Eart 110N The Dynamic Earth Lab
Lab#1—Relative Plate Velocities
Objective: To learn how to calculate the velocity relative to plate A of any point rotating with
plate B by using the appropriate Euler pole and rotation rate (i.e., the angular velocity vector).
Introduction: The Swiss mathematician Leonhard Euler proved in 1776 that any displacement
of a rigid body on a sphere is equivalent to a rotation about some pole. This theorem forms the
mathematical basis of plate tectonics. By pole we mean a vertical axis that pierces the sphere at
some point on its surface. On the earth we describe the location of the pole by latitude () and
longitude () and the rotation by an angle , usually in degrees. A positive value of  means that
the sense of rotation is counterclockwise—the same way that the fingers of your right hand curl
when your thumb points up at the pole.
In plate tectonics poles of rotation are often called Euler poles, and often they are used to
describe the rotation of one plate relative to another. Thus AEB means the Euler pole describing
the movement of plate B relative to plate A—that is, the movement of B with respect to the
reference frame in which A is fixed. Generally rotations do not commute, meaning that a rotation
R1 followed by a second R2 about a different axis does not yield the same result as if R2 is
performed first and R1 second. Thus, even though any rotation is described by a direction (the
axis) and a magnitude (the amount of rotation ), it is not in general a vector. But a very small
rotation d does have the properties of a vector, because infinitesimal rotations do commute. For
this reason rotation velocity of a plate,  = d/dt, is a vector, so we can use all the vector
operations such as addition, subtraction, dot and cross products that we have learned before. On
the other hand, finite rotations are more complicated to handle.
The rotation velocity of plate B relative to plate A is denoted AB. Knowing it, we can calculate
the velocity that a point X on plate B is moving relative to A in, say, cm/yr. Its speed is given by
v = ABRsin(a)
eq. 2.7 (Fowler p.14)
where R is the radius of the earth and a is the angle between the point and the pole. How to find
the angle a and also the direction that point X is moving relative to plate A is shown on p.14-16
of Fowler. In this lab we will work with angular rotation velocities and calculate linear velocities
of plates bounding the Pacific Plate.
In this lab the instructor will:
1. review the basic operations of vectors;
2. introduce the geometry of plate movements on a sphere;
3. discuss how to find relative velocities from the instantaneous rotation pole and angular
velocity (i.e., angular velocity vector) for a plate pair (see also p. 14-16 of Fowler);
4. review and demonstrate basic spreadsheet operations using Microsoft Excel;
5. and do an example using Excel to calculate the velocity of one plate relative to another at a
point on the globe.
In your handout for this lab you should have:
1. a map of the world showing the major plates;
2. Table 1 containing a list of points on the globe that lie on the boundary of the Pacific Plate;
3. Table 2 of relative angular velocity vectors for the Pacific Plate;
4. and copies of pages 14-16 of Fowler.
In lab period this is what you should do before leaving:
1. The first row of Table 1 is done for you as an example.
a. Plot the first point X (37 N, -122 E) on the map.
b. Given the angular velocity vector for the Pacific moving relative to the North America Plate,
use a calculator to verify the values for the five quantities shown in the first row of the table on
the right side that conclude with the plate direction and speed. (Note: the ‘test’ (+ or -?) refers to
the quantity (sinp - sinxcosa): if it is positive, the direction is 90 + C; if it is negative, the
direction is 270  C. This test is not given in Fowler.) Show your work!
c. Plot the rotation pole on the map. Check whether the calculated direction is in qualitative
agreement with the direction corresponding to the right hand rule (counterclockwise rotation). Is
the direction consistent with what you know to be the slip direction on the San Andreas Fault?
d. Mark with an arrow the direction of relative motion (~325 E) of the Pacific Plate and write the
speed in cm/yr beside it.
2. Do the same for the last row of Table 1, responding to and showing work for all parts a-d. Get
the rotation velocity vector from Table 2.
Finish the rest of the lab using a computer outside of the classroom and turn it in next lab period:
1. Go to the 110C website (http://www.es.ucsc.edu/%7Ercoe/eart110c/110c.htm) and download
the two spreadsheets Table1.xls and Table2.xls files. Table 1 has formulas entered in the last five
columns of its first row (i.e., in cells I7, J7, K7, L7, M7) that do all the calculations. Don’t
change them! But do look at them by selecting each cell in turn and see how the formulas
correspond to the equations you solved on pages 14 and 15 of Fowler. In future labs you will
make Excel formulas of your own.
2. Plot the location of point X for row 8 of the spreadsheet (cells B8 and C8) on the plate map.
Which plate adjoins the Pacific Plate there? Type its name into the Plate A column (cell D8).
3. Find the correct rotation velocity vector for this plate pair in Table 2 and enter the values into
Table 1 (cells F8, H8, G8).
4. Copy the five cells I7-M7 and paste them into cells I8-M8 immediately below. The correctly
calculated values for row 8 should appear there.
5. Plot the rotation pole on the map and check whether your velocity direction is reasonable in
the same way that you did in 1c above.
6. Mark the direction with an arrow and write the speed on the map, just as you did in 1d above.
7. Repeat for all the locations in Table 1. Note that your calculations for the last location (Juan de
Fuca Ridge) that you did by hand in the lab period can serve as a check on your spreadsheet
calculations (or perhaps vice-versa!).
As you do all this, visualize how the movements would look on the globe rather than on a flat
map, how the relative plate motion corresponds to rotation about a pole, and marvel about how
powerful this simple, descriptive plate tectonic model is for predicting what should be happening
at all the plate boundaries!
8. Print a hard copy of your completed spreadsheet Table 1.
9. Using your map and Table 1, write a coherent, grammatically correct abstract of 250 words or
less summarizing the lab: a sentence or so about what went into the calculation and the rest on
the types of boundaries (convergent=consuming, divergent=accreting, transform=conservative)
and relative motions that you found.
So here is what you should turn in at the next lab period:
A. Your hand written calculations done in the lab for the first and last locations in Table 1.
B. Your map showing the direction and speed of relative plate motion at each location.
C. Your printout of the complete Table 1 spreadsheet calculations.
D. Your concise, well written, correctly spelled abstract of the lab.