Math Problem

Discussion in 'Horological Misc' started by Tim Orr, Feb 13, 2020.

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  1. Tim Orr

    Tim Orr National Membership Chair
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    Good evening, all!

    I might have seen this somewhere, but can't remember if I did or where.

    There must be a formula for the 12 times that an hour hand and minute hand exactly coincide (High noon is one example). Or, on a 24-hour dial, the 24 times this occurs.

    Anybody know what the formula is or how to figure it out?

    Thanks!

    Best regards!

    Tim Orr
     
  2. Jerry Treiman

    Jerry Treiman Registered User
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    It's a bit late at night for me to try to work out a formula, but on a 12-hour dial the hands only match 11 times; for a 24-hour dial they only match 23 times. If you divide 60 by 11 that will give you how many minutes to add with each hour after noon on a 24-hour dial; divide 60 by 23 for the military dial.
     
  3. Ethan Lipsig

    Ethan Lipsig Registered User
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    Jerry's view that the hands coincide only 11 times on a 12 hour dial seemed odd to me at first blush, as did his apparent view that they would coincide approximately twice as often if there were twice as many hour markers on the dial and the hands rotated only once per day. Not being a mathematical sophisticate, I decided to run an analog test. So that Jerry would find the results credible, I used a 12-hour dialed Bigelow Kennard PL. Here is when the hands coincided:
    1. 12:00
    2. Slightly past 1:05
    3. Slightly past 2:10
    4. Slightly past 3:16
    5. 4:22
    6. Slightly past 5:27
    7. Slightly past 6:33
    8. 7:39
    9. Slightly past 8:44
    10. Slightly past 9:49
    11. Slightly past 10:54
    So, Jerry is correct that the hands coincide 11 times on a 12-hour dial, but I fail to see why they would coincide more or less often in one full rotation of the hands depending on how many hours are on the dial or on how long it takes for the hands to do one complete rotation. (The times at which the coincide during one full rotation would of course depend on the hour-marking of the dial.)

    If the question isn't how many times the hands coincide in one complete rotation, but rather how many times they coincide in a full day, then the answer would depend on how many rotations of the hands around the dial occur in 24 hours, but in this case, the hands on a 12-hour dial would coincide 22 times because there would be two complete rotations and the hands on a 24-hour dial watch would only coincide 11 times assuming the hands only do one complete rotation per day.

    I am no mathematician. Perhaps I am missing something.
     
  4. novicetimekeeper

    novicetimekeeper Registered User

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    " but I fail to see why they would coincide more or less often in one full rotation of the hands depending on how many hours are on the dial or on how long it takes for the hands to do one complete rotation. (The times at which the coincide during one full rotation would of course depend on the hour-marking of the dial.)"

    Funny it made sense to me. On a 12 hour dial in one rotation of the hour the minute rotates 12 times, in a 24 hour dial it rotates 24 times. I think Jerry was saying how many times it would match for a complete rotation of the hour hand.
     
  5. Ethan Lipsig

    Ethan Lipsig Registered User
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    Novicetimekeeper, you've nailed the fallacy that tripped me up -- my assumption that the hands would operate in the same relation to each other on one complete rotation no matter how many hours that rotation covered. In fact, I now recognize that while the minute hand completes 360 degrees of travel every hour on every normal watch, the hour hand's degree of travel depends on how many hours the hour hand takes to make a complete revolution. To state the most obvious case, if one had a one hour dial, the minute and hour hand would always both travel 360 degrees in an hour. This explains why the hands on a 24-hour watch would align themselves more often than on a 12-hour watch.
     
  6. Dick C

    Dick C Registered User

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  7. rmarkowitz1_cee4a1

    rmarkowitz1_cee4a1 Registered User
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    E=mc2?

    RM
     
    4mula1fan likes this.
  8. glenhead

    glenhead Registered User
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    #8 glenhead, Feb 13, 2020
    Last edited: Feb 13, 2020
    3600 seconds / 11 sections = 327.272727... seconds

    327.3/60 = 5 remainder 27.3

    5 minutes 27.3 seconds if you're looking for the time elapsed.

    Move the decimal a notch to find the elapsed angle, 32.73 degrees.

    Multiply by pi and divide by 180 to get 0.571 radians.

    Glen
     
  9. Tim Orr

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    Good evening, all!

    Many thanks to everyone! This has been puzzling me for some time.

    Best regards!

    Tim Orr
     
  10. novicetimekeeper

    novicetimekeeper Registered User

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    I have quite a few single handed clocks, so it was probably more obvious to me, I'm no mathematician.
     

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