The angular rate of rotation of clocks N and L about this line are identical. The angular momentum of each can be made to be identical by adjusting the mass of N up and the mass of L down until they are. Thus, there is identical angular momentum for each clock and no relative motion between them. Therefore there is no change in the synchronization of these clocks caused by either. The angular momentum of the two are different, but there is no relative motion.
Lets consider a clock on the edge of a a merry go round and another on the ground near by.
The two clocks also have differences in angular momentum. This is not exactly correct because the effect of acceleration in the box would be the same in all parts of the box but the effect of gravity would be slightly greater where the side of the box is closet to the gravity producing mass. However, let us assume for the sake of study that the effects of acceleration and gravity are the same. A person at the North Pole constantly feels the effect of being accelerated straight up at the rate of about 32 feet per second per second. Accelerating at 32 feet per second per second will get you to the speed of light in about a year's time.