You need to mark the center of whatever motor is causing the ball to rotate and/or whatever motor is causing the entire shell (housing the motors) to rotate. Note that in every case, the only reason the ball is turning rather than moving linearly, is because it is pulling against the center of its associated motor and not able to overcome the inertial contest, thus the ball can’t maintain it linear vector. The linear momentum of the ball is overcome by its bind to the inertia/momentum of the motor. The amount of momentum taken from the ball such as to cause it to turn, is given to the motor. As the ball increases its downward vector, the motor center increases its upward vector.
Up to that point, the goal is being achieved. The motor and its housing are gaining momentum in the upward direction. When the ball reached its lowest point, the center of the housing/motor will have shifted upward. The vibrating motor in your cell phone functions that same way as it jerks the entire housing one direction and then back again with a lopsided motor.
The catch comes in when it attempts to get back into the same condition in which it started. The ball must return to being on top, but without pulling the motor/housing downward.
IF the ball is free from affecting the motor then the ball will not be affected by the motor either, in which case, the ball will take a linear course off to the right side and leave the system. The momentum gained by the initial half cycle rotation will independently continue for both the ball and the motor, as the motor/housing continues to move upward and to the left. But such is not the desired result in that the ball gets lost and doesn’t cycle back to where it started relative to the housing. Most men are aware that they really don’t want their balls to leave their housing.
In order to get the ball back up to the top position, another contest of inertia must be imposed on the ball’s linear traveling to the right. The motor must pull the ball upward. The motor need not apply energy, but merely restrain the ball’s inclination to leave the system entirely. But of course in the process of pulling the ball upward, the motor gets pulled downward. This is not due to any energy being applied, but merely due to the attachment between the ball and the motor center or housing. This yields an effect counter to the goal.
The rub comes in when you realize that it is the exact same amount of inertial trade initially exchanged to get the ball to its lowest position that now must be applied to get that same ball to back to its original position. Thus the exact same amount of momentum that had been given to the motor, must be used up by the motor so as to return the ball back to the original position.
The energy being applied during the first half-cycle is for the purpose of causing the momentum exchange. During the second half-cycle, that energy is already in the balls momentum and thus need not come from the motor. But don’t confuse the energy cycle with the momentum cycle. The energy cycle between the motor and the moving parts is entirely different than the momentum cycle between the housing and the ball. In the second half-cycle, the energy that is in the ball, rather than from the motor, is used to cause another momentum exchange, but this time, it is reverse of the desired goal as the ball insists on moving out of the system, and the motor or housing insist that it turn upward - the exact same amount of inertial contest that started the turning in the first place.
In short;
The ball turned downward due to the inertial contest between it and the motor/housing, inspired by the motor.
The ball returned to the top due to the inertial contest between it and the motor/housing, inspired by the ball.
From where the energy came and where it is located within the system at any one time is irrelevant to the issue of total momentum.
Rather than work with the force and energy equations, look merely at the momentum equations for the ball and the housing/motor.
momentum = mass x velocity.
You can ignore the angular confusions if you want (I think) and look merely at the vertical vector components. Quickly it should be apparent that the only time the housing can gain any momentum is by taking it from the ball and vsvrsa. But during the first half the vector will always be positive and during the second half, it will always be equally negative.