Hubcapitualtion

As we were driving into Boston today... I know I've started a lot of posts this way, so I apologize if you're getting sick of car-related stuff, but it's all I've been doing lately.  Anyway, as we were driving into Boston today, we were behind a van on the Massachusetts Turnpike that suddenly lost one of its hubcaps.  Rather than simply falling to the ground, the hubcap continued to roll alongside the van, slowly losing speed.  It made me start thinking about the stabilizing effects of spinning, and wonder how long it would take to eventually fall over and stop.

Let's start by defining some coordinates:

The hubcap rolls in the x-direction, spinning about the z-axis.  Its angular momentum is

where I is its moment of inertia and ω is its angular velocity.  If we approximate the hubcap as a thin, solid disk, we can write

where m is the mass of the hubcap.  If we also assume it rolls without slipping, we can replace ω with a linear velocity, v:

The hubcap won't be perfectly vertical when it pops off the wheel, so gravity will apply a torque, tipping it over.  The amount of torque will depend on how far the wheel is already tipped.  We'll call the angle off the y-axis that the hubcap has tilted θ.  Then the torque applied by gravity is

Torque and angular momentum are related by

so we'll need to generalize our previous equation for L.  Taking the tilt of the wheel into account, we have

Plugging this into the relationship above, along with our equation for τ gives the differential equation

We're not interested in the z-axis rotation, since this will remain constant, but picking out the x rotation,

Solving this gives

where C is some constant.  The hubcap will be on its side when θ = 90°, so solving for t in this case gives

We can put C in terms of the initial tilt of the hubcap with

Plugging this into the equation for t gives

To get an idea of how this varies with θo, we can make a plot:

 

Although the time can go to infinity with a perfectly vertical tilt, it quickly drops off with any slight error.  Going at highway speeds, a tilt of just 0.00001° will cause the hubcap to fall over within 23 seconds.