1. b) The graph for 1b has been shown below.
Average Height Above Earth's Surface
y = 0.2333x
0 20 40 60 80 100
2. b) According to the article given, the French astronomer Jean Richer measured the period of
a pendulum at Paris and at Cayenne. Cayenne is almost at the Equator, and Paris is closer to the
poles than the Equator. The gravitational force in Cayenne was found to be less than that at
Paris, and G.D. Cassini found that the Earth was a prolate ellipsoid. The current knowledge of
the degree of flattening on Earth says that the poles have a slightly smaller radius than the
equator. The equation for the gravitational force is:
Given this equation, the gravitational force at the poles and the equator can be calculated. The
reason that the gravitational force in Cayenne was found to be less than that at Paris is because
the gravitational force is inversely proportional to the radius squared (Rowlinson, 2007). As the
equator has a large radius, the gravitational force is less.
In order to render the period of the pendulum exactly two seconds, it is known that the length
of the pendulum depends on the acceleration due to gravity, g, and the length of the string. It is
independent of mass. The formula for the period of a simple pendulum is
2π . Given this equation, at the equator, it is known that g=9.78 m/s 8
(Szeto, 2013). Therefore, π √ . To solve for L, this would be 9.691 m. The length of
the pendulum was 9.69 m in order to render a period of exactly 2.00 seconds.
3. a) "The length of day(LOD) is only approximately constant", and this is due to the Earth's
rotation. Earth does not have a uniform density, and the density of Earth is higher at the Earth's
core than at the Earth's surface. There are variations in the Earth's rotation rate, which have
been directly related to changes in atmospheric angular momentum. There is a LOD decrease of
2.4 milliseconds/century. The long-term changes in the LOD are caused by processes within the
Earth's core. The motion in Earth's core generates the magnetic field of the Earth, and the
changes in the fluid motion in the outer core that are inferred from the magnetic field match
the longer period changes in the LOD. There is a precession cycle of 25,730 years (Szeto, 2013).
LOD decreases long term because Earth's rotation slows down. Earth takes approximately
365.2422 days to orbit around the Sun, so every 400 years, 3 days are lost. Short-term changes
in LOD periods are accounted for by atmospheric angular momentum, including the tilt of
Earth's axis and the jet stream, but long-term changes are due to Earth's rotation. The Earth
takes 365.2422 days to orbit around Earth, which is why there is a leap year every four years to
synchronize the LOD.
b) Ideally, if the solar year was 365.25 days in length, then inserting an additional day every 4
years would be perfect. However, the true solar year is not 365.25, it is actually 365.2422 days.
Given this figure, there is a 0.0075 difference between this value and 365.25. In order to match
up the number of days to have perfect synchrony, three days will need to be removed every
400 years. 3/400=0.0075, and this is the difference between 365.2422 and 365.25. The value of
N would be 400/3=133.33 years. By this calculation, one day must be removed every 133.33
years to synchronize the number of days elapsed.
c) A sidereal day is the amount of time that it takes the Moon to orbit around the Earth. The
Moon goes around the Earth as the Earth goes around the Sun. The Moon issues a gravitational
force on Earth. Different parts of Earth are different distances on the Moon due to the effect of
differential forces. These differential forces form a bulge on Earth due to the deformation of
gravity. A sidereal day lasts from where the Moon is at a point on the Earth's meridian until it is
next at that point.
The Earth has a 23.5° tilt away from or towards the Sun, depending on the hemisphere and
time of year. Earth travels in a retrograde motion. The difference between the sidereal day and
solar day occurs because over a year, Earth does one revolution of the Sun and there is one less
solar day than sidereal day. The difference is 4 minutes a day. It takes Earth longer to go around 9
the Sun than the Moon. In one year, Earth takes 360° to travel around its orbital axis. The solar
day is 0.986° greater. For one second of solar time, Earth rotates 0.015 arcs.
A solar day is the period over a certain longitude, and it is the period that requires Earth to spin
from 1/365.2422 days. It is exactly 24 hours, or 86,400 s. A sidereal day is 23:56 hours, or given
that there are 3600 s in an hour, 86,160 s. The difference between a sidereal and a solar day is
240 s, or four minutes. To the second, taking the ratio of a solar day/sidereal day, the solar
second is 1.0028 s longer than a sidereal second. A diagram has been shown below to illustrate
the difference between a solar and a sidereal day (Haynes et al, 1999). As observable from the
figure, the reason that the solar day is longer is due to the degree difference of 0.986°, and the
Sun is further away from Earth by this additional angle.
d) If I see the Moon rising at 11:15 PM one evening, there will be a different time that the
Moon hits the Earth(this is called moonrise) the next evening. The Moon is 30 times slower
than Earth, and it takes one hour for Earth to rotate exactly 15°(Szeto, 2013). As the Moon hits
the Earth it introduces a lag angle, 2.