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.