Can't Sleep

See? Math, in Real Life

Mr. Toots
Capitola, CA
August 30, 2014
approximately 9:30pm

Question:

At what times of day will the hands of a clock be normal (form a right angle)?

Assume the clock is a twelve-hour clock.

Solution:

The minute hand of the clock sweeps through 2π radians per hour, or 3600 seconds.

The hour hand of the clock sweeps through 2π radians per 12 hours, or 43,200 seconds.

The angle φ formed by the two hands at time t seconds elapsed from 12 midnight is:

φ = (2π/3600)t – (2π/43200)t =
(1/3600 – 1/43,200) 2πt =
(11/43,200) 2πt =
(22πt/43,200)

The angle φ formed by the two hands will be a right angle whenever φ is an odd multiple of π/2:

φ = (22πt/43,200) = (2k+1)π/2 for k = 0, 1, 2, 3, …

Solving for t we have:

t = (2k+1)*43,200/44 for k = 0, 1, 2, 3, …

Another way of looking at this is to understand that the hands will be normal at every interval of 1963.62 seconds from any time where the hands were normal. So beginning at 3am, the hands will be normal at 3am + 1963.62 seconds or 3:32:44am

k φ t (seconds) Time
0 π/2 981.8181818 12:16:22 AM
1 3π/2 2945.454545 12:49:05 AM
2 5π/2 4909.090909 1:21:49 AM
3 7π/2 6872.727273 1:54:33 AM
4 9π/2 8836.363636 2:27:16 AM
5 11π/2 10800 3:00:00 AM
6 13π/2 12763.63636 3:32:44 AM
7 15π/2 14727.27273 4:05:27 AM
8 17π/2 16690.90909 4:38:11 AM
9 19π/2 18654.54545 5:10:55 AM
10 21π/2 20618.18182 5:43:38 AM
11 23π/2 22581.81818 6:16:22 AM
12 25π/2 24545.45455 6:49:05 AM
13 27π/2 26509.09091 7:21:49 AM
14 29π/2 28472.72727 7:54:33 AM
15 31π/2 30436.36364 8:27:16 AM
16 33π/2 32400 9:00:00 AM
17 35π/2 34363.63636 9:32:44 AM
18 37π/2 36327.27273 10:05:27 AM
19 39π/2 38290.90909 10:38:11 AM
20 41π/2 40254.54545 11:10:55 AM
21 43π/2 42218.18182 11:43:38 AM
22 45π/2 44181.81818 12:16:22 PM

All PM times after 11:43:38 AM will be the same as the any AM time plus 12 hours.

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