How Can a Table Float with Buckets?

[TheVarsoTMLA]

If the angle between the rope and the table is theta when the buckets are in contact with the table then the weight of each bucket (m) must be at least equal to the weight of the table (M) divided by 4sin(theta).

m = M/[4sin(theta)]

See attached PDF.

 

[EricL]

This is simple. The weight of the table is equal to that of the buckets. A little push and you could make the buckets float 1 inch above the table. It would still be a static system and would not move.

Okay, even being a person with feeble math skills compared to anyone else here, I believe I see a problem with this statement, and I think I see this very thing being accounted for mathematically in some of the other replies (but I'd have to spend a day or two learning the math before I could be sure!). So, here's my take: I believe that each bucket must weigh more than 1/4 of the table's weight on account of the fact that the angle of the ropes relative to vertical is not the same on both sides of each pulley. If the rope on both sides of the pulley were vertical, your statement would be true (each bucket could weigh as little as 1/4 of the table's weight), but since the pulled-by-the-bucket leg of each rope is vertical and the table-lifting leg of each rope is angled away from vertical, it can't be true. Using my visual-only approach, it's easy to imagine increasing that angle away from vertical until the tension in the rope that's needed to produce the necessary upward lift becomes extreme, yet clearly the tension must be the same on each side of the pulley. Therefore, one needs to account for that decreased vertical component of the lifting force that's due to the table-lifting leg not being vertical, and that means increasing the load in each bucket by some amount. I'm sure some of the math-based replies provide a way of figuring exactly how much extra weight is needed, and the method I'd try first would be simple trigonometry. But don't forget that just to get the system to hold the position shown in the photo, "too much weight" in the buckets works just as well as "just enough."

Here's a related thought about having the system in perfect balance (defined as having "just enough" weight in each bucket). You say that "with a little push you could make the buckets float one inch above the table", but that's not true as I see it. In real life this could be done since the pulleys will provide friction, but assuming friction-less pulleys, as soon as you change the height of the table, you change the angle of the table-lifting leg of each rope, thereby changing the vertical component of the lifting force. In this case, if the system were perfectly balanced when the buckets were in contact with the table, moving the table to create a gap between the buckets and the table would reduce the angle away from vertical of the table-lifting leg of each rope, increasing the lifting force, thereby causing the table and buckets to come together once again.

Taking that one step further, if the system were perfectly balanced at a particular table height with the buckets not in contact with the table, if you manually displaced the table either upward or downward, it would drift back to its original position, since the lifting force provided by the ropes would be reduced when the table was raised, and increased when it was lowered. In actual fact, this principle surely aids in providing stability to the system (that is, the table is not prone to tilting).

If I'm missing something basic, shoot me down. Also, I'm sorry if this repeats anything that already was explained by someone using math.

 

[TheVarsoTMLA]

So, here's my take: I believe that each bucket must weigh more than 1/4 of the table's weight on account of the fact that the angle of the ropes relative to vertical is not the same on both sides of each pulley. If the rope on both sides of the pulley were vertical, your statement would be true (each bucket could weigh as little as 1/4 of the table's weight), but since the pulled-by-the-bucket leg of each rope is vertical and the table-lifting leg of each rope is angled away from vertical, it can't be true.

as soon as you change the height of the table, you change the angle of the table-lifting leg of each rope, thereby changing the vertical component of the lifting force. In this case, if the system were perfectly balanced when the buckets were in contact with the table, moving the table to create a gap between the buckets and the table would reduce the angle away from vertical of the table-lifting leg of each rope, increasing the lifting force, thereby causing the table and buckets to come together once again.

You're right! The buckets must be more than 1/4 the mass of the table. In this case they must be 1/4sin(angle between rope and table) the mass of the table. So if the angle formed when the buckets are in contact with the table was 60 degrees then the weight of the buckets would have to be about 1/3 the mass of the table. Take a look at the PDF attached for the math explanation.

 

[jaiswalshrey07]

Yes. See... the buckets are on the table. So the downward force exerted by each bucket is balanced by normal rxns. So the only force on the table now is tension. The total tension = total downward force (eqbm) and therefore, Tension = Tg+4Bg (where T and B represent Table's mass and a single bucket's mass).

 

[sophiecentaur]

You've been doing some digging!

So the only force on the table now is tension.

You can't get away without the weight force too.
Read through the whole thread and you'll find that pretty much all the relevant points have been dealt with. You'll also find quite a lot of confusion about operating a Bosun's Chair; I've used one several times so I know of which I speak. As a non-superman I needed assistance to actually raise myself.
Fact is that the buckets need only to weigh just more (total) than the table (plus other relevant gubbins) for the trick to work and for the buckets to stay in contact with the table. If the table is heavier then it will just fall down. Stability is a problem and three buckets would be better than four.

Carry on digging; there's some good stuff back there but a lot of it has been locked and needs reviving with new threads.

 

[Jeff Bigler]

I built one of these with one of my physics classes back in 2018. Here's a picture of it when I got it out this year. I put 2.5 L of water in each bucket, so each one weighed 25 N (probably plus an additional 1-2 N for the bucket and rope). After taking this picture, I put spring scales on each of the ropes that were attached to the table in the center, and each spring scale read 18 N. This means the normal force between the table and each bucket must have been 7 N, and the table must have weighed 54 N. After I poured the water out of two of the buckets, the table and the two remaining buckets were close to perfectly balanced, which further supports that conclusion.

 

[Hornbein]

I can't resist trying to come up with the simplest explanation.

Imagine that each weighted bucket were replaced with an eye bolt that the rope attaches to. It's pretty obvious the table won't go anywhere. The buckets are heavy enough to simulate eyebolts.

How heavy? The sum of the weight of the buckets must be greater than the weight of the table. Suppose you pile weights on the table. When the weight of the weights plus table is greater than the buckets' weight then the buckets will rise and the table fall. (Ignoring friction and assuming everything is perfectly balanced.)

 

[sophiecentaur]

I built one of these with one of my physics classes back in 2018.

I like it. Such good educational value. You can guarantee that pretty much EVERY student will be diverted and confused by that demo. And that's not something you could say about most of the things we show them!!

 

[renormalize]

Such good educational value. You can guarantee that pretty much EVERY student will be diverted and confused by that demo. And that's not something you could say about most of the things we show them!!

A tensegrity table would also impress students. (And it takes up less space!)