(please click on a slide to view it bigger image)
The earth is spherical (almost) but we spend most of our life on a piece of land that is flat locally. Our common sense develops in a way that is appropriate for a flat earth. Even the geometry that we learn in school is Euclidean – that is – flat space geometry. We hardly ever think about what it means to have a spherical earth. Start delving into this question and a whole lot of fascinating insights spring up and that is the journey we shall make in this post – hope you will enjoy it and give some thought as to what life might be like on a flat earth. Very different for sure – it may not even be possible for life to survive on a flat earth.
General Background: Earth
is the 3rd planet in the Solar System – on average about 150 million km from
the Sun. It completes one revolution
round the Sun in about 365 days
The earth’s axis is tilted by about 23.4o
relative to the direction perpendicular to the plane of the earth’s
orbit. Average speed of the earth as it revolves round the Sun in its 970 million
km long orbital path is 30 km/second or 67,000 mph.
The next slide shows the spinning earth – it
completes one rotation in 24 hours. Because earth is spherical and it spins, we have nights and days -
light from the Sun falls as a parallel beam on the earth and illuminates the
part facing the Sun. The radius of the
earth is 6375 km; at the equator its circumference is 40,700 km and the speed
of spin is 1670 km/h (1000 mph). Because
of its spin, the earth bulged a little at the equator; but it is spherical to a
very good approximation and we shall consider it to be so in our
discussion.
Earth also has gravity and
attracts all objects towards its centre. The slide also shows how a person standing
upright at the north pole finds that somebody stands upside down at the south
pole; and sees people standing at all sorts of funny angles in between.
Points on the surface of the earth are
located by their latitude and longitudinal angles. The equatorial plane (the circle that divides
the earth in two equal halves) defines the zero degree latitude.
Latitude of a point on earth’s surface is the
angle that the line joining the point to the centre of the earth makes with the
equatorial plane. Longitudes run from
180oW to 180oE.
Half circles passing through the centre of the earth and the two poles are perpendicular
to the equatorial plane and define the longitudes.
0o longitude (the prime meridian) passes through Greenwich,
London. Longitudes (meridian) run through from 0 to
180o in the East and West directions from the prime meridian and they meet at the International Date Line in the Pacific.
Spherical Geometry: In order to really appreciate how spherical
earth is different from a flat earth, let us look at the difference between flat
(Euclidean) geometry and spherical (non-Euclidean) geometry. The following slides demonstrate this
difference. (click on the slide to
see its bigger image in a new window)
On a spherical surface, our experience
can be counter-intuitive and puzzling. I
have two puzzles here – see if you can figure them out. The answers are at the end of this post.
Puzzle
1. After lunch, Euclid leaves his tent and
walks 1 km to the South. He then turns
and walks 1 km to the East. After a
short tea break, Euclid walks 1 km to the North and finds himself at his
tent. How is it possible?
Puzzle
2. Riemann leaves his tent at 10 am and
start walking due North for 2 km. After
enjoying his lunch, he again walks due North for 2 km. Surprisingly, he finds
that he has returned to his tent. Can
you explain this?
How far is the Horizon?: The ground around us always feels flat because our field of
view is generally quite restricted – distances we see are only a few km
– and with the earth’s radius being 6375 km – a few km patch of land will
appear flat. Flat (Euclidean) geometry
will be a good approximation for our field of view and will work well over such
short distances. Even then, the
curvature of the earth restricts the distance we can see – how far is the
horizon? – In ancient times, this is how philosophers had concluded that the
earth is not flat.
Navigation:
When we travel from point A to point B, we
think of travelling along a straight line which is the shortest path between
any two points on a flat surface. This
is not so on a curved surface; the shortest path on a 2D curved surface, like
our earth, is a geodesic or the arc of the great circle passing through points
A and B. This is shown in the slide
A Parable: Life on a spherical surface can be very confusing!
Climate:
The earth receives most of its energy from the
Sun in the form of electromagnetic waves and lot of it in the visible region. Rays from the Sun fall on the earth and
deliver much more energy in the equatorial regions; amount of energy received
per unit area reduces as we travel towards the poles. This is shown in the following two
slides:
Consequences of Earth’s Spin: The earth and its atmosphere spin
(rotate) about the earth’s axis completing one rotation every 24 hours. The
rotation of the earth gives rise to some very interesting effects; we shall
look at a few of these here. For ease of
explanation, we shall talk in terms of centrifugal and Coriolis forces which are ‘fictitious forces’ observable
in the frame of reference of the rotating earth. They give the correct results and are widely
used in general discussions.
Earth bulges at the equator:
Because of the earth’s rotation, every object on the earth’s
surface is moving with the earth at a speed of 465
m/sec or 1000 miles per hour and feels an outward force – the
centrifugal force – directed away from the centre of the earth. Its magnitude is proportional to the square of the linear speed divided by the radius of the earth at that point. The linear speed of
rotation v and hence the magnitude of the force is a maximum at the equator,
decreases with latitude and is zero at the poles.
The centrifugal force
causes the earth at the equator to bulge out and the earth has a larger radius at the
equator than at the poles - it is an
oblate spheroid – the extent of the bulge is small; the earth’s radius measured across the equatorial plane is 6378 km
compared to its value at the poles of 6357 km.
You are less heavy at the equator: All objects
feel an attractive force directed towards the centre of the earth -
gravitational attraction. Gravitational
attraction is largest at the poles and you weigh more there than at the
equator. Your weight at any point on the earth is the product of your mass (m) and
the acceleration due to gravity (g) at that point.
