One particularly neat thing about the Ringworld is that it’s a great way to introduce the concepts of angular momentum, moments of inertia and the like. Force, mass, acceleration, momentum, velocity, and kinetic energy each have analogous terms in rotational mechanics – torque, moment of inertia, angular acceleration, angular momentum, angular velocity and angular kinetic energy. You can solve problems using either set of terms, but usually, for any particular problem it’s much easier to do the work in one set of terms than in the other. Ringworld, however, is different – it’s nearly as easy to calculate the kinetic energy of the Ring (due to its rotation) as the mass of the Ring times 770 (miles/second)2, or to calculate the moment of inertia of the Ring around Axis 1, and calculate the kinetic energy as the moment of inertia times the angular rate of rotation squared. The Ring is stable around its usual axis of rotation – any slight wobble will not tend to grow. If the Ring was rotating around one of its other axes, wobbles would tend to grow until the Ring was tumbling.
The moment of inertia for Ringworld is approximately 4.5x1049 kg*meters*meters, and the kinetic energy associated with the rotation of RW is therefore 1.5 *1039 kg*(meters/second)2. This amount of K.E. is equivalent to a rest mass of a quarter of the moon. For comparison, one gallon of gasoline masses about 3 kg, and thus has a rest energy of 2.5*1017 kg*(meters/second)2 which is enough energy to accelerate a 1000-kg automobile to more than 5% of the speed of light, so imagine the energy in one-quarter of 7.35 ×1022 kg
By the way, this talk of angular dynamics raises the issue of another kind of stability – the stability of a rotating object with respect to small perturbations in the spin axis. It turns out that because the Ring is rotating around axis 1, it is stable with respect to perturbations of that axis – if you tried to spin the Ring around one of the other axes, it would wobble and eventually end up rotating around axis 1 (and I wouldn’t want to be living on it during the transition!)