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Re: solving a periodic function
*To*: mathgroup at smc.vnet.net
*Subject*: [mg114005] Re: solving a periodic function
*From*: Fred Klingener <gigabitbucket at BrockEng.com>
*Date*: Sat, 20 Nov 2010 06:14:36 -0500 (EST)
*References*: <ic34rb$5s7$1@smc.vnet.net>
On Nov 18, 7:06 am, Michael Stern <nycst... at gmail.com> wrote:
> I have a fun little puzzle that I solved via other methods, but would
> like to know if there might have been a more elegant method via
> Mathematica.
>
> I wanted to figure out the times during the day at which the hour hand
> and minute hand of a clock line up perfectly. I worked in terms of "unit
> time", in which the 12 hours of the clock are represented as ranging
> from 0 to 1. It is then easy to arrive at
To get a grip on what was going on (and maybe because DateList takes
seconds at the end to cast the answer in {HH,MM,SS}), I started by
calculating the hand positions in radians clockwise from 12 as a
function of seconds past midnight:
hourHandPosition[t_] := (2 Pi/12) t/( 60 60)
minuteHandPosition[t_] := (2 Pi/60) Mod[t/60, 60]
Plot[{hourHandPosition[t], minuteHandPosition[t]}, {t, 0, 12 60 60}]
Solve can handle this in the following form, getting times when hour
and minute hand are coincident, trying first for a solution in the
Rationals:
sols = Solve[hourHandPosition[t] == minuteHandPosition[t], t,
Rationals];
and it gives answers in the form 43200 (n - 1) / 11, {n, 0, 11}
The {HH, MM, SS} form can be constructed as follows (using now NSolve
to generate forms that fit DateList):
Drop[#, 3] & /@
DateList /@ (t /.
NSolve[hourHandPosition[t] == minuteHandPosition[t], t,
Reals]) // TableForm
0 0 0.
1 5 27.2727
2 10 54.5455
3 16 21.8182
4 21 49.0909
5 27 16.3636
6 32 43.6364
7 38 10.9091
8 43 38.1818
9 49 5.45455
10 54 32.7273
The case where the hands are opposed is
Drop[#, 3] & /@
DateList /@ (t /.
NSolve[hourHandPosition[t] == minuteHandPosition[t] + Pi, t,
Reals]) // TableForm
6 0 0.
7 5 27.2727
8 10 54.5455
9 16 21.8182
10 21 49.0909
11 27 16.3636
12 32 43.6364
13 38 10.9091
14 43 38.1818
15 49 5.45455
16 54 32.7273
, which would match your array if I Mod[]ed the hours and sorted.
>
> p.s. the solutions are evenly spaced at the following times
> {
> {"12:32:43.6364"},
> {"1:38:10.9091"},
> {"2:43:38.1818"},
> {"3:49:5.45455"},
> {"4:54:32.7273"},
> {"6:00:00."},
> {"7:5:27.2727"},
> {"8:10:54.5455"},
> {"9:16:21.8182"},
> {"10:21:49.0909"},
> {"11:27:16.3636"}
>
> }
I frankly don't know why the ___Solves don't work on
unstraightness[t]==0, but maybe there's a clue in the fact that
something like this works:
Drop[#,3]&/@
DateList/@
(12 3600 t/.
Table[
FindRoot[
unstraightness[t]==0
,{t, j/12}
]
,{j,1,11}
]
)//TableForm
Fred Klingener
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