Drifting Apart

I'm finally settled here in my new home of Ann Arbor, MI, although last night I lost power in a large thunderstorm, and it's not due back until Monday night.  I'm sitting in a local coffee shop with nothing much to do, so I figured I'd end my silence here by talking about an interesting feature of my new apartment.

I've found that the floor where I have my computer desk is slightly sloped, which sometimes causes my wheeled office chair to drift away from the table.  What's interesting is that it only happens under certain circumstances.  In order to roll back, the wheels must be pointed close enough to the direction of the slope that gravity, the force rolling it down the slope, can overcome the rolling friction created by imperfections in the wheels and the floor.

We can define an effective slope for the floor based on the direction relative to the full slope that the wheels point.  If we call the full slope θ and the deviation from the straight backward direction φ, then the effective slope is

Using this, the force exerted by gravity is

where m is the mass of the chair (plus anyone in it).  Frictional forces generally depend on the normal force between two surfaces.  In this case, that's

Plotting these together as functions of φ gives something like

 

where the green line is F and the red line is N.  Anywhere the green is over the red, the chair will roll.  Varying θ changes the crossing point; for sufficiently small angles, the chair will not move at all.  Also note that frictional forces are usually some fraction of N, so the red line will likely be lower than shown.

With luck, my power will be back on sooner than scheduled, but I'll try to keep posting regardless...

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...