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ConstructingConstructingAA
SundialSundial
ContentContent
• What is a sundial?• Sundial history• Types of sundial• Experiment & results• Math properties of our sundial• Other application of sun angle
introductionintroduction
• Aim of our project: To investigate the theories and
principles that makes a sundial work, and thus to construct a functioning sundial in NUS High School campus.
IntroductionIntroduction
• What is a sundial?• “The earliest type of timekeeping device,
which indicates the time of day by the position of the shadow of some object exposed to the Sun's rays. As the day progresses, the Sun moves across the sky, causing the shadow of the object to move and indicating the passage of time.”
– Encyclopædia Britannica
IntroductionIntroduction
• Components of a sundial:
– Dial face– Gnomon dial– Dial calibration
History of sundial developmentHistory of sundial development• Rudiment: 5000-3500 BC• First known sundial with calibration:
800 BC, Egypt• 250 BC onwards: more complex
sundials were built by the Greeks– correct for season change– Portable
• The geometry knowledge was firstly applied in constructing these sundials.
History of sundialHistory of sundial• 100 AD: people found out that a
slanting gnomon is more precise than a vertical gnomon
• 150 AD: trigonometry was introduced by the Greek.– Trigonometry is much easier than
geometry.
• Sundial’s fate after mechanical clock
Types of sundialTypes of sundial
• Equatorial sundial
• Horizontal sundial
• Vertical sundial
Equatorial sundialsEquatorial sundials • Gnomon in the
center of the plate • Parallel to the
Earth's axis and points to the north celestial pole
Horizontal sundialHorizontal sundial• Flat horizontal dial
plate with hour lines • O towards south and
A towards north• The shadow of
gnomon placing on the hour lines indicates the time
Mathematics behindMathematics behind•OP is pointing to the pole, P.
•PNS is the meridian,
• NPT is the hour angle
• TON is the shadow angle.
Mathematics behindMathematics behind
cos NP cos PNT = sin NP cot TON - sin PNT cot NPT
Mathematics behindMathematics behindcos NP cos PNT = sin NP cot TON - sin PNT cot NPTIn which, PNT = 90º, NP = Ø and TON = Since cos PNT = 0, sin PNT = 1,
therefore 0 = sin Ø cot - cot (HA)tan = sin Ø tan (HA)Where, is the Shadow Angle for a given time.Ø is the Latitude of the sundial.
Mathematics behindMathematics behindtan = sin Ø tan (HA)
HA is the hour required.– multiplied by 15 – 24 hour solar day– Sun appears to go once around the Earth
(360°). – This means that 360° is equivalent to 24 hours– making 15° equivalent to one hour (360 / 24 =
15).
ExampleExample
The calculated shadow angles is at latitude 51ºN.
Types of sundialTypes of sundial• Vertical sundials can be divided into five
groups due to the direction they face.1. Vertical direct north sundials - early morning and late
evening hour 2. Vertical direct south sundials - greater duration of time 3. Vertical direct east sundials - the morning hours 4. Vertical direct west sundials - the afternoon hours 5. Vertical declining sundials - Southwest decliners,
Southeast decliners, Northwest decliners and Northeast decliners
Vertical direct south sundials Vertical direct south sundials - - greater duration of timegreater duration of time
- - hour lines run anti-clockwisehour lines run anti-clockwise
Vertical direct east sundialsVertical direct east sundials• Greater duration of
time• Dial plate lies in the
meridian• Gnomon is parallel to
the dial plate, thus parallel to the Earth's axis
Vertical direct west sundialsVertical direct west sundials
• Only the afternoon hours
• Can be used on any latitude
• Dial plate lies in the meridian
• Gnomon is parallel to the dial plate, thus parallel to the Earth's axis.
Experiment Experiment Aim: To construct a functional sundial
locating in NUS High School campus.
Mathematical model used:
tan
hd
Underlying assumptionsUnderlying assumptions• Since Singapore is near to the earth
equator(1 °22 ' N, 103° 48' ) if we put the gnomon parallel to the earth axis, approximately the path of the sun relative to the gnomon will be a semi-circle in the equatorial plane
• Assuming the angular velocity of the sun relative to a point on earth is constant
Model Model
Data collectedData collectedClock time Time / min Length of shadow / cm tanθ θ / °
0745 465 173.0 0.161 9.129
0800 480 127.5 0.218 12.300
0815 495 100.1 0.278 15.512
0830 510 81.1 0.343 18.921
0845 525 66.8 0.416 22.596
0915 555 48.5 0.573 29.821
0930 570 45.3 0.614 31.537
0945 585 43.5 0.639 32.582
1015 615 30.3 0.917 42.536
1030 630 25.9 1.073 47.026
1045 645 23.1 1.203 50.276
1100 660 20.8 1.337 53.196
1115 675 18.6 1.495 56.215
1145 705 13.8 2.014 63.600
1200 720 11.7 2.376 67.176
1245 765 5.3 5.245 79.206
1300 780 3.2 8.688 83.434
1315 795 0.0 Undefined 90.000
Analysis Analysis After some analyze of the data, we
had observed there is a relationship between time and θ. By plotting time vs. θ graph, we get the following result:
Graph of θ vs Ti me
020406080
100
450 550 650 750 850
Ti me / mi n
θ /
°
(7:30a.m.)
TrendTrend• Linear• Exponential• Introducing regression line Used to depict the relationship
between the independent variable and the dependent variable
Regression lineRegression line To determine the best fitting line, we need
to calculate the correlation coefficient and coefficient for each line.
where y is the value predicted by the graph
22
2
( ' )
( )
y yR
y y
Graph of θ vs Ti me wi th a l i nearregressi on l i ne
R2 = 0. 9966
0
50
100
450 550 650 750 850
Ti me / mi n
θ /
°
Graph of θ vs Ti me wi th aexponent i al regressi on l i ne
R2 = 0. 9626
0
50
100
150
450 550 650 750 850
Ti me / mi n
θ /
°
DiscussionDiscussion• Inaccurate measure of shadow
length• Weather condition• The position of the shadow of the
gnomon has a displacement of zero from the origin at 1:15p.m.
• Interpolation and extrapolation
Conclusion Conclusion • Hypothesis is valid
• Further application that we can explore
Q & AQ & AAny questions?Any questions?
If not….If not….
The EndThe EndThank you for your Thank you for your
attentionattention