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RE: Which cup will fill up First ?
Bucket 3. If you look closely, some buckets have no openings. Bucket 3 does not open into 4, and the pipe to 5 is also closed off. So it would fill up first. Then it would over flow and the second bucket filled would be bucket 4 followed by 5&6, 7, 2, and 1 (assuming all the buckets are in a room with a watertight floor and walls.
Bucket 3 will be filled first followed by bucket 2 and then bucket that I that is supplying. Bucket 7 is not going to be filled at all but water will still come into it but will not stay.. It has an opening at the bottom. 4,5, 6 is the last because no drop of water will even come to it.
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Depending on the viscosity of the fluid filling the buckets after the cup labled 3 has filled I believe 5 may be the next bucket filled. Say the fluid is water, 5 would fill next depending on the rate of wich the water is flowing from the spicket due to water streaming along the surface area of the pipe and reaching cup 5 as it has the closest proximity to the first cup being filled (cup 3). As I said this only is possible assuming the fluid has a viscosity simmilar to that of water.
But there are holes at the top of bucket 3 so it will just overflow. It will be the only one to fill up unless the whole box fills up, in which case, the scenario I laid out will play out.
If the fluid has a viscosity simmilar to that of watet it would over flow the third recepticle and the overflow may travel along the outside of the tube leading to the 5th and 4th recepticle. This all depends on what the fluids viscosity is and the rate at which it is being released from the spicket.
Viscosity is relevant to flow. Even if it is water, if the flow is fast enough, it won't matter.
Cup 5 will not have a drop no matter what the fluid may be. Reason why? If you look closely, you will see that the pipe did not open to it. It's just pointing.
The top to bucket 5 is open. If the room floods with water, bucket 5 will fill once water rises above the top of the bucket.