MPE2013 gives us an opportunity to learn more about our planet. There are interesting features to be explored that require simple but deep principles of physics and that can become the basis of a discussion in the classroom. I frequently teach future high-school teachers and like to start by exploring questions that come from many different directions. Here is one.
During an earthquake, the mass distribution in the Earth’s crust changes. This changes the Earth’s moment of inertia, which is the sum of the moments of inertia of each point. The moment of inertia of a point mass is the product of its mass and the square of its distance to the axis of rotation. Meanwhile the angular momentum is preserved: this angular momentum is the sum of the moments of inertia of each point mass times its angular velocity. Hence, if the moment of inertia of the Earth decreases (increases), the angular velocity of the Earth increases (decreases). The simple physical principle of conservation of angular momentum thus allows us to explain disparate phenomena such as the Earth’s changing rotation rate, figure skaters spinning, spinning tops, and gyroscopic compasses.
The major earthquakes in Chile (2010) and Japan (2011) increased the Earth’s spin and hence decreased the length of the day. We could imagine that the length of the day would be measured by taking, for instance, a star as some fixed point of reference. But instead, geophysicists use seismic estimates, through GPS measurements, of the movements of the fault to compute how the mass distribution and thus the length of the day has changed. According to Richard Gross, a geophysicist working at NASA’s Jet Propulsion Laboratory, the length of the day has decreased by 1.8 microseconds as a result of the 2011 Japan earthquake.
Instead of continuing to read Richard Gross’ interview, you can start playing the game and discover for yourself the answer to the next questions. For instance, earthquakes closer to the Equator have a larger effect on the Earth’s spin than those close to the poles. Similarly, those with vertical motion have a larger effect than those with horizontal transversal motion.
It is also an opportunity to start discussing the motion of a solid (here the Earth) in space. On short intervals of time, the speed of the center of mass is approximately constant. Hence, if we consider a reference frame centered at the center of mass and moving uniformly, then we are left with three degrees of freedom for the movement of the solid. The derivative of this movement is a linear orthogonal transformation preserving the orientation. Hence, it is a rotation around an axis: the north-south axis. The three degrees of freedom describe the position of the axis and the angular velocity around it. But there is a second axis, which is very important: it is the Earth’s figure axis, about which the Earth’s mass is balanced. This axis is about 10 meters offset off the north-south axis. Large earthquakes abruptly move the position of this axis. For the 2011 Japan earthquake, the shift has been estimated at 17 centimeters.
Earthquakes are far from being the only phenomena changing the angular speed of rotation and the position of the Earth’s figure axis. Indeed, they change with atmospheric winds and oceanic currents, but these changes are smoother than the ones observed with earthquakes.
Should we care for such small changes? According to Richard Gross, we should if we work for NASA and, for instance, intend to send a spacecraft to Mars and land a rover on it. Any angular error may send us very far from our target.