How big would a clock have to be for the motion of the tip of its hour hand to be visible to the naked eye?
It’s going to depend on how close you are to the tip of the hour hand and if there are fixed references you can use to better detect motion. Say you can detect 1/10 of an inch per second motion. The hour hand takes 43200 seconds to go full circle, so a circumference of 4320 inches, or 360 feet would be needed, having a diameter of approx. 114.6 feet. If you can get close enough, and there are fixed references points on the clock face behind the hand, you could detect pretty slow motion. If the clock face was flat black (eliminating shadows), or if you are far enough away, the motion might need to be a lot faster. If you assume the entire clock face has to be in your field of view, it’s going to have to move faster, and probably end up larger than 114.6 feet.
I’m not sure how many clock hour hands travel at a constant (very slow) speed. Some advance and stop based on some gearing mechanism. Second hands are the same, they move smoothly or they advance in steps, but either movement is more apparent because it’s traveling at 60 timese the speed of the hour hand.
I have on my wall a clock that is just over 14" in diameter. (Actually, its 14" between the numbers, where the minute hand passes.) On this particular clock the movement of the minute hand is visible – it travels constantly, does not ‘click’ forward with each minute.
So. The minute hand on my clock travels 43.98226" per hour (14 X 3.14159).
Assuming you need this much distance of travel per hour for the hour hand movement to be visible, then the circumference would be 527.78712" (43.98226 X 12), or a diameter of 168", or 14’.
That is, the area encircled by the hour hand’s movement needs to be 14’ in diameter. I guess the face of the clock and the area covered by the minute hand can be much larger.
So, unless I screwed up my math, the answer is 14 feet in diameter.
Boyo Jim makes a very important point. The old synchronous motor electric clocks were about the only ones I can think of that move with a constant motion.
Modern crystal controlled clocks and watches tick once a second, and the movement of the minute hand is perceptible on ordinary wall clocks - which if scaled up would mean 12 times size to account for the rotational speed, and maybe an additional doubling for the shorter hour hand relative to the minute hand. So maybe 24 feet diameter.
My watch moves the hour hand once every 3 minutes, and the jump is visible. But it is an unusual design with a stepper motor whose only purpose is to drive that hand. This watch has five stepper motors, and can do some remarkable gymnastics with its hands. Most quartz crystal watches have a single stepper and a gear train for the various hands. Thus all the hands step once a second. Mechanical watches beat (half a cycle of the balance wheel) with frequencies ranging from 5 to 10 Hz.