 0.0001 to 1 x the moon's mass
But instead, I get 1.27 x 10^-11 which is obviously very wrong for some reason

 

[Dick]

That's what I'm wondering too. But here's what I did:
I used ∏(R² - r²) to find the area of the ring
R being 138,232 and r being 65,232, the area I get is then = 4.6662 x 10¹⁰ km² (which is supposedly right), then I multiply it by the thickness (0.02 km) to get a volume of 933,240,000 km³.
Then I multiplied the volume by the density which is 1000 kg/m³, to get a mass of 933,240,000,000 (or 9.332 x 10¹¹)
Lastly, I divide the mass of saturn's rings by the moon's mass, therefore it looks like this: (9.332 x 10¹¹ kg)/(7.348 x 10²² kg) to give me the final answer of something ranging from 0.0001 to 1 x the moon's mass
But instead, I get 1.27 x 10^-11 which is obviously very wrong for some reason

Now you've got the volume right as well. In terms of km^3. That's not the same as m^3. If you want to multiply by 1000kg/m^3 you had better either convert the km^3 to m^3 or do vice versa for the density. You still aren't tracking the units correctly until the end. I'll give you a big hint. There are 10^9m^3 per km^3.

 

[Remon]

Now you've got the volume right as well. In terms of km^3. That's not the same as m^3. If you want to multiply by 1000kg/m^3 you had better either convert the km^3 to m^3 or do vice versa for the density. You still aren't tracking the units correctly until the end.

Oh ok, so after I multiplied 933,240,000 km³ by 10⁹, I get a volume of 9.3324 x 10¹⁷ km² instead, then I multiply it by the density of 1000 kg/m³ to get a mass of 9.3324 x 10²⁰ for the rings. Then I divide it by the moon's mass and get ≈ 0.01 which is one of the answers. I hope this is right and thank you for helping.

 

[Dick]

Oh ok, so after I multiplied 933,240,000 km³ by 10⁹, I get a volume of 9.3324 x 10¹⁷ km² instead, then I multiply it by the density of 1000 kg/m³ to get a mass of 9.3324 x 10²⁰ for the rings. Then I divide it by the moon's mass and get ≈ 0.01 which is one of the answers. I hope this is right and thank you for helping.

You mean you get a volume of 9.3324 x 10¹⁷ m³, I hope. You are still messing the units up on intermediate steps. But, yes, the final answer is pretty right. See how important units are? If you had realized that you should have put a unit on the final mass.

 

[Remon]

You mean you get a volume of 9.3324 x 10¹⁷ m³, I hope. You are still messing the units up on intermediate steps. But, yes, the final answer is pretty right. See how important units are? If you had realized that you should have put a unit on the final mass.

Oh yeah, I meant m³, not km², I just got confused with all the units, and I forgot to put the kg for the mass at the end but I was just focusing more on the number for now