A Tense Moment

A few days ago, I was sitting on the balcony at my parents' condo, and I noticed a water droplet on the leaf of their banana tree.

 

Seeing it got me thinking about surface tension, the force that gives it the characteristic shape that small drops have.

A simple way to think about surface tension is this: each molecule of water has a certain amount of attractive force between it and its neighbors.  However, at the surface, there are no molecules on one side, so the force associated with the other molecules is stronger.  Here's a diagram (from Wikipedia):

When a droplet like this sits on a solid surface, it will be spread out slightly by the force of gravity.  The height is given by

where ϒ is the surface tension between air and water, θ is the angle the water's surface makes with the leaf, and ρ is the density of water.  Plugging in the values we have, h = 4.1 mm.  Unfortunately, my photo doesn't include any scale, so I can't check how close this is, but it sounds reasonable.

I'll be leaving for Michigan in a few days to start grad school, so there will be a pause in posting while I move in.

Wheelchair Wheelies

Recently, my mother Sally bought her father a wheelchair.  It reminded me of something I've often wondered about wheelchairs: what would it take to go up on the back two wheels?  You could manage it by accelerating forward rapidly, since the wheels would essentially leave the rest of you behind, but how fast would you need to accelerate?

There are two main parameters we need to consider: y, the vertical distance between the center of mass and the rear wheel axis; and θ, the angle between those two points.  They are related by

where r is the distance between the two points.  Differentiating this twice gives

where dots indicate time derivatives.  We know that the CM's acceleration will come from the forward acceleration of the wheels, and from gravity, so we can write

where a is the forward acceleration.  Combining these gives

This is a pretty complex differential equation.  I couldn't come up with an analytical solution, so I decided to do it numerically.  For that though, we'll need some values.  I found a helpful diagram on a Texas Accessibility Standards page, and picked out the relevant bits (measurements in mm):

I plugged in these numbers, along with a guess for a, and put it through WolframAlpha for a numerical solution.  Tweaking a to find when the wheelchair lifts, I found an acceleration of about 30 m/s^2.  To get a sense of scale, that's about the same acceleration involved in going 0 to 60 mph in 1 second.  That would be pretty tough; I think people who do this trick press back on the seat to help.

I'm still feeling pretty crummy; yesterday I had minor surgery to have my vascular access port removed, so expect slow posting in the near future...

Suction Cups

Sorry, no clever title this time; I've been feeling pretty crummy the last few days, so I haven't been as active here as I'd like.  Last time I promised a little talk about suction cups, so here we go...

Suction cups work by creating a low-pressure (or ideally vacuum) chamber, which is then maintained by the outside air pressure.  We'll make a few simplifying assumptions about the suction cup: it stays perfectly conical, and its surface area remains the same.

The surface area of the cone is given by

or, solving for r^2,

Meanwhile, the volume is

Combining these and simplifying gives

Plotting this gives

 

The maximum on the plot will be the height with the smallest internal pressure (volume and pressure are inversely related).  That means that after pressing a suction cup down to evacuate as much air as possible, pulling it up slightly will create a stronger grip.

I'm actually feeling a bit better having written this up, so maybe I just needed more brain-exercise...

Depressing Sleeping Arrangements

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...

Extruded Plastic Dingus

(Title from a favorite movie of mine, The Hudsucker Proxy)

While I was visiting my brother Nate, we went to the Baltimore Farmers' Market, where a local performance group had brought a large selection of hula hoops.  He felt the need to show up our mother, who had also come across a group lending out hula hoops back here in Northampton.  I've never been any good with them, but I thought I'd explain a bit about how they work.  I had a basic understanding, but I figured I should try to read up a bit on the details.  Since I'm between schools, I no longer have access to the pricey journal databases, but I did manage to find a paper on the free physics article site, arXiv.  As it turns out, the dynamics involved are rather complex, so I thought I'd try to give a basic explanation and highlight some interesting aspects.

