A Ball in a Groove

We study the statics of an elastic sphere held in a rigid groove by gravity and friction, as determined by contact mechanics. As a function of the groove's opening angle and its tilt with respect to the vertical, we identify two regimes of static equilibrium for the ball. At large opening angle or small tilt, the ball rolls at both contacts as it is loaded. This is an analog of the ``elastic'' regime in the mechanics of granular media. At small opening angle or large tilt, the ball rolls at one contact and slides at the other as it is loaded, analogously with the ``plastic'' regime in the mechanics of granular media. In the elastic regime, the stress indeterminacy is resolved by the kinetics of the response to loading.