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