Sliding down over a horizontally moving semi-sphere
Roberto A. Lineros

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 . 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 , 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: and , and the initial velocity of the sliding object . 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 is that coincides with the case of a fixed semi-sphere. On the opposite case, the minimum value of…
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