Sliding down over a horizontally moving semi-sphere

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 $m_B$. 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.

Comment on ‘Sliding down over a horizontally moving semi-sphere’

“…Incidentally, the reason why the q * is determined only by the ratio of m A /m B = 1/(γ −1 − 1) and T 0 /m A Rg = ò is that q * is dimensionless, and there are only two independent dimensionless variables that can be constructed out of the parameters of this system. In particular, when T 0 = 0, the answer can only depend on m A /m B and not on g and R, as noted in the Conclusions of [1].…”

“…(Some figures may appear in colour only in the online journal) Lineros [1] studied a frictionless system where an object A of mass m A slides on a movable semi-sphere B of radius R and mass m B , as shown in figure 1, and determined the condition when the object A loses contact with the semi-sphere. The derivation presented in this paper was somewhat complex, and relies among other things on obtaining the solution of a differential equation.…”

“…In this comment, I will employ the notation used in [1]. The angle that object A makes with respect to the vertical line at the center of the semi-sphere is q, and the angular speed of object A (with respect to semi-sphere B) is ω (  q = ).…”

“…Defining q * to be the angle when the block loses contact with the semi-sphere, at that angle the normal force of the sphere on the block N A (q * ) = 0 so equation (2) gives a Bx (q * )= 0. Hence equation (1) gives…”

“…⎛ ⎝ ⎞ ⎠ To reproduce the result in [1], we need to relate ¢ T 0 to T 0 , the kinetic energy of the block in the 'laboratory' frame. For this we use (see e.g.…”