A sphere moving down the surface of a static sphere and a simple phase diagram

TL;DR

This paper analyzes the motion of a small sphere on a larger static sphere, deriving a phase diagram that categorizes pure rolling, slipping, and detachment states based on static friction and initial conditions.

Contribution

It introduces a simple phase diagram delineating different motion regimes of a sphere on a sphere, considering static friction and initial conditions.

Findings

Derived critical angles for detachment with and without friction.

Identified phase boundary between pure rolling and slipping.

Presented a phase diagram mapping motion states based on static friction.

Abstract

A small sphere placed on the top of a big static frictionless sphere, slips until it leaves the surface at an angle $\theta_{l}=\cos^{-1}{2/3}$. On the other extreme, if the surface of the big sphere has coefficient of static friction, $\mu_s\to\infty$, the small sphere starts rolling and continues to do so until it leaves the surface at an angle $\theta_{l} =\cos^{-1}{10/17}$. In the case where, $0\leq\mu_s<\infty$, we get a simple phase diagram. The three phases are pure rolling, rolling with slipping and detached state. One phase line separates pure rolling from rolling with slipping. This diagram is obtained when stopping angles for pure rolling are plotted against static friction coefficients $\mu_s$. Study in this article is restricted to the case when the mobile sphere starts at the top of the static sphere with infinitesimal kinetic energy.