Posts Tagged math

Showing My Math

So here’s the deal. I’m working on a new project and I want to make sure at least some of the details have actual technical grounding. I’m okay with a little handwavium, it’s probably unavoidable, but I want to at least have some grounding in reality. Problem is, I’m not all that great at actual technical grounding, as the last physics class I took was non-AP physics in high school where I barely got a B. So I might occasionally make these posts, I might make just this one, in an attempt to crowdsource some of my equations. The questions I have are whether I’ve got the right equations, and whether I’m using them correctly, and to also play around with LaTeX a little. But mostly the first two things.

So here’s today’s problem: Given a cylindrical space ship with an internal radius of 6km, how fast must it be rotating to create a centripetal acceleration equivalent to earth gravity for someone standing on the inside surface? I didn’t know any of these equations, but found them at this rather helpful forum post. First, we must find the speed at that 6km point that would produce an acceleration of 9.8m/s²:

a=\frac{v^2}{r}
a=9.8 \frac{m}{s^2}
r=6000 m
9.8=\frac{v^2}{6000}
9.8*6000=v^2
58800=v^2
v=242.49 \frac{m}{s}

That number sounds awfully damn fast, but consider the speed of rotation of the earth at sea level on the equator is roughly 465 m/s. Next step, at least what I’m assured is the next step, is converting this into radians/second:

W=\frac{v}{r}
v=242.49
r=6000
W=\frac{242.49}{6000}=0.0404

Finally this can be converted to revolutions per minute. The conversion formula I found is:

1 \frac{rad}{s} = \frac{60}{2pi} = \frac{30}{pi} rpm
0.0404 \frac{rad}{s} = \frac{30*0.0404}{pi}=\frac{1.212}{pi}=0.386 rpm

Therefore the ship is rotating at a rate slightly faster than once every three minutes. What I didn’t expect is that, since the rate of rotation is a constant, centripetal acceleration increases linearly from the axis of rotation. I’m so accustomed to formulas for gravity having squares all over the place, but this isn’t, strictly speaking, gravity. It’s an acceleration equal to gravity. So at half the distance from the axis of rotation, we can work backwards with W as a constant…

W=\frac{v}{r}
0.0404=\frac{v}{3000}
v=0.0404*3000=121.2\frac{m}{s}

a=\frac{v^2}{r}
a=\frac{121.2^2}{3000} = \frac{14689.44}{3000} = 4.9 \frac{m}{s^2}

Which is equivalent to half gravity.

My next trick will be to find a formula that describes the rate of descent for a body falling through linearly increasing gravity. That’s less likely to come up in-story, but more for my own curiosity.

Edit: Some further poking around (which, I’m ashamed to say, has mostly been at Wikipedia so far) suggests that 2rpm is about the maximum rotation that most humans can adjust to with no ill effects, so my rotation of nearly 1/6 that rate is shockingly safe in and of itself. So that’s good to know. Now if only it didn’t have a “citation needed” tag.

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Two Quick Parenting Tips

Yesterday my daughter got her first batch of vaccinations, because we’re strong believers in the power of herd immunity. Go vaccines! When all the shots were done my wife was told the baby might suffer some discomfort or feverishness, and if she did to give her some Acetaminophen (aka Tylenol). We got a handy little chart showing recommended dosages for two available concentrations of Acetaminophen, 32 mg/1 ml and 160 mg/5 ml. The dosages for the former are in milliliters, for the latter in teaspoons.

I hadn’t seen this chart before I hit the first pharmacy yesterday. I was acting only on a text from my wife asking me to pick up some children’s liquid Acetaminophen. So I go to Target, get the box marked “infant” rather than “2-11 years” and pick the grape flavor because they all taste the same flavor of chemical sweet yuck anyway. I bring this box home, show it to my wife, and get sent immediately back out because the chart lacked 0-5 month dosing information for the 160/5 concentration, suggesting only only get the 32/1 concentration.

Some people already know where this story is going.

Out I go. First stop is CVS, the nearest drug store to the house. My logic is that they would have a better selection of children’s medications than a grocery store. And they do. They, in fact, have children’s Acetaminophen in two locations in the store, but only in the 160/5 concentration. Nowhere do they have 32/1. At this point I bail and head towards Babies R Us. Surely they would have a full selection of medications in any and all concentrations. Again, no 32/1, only 160/5.

Now I’m in a panic, but I finally examine the chart more closely. Looking at the different dosages, specifically down the chart where it says children ## months and up should get 5 ml of 32/1 or 1 tsp of 160/5. I pull up the calculator on my phone. The nI pull up the unit converter. And then I go home after an hour of panicked wild goose chasing. So if I can offer two quick bits of parenting advice so that you can learn from my mistakes:

160/5=32.

1 tsp ≈ 5ml.

Right down the chart it gave dosages in two different units that are a rounding error apart. For two concentrations that were actually the same concentration, just one with the fraction reduced. Sadly, I was in a panic because I was certain my baby was in pain, and the math portion of my brain utterly failed me. But…that’s fatherhood, I suppose.

Next week I start two weeks of stay-at-home fatherhood as I bridge the gap between my wife’s maternity leave and the baby starting day care. I’m hoping to get some work done on two short stories during her various naps, and perhaps start in on a secret project running around the back of my head. It involves the World Building Earth series, which is the most I’m comfortable saying right now as it may not even happen.

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