The Rigid Sphere and The Ant

• A rigid sphere of mass $M$, radius $a$, and isotropic moment of inertia $I = \beta M a^2$ has an ant of mass $m=\alpha M$ moving on it; otherwise, the two objects are freely moving and rotating in space. The ant is constrained to move on the surface of the sphere, but can choose its path on the sphere arbitrarily. Let’s assume that it follows a path determined by $\theta(t)$ and $\varphi(t)$ coordinates on the sphere where $\theta(0) = \theta_0$ and $\varphi(0) =0$. In the beginning, the ant is not moving relative to the sphere, and the sphere is rotating with $\boldsymbol{\omega}(0) = \omega_0\hat{\mathbf{z}}$. The position of the center of the sphere at time $t=0$ is chosen as the origin of the coordinates. You can express the results in the system of units where $M = a = \omega_0 = 1$.
  1. Let the rotation matrix $R(t)$ define the net rotation of the sphere from the initial time to some arbitrary time $t$. Find an equation that describes the evolution of $R(t)$.
  2. If the surface of the sphere is frictionless, then the ant has no control over its motion; its position will therefore be a dynamical variable and not an input of the problem. Find equations governing both $R(t)$ and the path $\theta(t)$, $\varphi(t)$.
  3. Simplify the dynamics in part (1) by keeping only the first order terms in $\alpha$.
  4. Estimate the change in the length of a day due to the introduction of road transportation. Note that in most countries, cars drive on the right. Can road transportation change the seasons? How long does it take for it to do so? 😃

Solution notes (PDF)