I and my kids have become interested in potentially making a spinning ice ring like has been going around the midwest the last two years. For example, Oldest spinning ring on YouTube
The concept is fairly straightforward: set a middle point and use a chainsaw to cut through the ice in one or two concentric circles (the second is usually just a bit smaller and is used to remove an inch or two of ice to allow the easier spinning of the ice. But the larger the circle, the greater amount of time to cut and the harder to keep a perfect radius (or fix any flaws). So,
Assuming 4" (10 cm) thickness, what is the minimal circle radius to hold up 150kg (~2 humans) of people on the ice? Or really it would be good to just come up with a reasonable formula to use! I know that having two people stand near the edge would mess everything up, but I’ve got to start somewhere. So here is my best attempt at doing so but would appreciate any feedback:
Ice is ~8% less dense than water (0.92 g/mL vs. 1.00 g/mL) so I need to have 150 kg of buoyancy. So I need 150kg = 150,000 mL excess ice buoyancy.
150,000mL / (1.00g/L - 0.92g/L) = 1,875,000 g of ice
1,875,000 g of ice /(0.92 g/mL) = 2,038,000 mL of ice
Assuming a cylinder with a height of 10 cm: V=1/2hr2 2,038,000 mL (cm3) = 1/2 *10cm * r2. So r2 = 2,038,000 cm3 * 2 / 10cm = 407,600 cm2. So r= 638cm = 6.38m
So 6m radius and 12m diameter, but this just seems really a lot larger than I would expect especially based on seeing videos and even people standing on small icebergs. Even doubling the thickness of the ice doesn’t dramatically decrease the radius needed. I was really hoping that I could make a ~2m radius circle on our small pond that would work. Do I not need full neutral buoyancy?
Another way to look at it is that the “float capacity” is the amount by which the iceberg sits above the water. If 2" of ice sticks aove the water, then you can add the weight of a disc 2" thick of water before the ice is overloaded.
I think my concern with smaller ice discs would be their tendency to tilt as you approach the edge. 7 feet dia x 4 inches of ice is 12.7 cu ft of water/ice, which is about 750lb. A person standing near the edge weighing 100lb would be a noticeable imbalance. 2 or more in one area, bad idea.
Meanwhile, using your 8% weight difference - 7’dia x 4" ->750lbs water, 8% of that is about 60lb. Suggests to me anyone over 60lb may sink the float?
For any ice circle there needs to be sufficient buoyancy to stop tipping if all the people on it were standing together near one point on the edge unless you want to risk numerous people drowning in freezing water. If the ice circle tilts below the level of the adjacent ice people can slip through the gap and then be trapped under the circle. If enough of the people fall in that way the rest will not weigh enough to tip the circle back up to extract any of the people that fell in.
For a small circle that can only support 2-3 people this is a serious problem when anyone is stepping on or off the circle.
Heck, we used to stand on icebergs the size of the outline of a car. Tipping was part of the fun. Really stupid. I have chills just thinking about it. I don’t know how we survived childhood near Lake Erie.
Or getting pinned at the waist, alone, legs and cell phone dangling below the surface, inaccessible and chilling fast in the freezing fluid. Head and torso dressed for the weather, you’d have quite a while to think about the mistakes but not too long.
For an iceberg with enough payload on it to completely submerge said iceberg, the total buoyancy provided is the volume of the ice multiplied by the difference in density of the ice and the water:
Fb = Vice * (ρwater - ρice)
Your target buoyancy is 150 kg, so:
150 kg = Vice * (1000 kg/m3 - 920 kg/m3)
150 kg = Vice * 80 kg/m3
Vice = 150/80 m3
Vice = 1.875 m3
You said the thickness of your spinning disc of ice would be 4 inches (0.102 meters), so the top surface area would be 18.38 m2, and the diameter would need to be 4.84 m.
That represents the theoretical absolute minimum to just barely keep 150 kilograms of meat from slowly sinking to the bottom of the lake. If you’re actually planning to do this, you need to add in a generous safety factor, since you don’t truly know the thickness of the ice, and add in another safety factor to absolutely assure stability when two knuckleheads are standing together on one edge of the disc. The principles of static stability for floating bodies are fairly easy to understand, but the math for real-world shapes (like a broad, partially-submerged disc that’s tilted at an arbitrary angle relative to the surface of the water) is harder. You could do some trial-and-error, starting with a very generously sized disc, and see how stable it is when tested by people wearing strap-on ice traction cleats on their boots (so they can quickly move away from the edge of the disc and/or jump to safe ice), and safety ropes attached to them that are held by other folks standing on the stationary ice.
Thanks! Your math makes it far clearer! I should note that the pond is only 4 feet deep at the most (well there is a much deeper corner but it is <10% of the area). I and others have crashed through the ice in late spring multiple times and just walked out so safety is only a concern so far as to make sure that someone is on shore and ready to respond.
But as mixdenny pointed out, we all have experience with fun on smaller pieces of ice and I have to think that maybe their having a lot of submerged ice volume helps to keep them more stable and floating than one would guess!
That’s not the volume of a cylinder. The right formula is V = \pi h r^2 . Using that with your 2038000 mL number gives a radius of 255 cm, which is roughly in line with Machine_Elf’s numbers.
That must be* the funnest part. Anything that generates motive power! People with poles, the chainsaw motor, auger motor, snowmobile, tractor, Silverado pickup, Toyota Tercel.
Reminds me of a couple of “Fail” videos where some guy holds the rear tire of a motorcycle against one of those playground merry go rounds that hold like 4 people. Get a couple of your buddies on the merry go round and rev 'er up! They get flung off after the G forces get too high to hang on. Chaos ensues. And injuries.