Sliding down over a horizontally moving semi-sphere

TL;DR

This study analyzes the dynamics of an object sliding down a semi-sphere that can move freely, revealing how the last contact angle depends on mass ratios and initial conditions, with specific bounds identified.

Contribution

It provides a theoretical analysis of the last contact angle in a frictionless semi-sphere system, showing its dependence solely on mass ratios and initial energy conditions.

Findings

Last contact angle depends only on mass ratio

Maximum angle is 48.19°, matching fixed semi-sphere case

Object detaches at the top when initial kinetic energy is half potential energy

Abstract

We studied the dynamics of an object sliding down on a semi-sphere with radius $R$. We consider the physical setup where the semi-sphere is free to move over a flat surface. For simplicity, we assume that all surfaces are friction-less. We analyze the values for the last contact angle $\theta^\star$, corresponding to the angle when the object and the semi-sphere detach one of each other. We consider all possible scenarios with different combination of mass values: $m_A$ and $m_B$, and the initial velocity of the sliding object $A$. We found that the last contact angle only depends on the ratio between the masses, and it is independent of the acceleration of gravity and semi-sphere's radius. In addition, we found that the largest possible value of $\theta^\star$ is $48.19^{\circ}$ that coincides with the case of a fixed semi-sphere. On the opposite case, the minimum value of $\theta^\star$ is $0^\circ$ and it occurs then the object on the semi-sphere is extremely heavy, occurring the detachment as soon as the sliding body touches the semi-sphere. In addition, we found that if the initial kinetic energy of the sliding object $A$ is half the value of the potential energy with respect to the floor. The object detaches at the top of the semi-sphere.