It is well known that a stopped clock gives the exact time twice a day, while a clock that gains or loses time may not be right more than once over a period of months.

My clever father adjusted his clock to give the correct time at least twice a day, while running at the normal rate.

Assuming he was not able to set it perfectly (a reasonable assumption) how did he do it?

The father could have added a periodic motion to the general motion--say in the form of a sine function with an amplitude of say 3 minutes either side of the normal time.  So, while during the day the clock would initially be as much as 3 minutes ahead and as much as 3 minutes behind at some times during the day, it would pass through the correct time a couple of times also, and its average motion would indeed be the normal rate.

It would keep being right twice a day (or more if the cyclic interval were shorter than one day), until the inaccuracy of the "normal rate" brought the 6-minute span completely outside the true time moment.  Then the clock would need to be reset, but this is just like normal clocks that need to be reset once in a while when they deviate by too much from the actual time.

Posted by Charlie on 2005-06-24 13:20:27