My sister Rachel has been visiting for the past few days (hence the long silence), and I gave up the guest room to her and her husband Dave. Instead, I've been staying in the living room on an inflatable mattress. I noticed that sitting on the bed makes me sink much lower than lying on it, so I thought I would look into exactly what the relationship is.
We'll assume a simple shape for the depression, like this:
Where
d is the depth sunk, and
A is the area that the weight is spread over. We can describe the situation with three equations:
where
m is the mass of whatever is on the bed,
V is the volume of air in the bed,
p is the air pressure in the bed, and subscripts
i and
f indicate values before and after the weight is placed on the bed. Combining these and solving for
d gives
A couple interesting things about this equation:
- It is possible to spread out enough that you don't sink at all.
- If you're infinitely heavy, the negative term goes to zero. At first this surprised me, since it seemed to say there was a maximum amount you could sink, but looking at the leftover term, you'd be crushing the mattress flat.
- Even with the same pressure, a larger mattress will make you sink more.
Thinking about pressure made me wonder about the ideal suction cup. I'll try to get something up about that tomorrow...
Being able to spread out enough to not sink doesn't sound right to me. Are you assuming that the sides of the bed are perfectly rigid?
ReplyDeleteNo, it is possible. The reason you sink is that the pressure isn't great enough to hold you up, but sinking increases the pressure until it is. If you're light enough, or spread enough, the pressure already in the mattress can be enough.
ReplyDeleteIf you try to make the sinking depth vanish, you get p_i=mg/A, which is equal to p_f by your first equation. Isn't p_i=p_f equivalent to having nothing at all on the mattress? If you put a mass on the mattress and the pressure doesn't increase, you don't have a balance of forces.
ReplyDeleteI think Nate's intuition is right. The only way to get the extra pressure needed to support an object is by reducing the volume of air in the mattress, as per p_i V_i=p_f V_f.
Actually, p_i=p_f just means you don't sink, which is consistent with what I was saying earlier. Theoretically, the mattress could have the same pressure as the surrounding air, and stay inflated, but since we put extra air in, I'm pretty sure it should support some weight without deforming.
ReplyDelete