# Homework Help: Floating Cylinder of Uniform Density

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1. Nov 6, 2015

### ScrubTier

A solid cylinder of uniform density of 0.85 g/cm3 floats in a glass of water tinted light blue by food coloring.
https://s4.lite.msu.edu/enc/74/b3c49e2ca8cb7c509e3042e72ba7ea7017a9952689b6f0e427ff558774d5f3fbf098898cb4b4f2de9035623ceb3cd36ea31f7d51c48cb6ba67d249dfd4687472f9dc5f4ac2ae0c859d1d39714a7373604f270a8c685ef41773164e58fe8763aa.gif
Its circular surfaces are horizontal. What effect will the following changes, each made to the initial system, have on X, the height of the upper surface above the water? The liquids added do not mix with the water, and the cylinder never hits the bottom.

1. More tinted water is added to the glass.
2. A liquid with a density of 0.76 g/cm3 is poured into the glass.
3. The cylinder is replaced with one that has the same density and diameter, but with 1.5× the height.
4. The cylinder is replaced with one that has the same density and height, but 1.5× the diameter.
5. The cylinder is replaced with one that has the same height and diameter, but with density of 0.83 g/cm3.
6. A liquid with a density of 1.06 g/cm3 is poured into the glass.
2. Relevant equations
Density = Mass/Volume

3. The attempt at a solution

1. No Change- Both densities are the same so no change occurs
2. X Decreases- The lower density liquid will sit on top of the water and the cylinder
3. X increases- A proportional amount to before will be above water and because it is longer more will stick out
4. No Change- Diameter has no effect on density
5. X increases- Since it is now less dense it will float higher
6. No Change- The denser liquid will sink to the bottom and not effect the cylinder

2. Nov 6, 2015

### ScrubTier

https://s4.lite.msu.edu/enc/74/b3c49e2ca8cb7c509e3042e72ba7ea7017a9952689b6f0e427ff558774d5f3fbf098898cb4b4f2de9035623ceb3cd36ea31f7d51c48cb6ba67d249dfd4687472f9dc5f4ac2ae0c859d1d39714a7373604f270a8c685ef41773164e58fe8763aa.gif sorry. This is the picture for the question

3. Nov 6, 2015

4. Nov 6, 2015

### insightful

EDIT: Forget the following; see discussion below.

Think what would happen if you cut the cylinder in half, making two cylinders at half the length ("height")?

What about 2x the diameter or 5x?

Last edited: Nov 6, 2015
5. Nov 6, 2015

### haruspex

@ScrubTier , I agree with all your answers except #1. Read very carefully the definition of X.

Edit: typo, I meant all except #2.

@insightful , you seem to disagree with answers 3 and 4. They look right to me. Are you perhaps taking the circular faces as vertical?

Last edited: Nov 6, 2015
6. Nov 6, 2015

### insightful

I'm taking it as a cylinder on its side, like a log floating.

7. Nov 6, 2015

### haruspex

Then read the question statement again. It is floating upright ("circular surfaces are horizontal").

8. Nov 6, 2015

### insightful

Well, I interpreted the "circular surfaces" as the " curved surfaces." OP?

9. Nov 6, 2015

### haruspex

I suppose you could argue that is ambiguous, but it is resolved by this:

10. Nov 6, 2015

### insightful

I could argue that a floating log has an "upper surface" too, but I do see your point (that and the use of the word "height").

Note my EDIT above.

11. Nov 6, 2015

### insightful

Are you assuming that the added water which "does not mix" somehow distributes itself uniformly on the surface?

12. Nov 6, 2015

### insightful

What would happen if a liquid with a density of 0.85 g/cm3 were poured into the glass?

13. Nov 6, 2015

### haruspex

My mistake, I meant all except #2. Edited above.

14. Nov 6, 2015

### ScrubTier

I was thinking that it would distribute itself evenly yes, is that incorrect? And are 4 and 5 wrong? Or not?

15. Nov 6, 2015

### haruspex

Insightful was responding to my initial post, but I had a typo there.
As I meant to say there, I agree with all of your answers except #2. Check very carefully how X is defined.

16. Nov 6, 2015

### ScrubTier

I was thinking since the new liquid added is less dense than the water it would sit on top of the water and because the cylinder is heavier that it would not be able to float. I just tried all the same answers except with 2. as No Change and it was wrong

17. Nov 6, 2015

### insightful

Well, you have one more possibility for 2..........
Also, what about my question in #12?

18. Nov 6, 2015

### ScrubTier

If it was the same then would the x increase?

19. Nov 6, 2015

### ScrubTier

I just tried it with all of the same answers except #2 increase. Something besides #2 must be wrong

20. Nov 6, 2015

### haruspex

I cannot think what else could be wrong.
Yes, for #2, it should be an increase, but do you see why? How is X defined?

21. Nov 6, 2015

### insightful

For #1, if the added "new" water actually did distribute itself over the "old" water (not mixing), and x is measured from the "old" water level, x would increase, but this is a stretch.

22. Nov 6, 2015

### ScrubTier

I discovered a typing error! You guys were right :) Thank you both very much