Centrifugal force acts
in exactly opposite direction and reduces the amount of earth’s gravitational
attraction at all places – hence your weight is reduced. This effect is greatest
at the equator and your weight is reduced by about 0.5% relative to what you
will weigh at the poles. For a more detailed analysis, see the slide (your weight = mass x g)
Coriolis Effect (Force): Is a force experienced by moving objects in a spinning earth. The force depends on the speed of the object, the direction it is moving and on the earth's rotaional (angular) velocity. It is mathematically difficult to explain but for the particular case of an object moving from
the Equator to North Pole, Coriolis Effect may be understood as follows:
All points on the earth complete one
full rotation through 360 degrees in 24 hours (constant angular velocity) but a
point at the equator has to go farther in a day than a point in Glasgow at a
latitude of 56oN. Object at the equator have a bigger eastward
velocity than objects at more northerly latitudes.
As the object travels north, it will be going
east at a faster speed (easterly speed it had at the equator) than the ground beneath and its path will deflect east - it will seem as if some mysterious force is acting on the object.
Objects travelling towards the equator
will have a smaller eastward speed than the ground beneath them and they will seem to
be forced west.
The angular velocity of Earth
is 360 degrees per day (one rotation per day), or 73 microradians/s. At wind
speeds of 20 m/s (or 50 m/h), the Coriolis Effect generates an acceleration of about
0.003 m/s2. Over an hour, this is a total deflection of
about 18 km.
Coriolis effect is responsible for the way atmospheric air circulation and ocean currents are. Because of the uneven heating of the different regions of the earth by the Sun, heat transfer happens and air and water masses move from hot equatorial regions towards the polar regions. Coriolis effect causes the flow of air to be modified as explained above and creates a much more complex air circulation patterns than would exist if the earth were not spinning. This is explained in the following slides:
With Coriolis force the pattern looks as in the above slide.
Earths Tilt: The earth's axis is tilted at 23.4 degrees with respect to the ecliptic plane (the plane of earth's orbit around the Sun) and maintains this tilt during its orbital journey.
At any time of the year, the northern and southern hemispheres receive energy from the Sun in different amounts. This is responsible for the earth to have seasons. I have discussed this topic in detail in my post and I refer you to that post.
Another interesting aspect of the earth's tilt is that the tilt changes slowly with a period of 41000 years.
As the tilt changes, the amount of sunlight falling on the earth changes over the course of its rotation. This can act as a trigger for the onset of ice ages. The cause of ice ages is a bigger topic and is discussed under Milankovich Cycles in my post.
SOLUTION TO PUZZLES:
Coriolis effect is responsible for the way atmospheric air circulation and ocean currents are. Because of the uneven heating of the different regions of the earth by the Sun, heat transfer happens and air and water masses move from hot equatorial regions towards the polar regions. Coriolis effect causes the flow of air to be modified as explained above and creates a much more complex air circulation patterns than would exist if the earth were not spinning. This is explained in the following slides:
Earths Tilt: The earth's axis is tilted at 23.4 degrees with respect to the ecliptic plane (the plane of earth's orbit around the Sun) and maintains this tilt during its orbital journey.
At any time of the year, the northern and southern hemispheres receive energy from the Sun in different amounts. This is responsible for the earth to have seasons. I have discussed this topic in detail in my post and I refer you to that post.
Another interesting aspect of the earth's tilt is that the tilt changes slowly with a period of 41000 years.
As the tilt changes, the amount of sunlight falling on the earth changes over the course of its rotation. This can act as a trigger for the onset of ice ages. The cause of ice ages is a bigger topic and is discussed under Milankovich Cycles in my post.
SOLUTION TO PUZZLES:
POST SCRIPT: We have gone over some of the fascinating aspects of life on a rotating spherical earth and looked at how this creates a large variety of events both over short and long time scales.
The earth is spherical was established historically by observations of length of shadows, finite horizon distances, shape of earth's shadow in lunar eclipses and now by actually photographing the earth from space. Against this background, I find it amazing that there are people who claim that the earth is flat. This is a matter that I think falls in the realm of mental aberration - I am not a psychologist but when otherwise sensible humans who lead a normal life can still believe that the earth is flat then I start to wonder about how our minds are organised. To me, this represents a bipolar mind - switching between logical and illogical states . Is bipolar mind an illness or human mind is naturally organised in a multipolar mode? I can think of many aspects of life - religious belief is another example - where a multipolar mind can quite happily live with contradictions. Human society throws many contradictions; is multipolar mind just a strategy to cope or is it genetic or is this a real disorder (illness)? I would like to understand.
The earth is spherical was established historically by observations of length of shadows, finite horizon distances, shape of earth's shadow in lunar eclipses and now by actually photographing the earth from space. Against this background, I find it amazing that there are people who claim that the earth is flat. This is a matter that I think falls in the realm of mental aberration - I am not a psychologist but when otherwise sensible humans who lead a normal life can still believe that the earth is flat then I start to wonder about how our minds are organised. To me, this represents a bipolar mind - switching between logical and illogical states . Is bipolar mind an illness or human mind is naturally organised in a multipolar mode? I can think of many aspects of life - religious belief is another example - where a multipolar mind can quite happily live with contradictions. Human society throws many contradictions; is multipolar mind just a strategy to cope or is it genetic or is this a real disorder (illness)? I would like to understand.
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