The point of departure of a particle sliding on a curved surface

TL;DR

This paper investigates the conditions under which a particle thrown tangentially on a curved surface leaves or remains on the surface, analyzing specific surface shapes and the influence of friction.

Contribution

It introduces a detailed analysis of particle departure conditions on curved surfaces, including special surface shapes and the effect of friction.

Findings

Particles can leave or stay on surfaces depending on initial velocity and surface shape.

Specific conditions for departure are derived for surfaces with equations like y= -αx^k.

Friction effects alter the departure conditions.

Abstract

A particle is thrown tangentially on a surface. It is shown that for some surfaces and for special initial velocities the thrown particle leaves immediately the surface, and for special conditions it never leaves the surface. The conditions for leaving the surface is investigated. The problem is studied for a surface with the cross section $y=f(x)$. The surfaces with the equations $f(x)= -\alpha x^{k}\ (\alpha, k>0)$ is considered in more detail. At the end the effect of friction is also considered.