The article presents the situation with this diagram:

The large gray circle is the hula hoop, and the smaller filled circle is your waist.  The dotted ellipse represents the hip movements that keep the hoop going.  The important factors that determine the movement of the hula hoop are: the friction between you and the hoop, the shape that you move your hips in, the distance that you move your hips, and the size of you and the hoop.  Rather than get into the details, I'll just summarize some of the article's more interesting conclusions:

• More friction or a large hoop requires you to move your hips more
• Low friction allows you to twirl the hoop in the opposite direction that you swing your hips
• Theoretically, there are unstable solutions to the hoop's movement, meaning if you were able to get everything perfectly lined up, it would work, but in reality, you'd never manage it

I'm not sure any of this has helped my ability to actually hula hoop, but at least I know how it works now, and to a theorist, that's all that really matters.  As always, you're welcome to ask for more details about anything that piques your curiosity in the comments.

Doff Your Cap

While I was on my trip, I carried a water bottle with me everywhere to survive the heat.  It had the typical screw top, and I wondered whether squeezing the bottle could cause the top to unscrew, or if the friction would be too great.

There are two forces acting on the cap when you squeeze the bottle: the increased air pressure in the bottle, and the normal force from the screw threads.  We can diagram them like this:

where A is the area of the bottle cap, p is the pressure increase in the bottle, N is the normal force applied by the threads, and θ is the angle the threads are tilted at.  The magnitude of N is equal to the force applied directly against the threads, so we can write

Using this, we can find the torque on the cap applied by friction.

where r is the radius of the cap, μ is the coefficient of friction, and we have assumed that A is given by the usual area of a circle.

In addition to the frictional torque, the tilt of the normal force will also apply a torque, given by

Combining these into a net torque gives, with some rearranging,

This says that any amount of pressure will cause the cap to unscrew, as long as

Plugging in a typical coefficient of friction for plastic, 0.2, gives an angle of about 11.5°.  Looking at the water bottle I was using,

 

we can see that although my bottle would not work, one with only slightly steeper threads could.  I think it's interesting that the pressure doesn't factor in – as long as the angle is correct, any amount of pressure can make the cap unscrew.

Light Rail

I have returned from my trip; big thanks to everyone who hosted me, and showed me a wonderful time.  I have a couple post ideas kicking around from my travels, but I thought I'd take a minute to give a quick primer on the special relativity issues I mentioned last time.

Special relativity rests on the idea that the speed of light is constant.  This may not seem like such a significant statement, but you have to consider that it means the speed is always constant.  Normally if you were chasing after something, all you need to do is go faster than it, and eventually you'd catch up.  However, if you tried that with light, no matter how fast you went, its speed would always appear to be the same amount faster than you.  This leads to some interesting effects.

Imagine we construct a clock that uses light as its timekeeper – it contains a hollow tube with a beam of light that travels down it, bouncing off each end; each bounce is a tick.  It might look something like this:

 

Now suppose we strap this clock to a rocket, and watch as it flies by.  It would look something like this:

 

The light is still going at the same speed, but it has to travel a greater distance, so the tick that we measured earlier has been made longer by the movement.  This is the relativistic effect known as time dilation.  Most other interesting effects follow from similar thought experiments.

One of the important restrictions of special relativity is that it only applies in an inertial frame, that is, a perspective which is moving at a constant velocity.  Technically, it doesn't quite apply to the situation I talked about last time, since my train had to accelerate up to speed, then slow down again, but we can adapt it as long as we're careful.

While on the train, clocks outside will appear to be running slower than my own, but since I accelerate before and after such observations, they are potentially invalid.  However, someone outside looking at my clock would be correct in noting that it appears slower than theirs.  Since the person outside stays still during my entire trip, their observations must be correct.  If the train takes a time t according to the stationary person, then they will see a time t/γ pass on my clock.  Plugging in the numbers from last time gives the 95 picosecond difference I mentioned.