Re: Sfera in una doppia guida
Inviato: 9 set 2023, 5:36
According to the condition of the problem, the ball rolls without slipping, therefore, the speeds of those points of the ball that at a given moment of time touch the metal at AB (Fig.) are equal to zero. Considering the ball to be an absolutely rigid body (that is, the distance between any two points of the ball is unchanged), we conclude that at a given moment in time all points of the ball lying on the segment AB are motionless. And this means that at each moment of time the movement of the ball is a rotation about the axis AB. (It is clear that points A and B - the points where the ball touches the metal - are moving with the speed .)
The instantaneous velocity of any point of the ball is , where is the angular velocity of rotation, is the distance from the point to the axis AB. The speed of the center of the ball (point O in the figure) is equal to ; the distance from point O to axis AB is . Hence,
It is clear that the points of the ball most distant from the axis AB have the maximum speed. From geometric considerations it is clear that at any moment there is only one point that is maximally distant from the axis - in the figure this is point Q. The distance from point Q to the axis of rotation is , and speed of Q is:
The instantaneous velocity of any point of the ball is , where is the angular velocity of rotation, is the distance from the point to the axis AB. The speed of the center of the ball (point O in the figure) is equal to ; the distance from point O to axis AB is . Hence,
It is clear that the points of the ball most distant from the axis AB have the maximum speed. From geometric considerations it is clear that at any moment there is only one point that is maximally distant from the axis - in the figure this is point Q. The distance from point Q to the axis of rotation is , and speed of Q is: