A Clock Angles Problem

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Here is an e-mail inquiry  and my reply based on the Clock Angle program.  I recently posted an update to Clock Angles which animates the answer to the question posed. 



-----Original Message-----
From: 
Sent: Monday, October 18, 2004 8:28 AM
To: XXX @delphiforfun.org
Subject: Feedback


*******************************************************************************
Username:         val
UserEmail:       ..............
ContactRequested: ContactRequested


Date:             October 18, 2004
Time:             12:27

Comments:

how do you figure out how many times the hands are at a 90 % angle in a 24 hour period???? please respond, it's making me crazy!!
thanks

 

... "Spoiler " reply below  ...

 

 

Val.

The answer is 22 times each 12 hours.  Just like the earth revolves 366 times each year but we only see 365 sunrises, the hour hand makes one revolution in in 12 hours while the  minute hand revolves 12 times, but the hour hand only sees the minute hand 11 times because it creeps forward 1/12 of a revolution for each revolution of the minute hand.  And, since the 90 degree angle applies for minute hand 90 degrees ahead or minute hand 90 degrees behind the hour hand, the answer per 12 hours is doubled, or 22 times. 

Mathematically, we can calculate  the hand angles in terms of time expressed in hours and minutes (h:m).  Since the minute hand moves 360 degrees in 60 minutes, it moves 6 degrees per minute, and angle of minute hand (A
m) =6m.  The hour hand make a complete revolution in 12 hours (720 minutes).  So the hour hand moves 1/12 of a revolution(30 degrees) for each hour plus 1/720 of a revolution (1/2 degree) for each minute.  .  So the angle of hour hand (Ah)=30h + m / 2.  If we set   Am = Ah + 90 to calculate when the minutes hand is 90 degrees ahead, we get 6m = 30h + m/2 +90.  Solving for m, we get m = (60h+180) / 11.  At 12 o'clock (call it 0) we have to wait 16.36 minutes before the minute hand gets 90 degrees ahead, etc.  Between 8 and 9, the minute hand never gets 90 degrees ahead because by that time, it's 9 o'clock!.  If you make a table of values of m for each value of h, the value for h=8 will be m=(60x8+180)/11=660/11=60.  But when m=60, h=9 so no 90 degree solution for 8 o'clock..  Get it?  In 12 hours the minute hand is 90 degrees ahead 11 times.  Same logic applies for 90 degrees behind.  Between 2 and 3 o'clock, the minute hand is never 90 degrees behind, so again 11 times in 12 hours, making 22 times altogether in 12 hours. 

And of course, 44 times in 24 hours. 

Hope that all helps

Gary

Addendum Jun 23, 2006: Here's a table of results for each hour which may help a viewer who was confused recently because hour 9 did not return 0 minutes for the leading minute hand case.

.

Hour
Minute hand trailing    hour hand HH:MM time Minute hand leading hour hand HH:MM time
1 -10.91 12:49.09 21.82 1:21.82
2 -5.45 1:54.55 27.27 2:27.27
3 0.00 3:0.00 32.73 3:32.73
4 5.45 4:5.45 38.18 4:38.18
5 10.91 5:10.91 43.64 5:43.64
6 16.36 6:16.36 49.09 6:49.09
7 21.82 7:21.82 54.55 7:54.55
8 27.27 8:27.27 60.00 9:00
9 32.73 8:32.73 65.45 10:05.45
10 38.18 10:38.18 70.91 11:00.91
11 43.64 11:43.64 76.36 12:16.36
0 -16.36 11:43.64 16.36 12:16.36
12 49.09 12:49.09 81.82 1:21.82

Click here for the page where the program may be downloaded.

 

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