Sliding down over a horizontally moving semi-sphere

Abstract

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 θ ⋆, 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 θ ⋆ is 48.19° that coincides with the case of a fixed semi-sphere. On the opposite case, the minimum value of θ ⋆ is 0° 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.