CLARK ATLANTA UNIVERSITY DEPARTMENT OF PHYSICS
PHY 104: INTRODUCTION TO EARTH SYSTEM SCIENCE
FALL 2006
EXTRA CREDIT PROJECT 1: MAGNITUDE OF LUNAR AND SOLAR TIDAL FORCES
Due on 1 November 2006
I.	Go to http://www.wikipedia.org or an astronomy textbook to find the law and the numeric value parameters listed.
Newton's law of universal gravitation
Value of the gravitational constant, G
Orbital radius of the earth (semi-major axis)
Mass of the earth
Radius or diameter of the earth
Mass of the moon
Orbital radius of the moon (semi-major axis)
Mass of the sun

Example for #1

Sun

Earth

II.	Your textbook1 on p. 213 states that the effect of the sun on earth tides is less than the effect of the moon on these tides. Stacey2 offers a first-order analysis of the tidal force exerted by these two bodies and shows how this force can act to deform the earth into the approximate shape of a prolate ellipsoid (squished elliptical solid). Figure 9.21 on p. 214 of your textbook1 shows the geometry used to calculate the magnitude of the tidal effect due to the moon and sun in two different configurations. One configuration shows how the combined gravitational attractions of the moon and sun generate neap tides, while the other shows how spring tides are generated. Figure 9.20 on p. 213 shows how different lunar/solar geometries, the shape of ocean basins, and coastal factors can produce complicated tidal patterns in different localities around the world.
1.	Draw a diagram which illustrates the force of gravitational attraction of the sun on the earth.
2.	Draw a diagram which illustrates the force of gravitational attraction of the earth on the moon.
3.	Draw a diagram which illustrates the force ("centrifugal") which counteracts the sun's gravitational attraction on the earth, thereby maintaining the earth's orbit around the sun.
4.	Draw a diagram which illustrates the force ("centrifugal") which counteracts the earth's gravitational attraction on the moon, thereby maintaining the moon's orbit around the earth.
5.	Draw a gravitational force diagram with the moon to the left of the earth and the sun to the right of the earth.
6.	For #5, would the net effect of the gravitational (pulling towards center of attracting object) and centrifugal (pushing away from axis of rotation) forces be to stretch the earth at its poles or at its equator? Hint: look at your diagram in #5 for the answer.
III.	Problem IV.1. below is solved in three steps.
1.	Use Newton's law of universal gravitation to calculate separately the gravitational attraction of the moon on the near side of the earth and the gravitational attraction of the moon on the far side of the earth. Do the same for the gravitational attraction of the sun. This will result in four forces: (i) gravitational force of the moon on the near side of the earth, (ii) gravitational force of the moon on the far side of the earth, (iii) gravitational force of the sun on the near side of the earth, and (iv) gravitational force of the sun on the far side of the earth.
2.	Neglecting  the centrifugal force which maintains the earth in orbit around the sun, find the difference in gravitational attraction caused by the moon on the near side and far side of the earth. Find the same difference in gravitational attraction caused by the sun. These calculations will result in two differential forces: (i) the lunar gravitational attraction across the earth, and (ii) the solar gravitational attraction across the earth.
3.	The ratio of the difference in lunar gravitational attraction across the earth to the difference in solar gravitational attraction across the earth represents the strength of the sun's tidal effect relative to the moon's tidal effect.
IV.	Use Newton's law of universal gravitation to show the following:
1.	The sun's tidal effect is about 46% that of the moon's.
2.	The sun's tidal effect is slightly less than one-third (33.3%) of the combined effect of the moon and the sun when they are aligned with each other and the earth.
3.	Does #2 represent spring tide or neap tide conditions?
V.	Problem IV.2. is solved by calculating a ratio similar to that found in IV.3.
This extra credit project can add up to two points to your final grade.
References
1Earth Science: Understanding Environmental Systems, 2003, McGraw-Hill, by Edgar W. Spencer, pp. 212-215.
2Physics of the Earth, 1992, 3rd ed., Brookfield Press, by Frank D. Stacey, pp. 83-90, 115-118.