Energy and Magnitude of the Sliding Block

The following unit definitions are from: http://www.unc.edu/~rowlett/units/index.html

newton (N) the SI unit of force.

A force of one newton will accelerate a mass of one kilogram at the rate of one meter per second per second. The corresponding unit in the CGS system is the dyne; there are 105 dynes in one newton. In traditional English terms, one newton is about 0.224 809 pounds of force. Nineteenth-century engineers created another English unit of force, the poundal; there are approximately 7.233 01 poundals in a newton. The newton is named for Isaac Newton (1642-1727), the British mathematician, physicist, and natural philosopher. He was the first person to understand clearly the relationship between force (F), mass (m), and acceleration (a), expressed by the formula F = ma.

newton meter (N m)  the SI unit of torque.

Torque, the tendency of a force to cause a rotation, is the product of the force and the distance from the center of rotation to the point where the force is applied. Torque has the same units as work or energy, but it is a different physical concept. To stress the difference, scientists measure torque in newton meters rather than in joules, the SI unit of work. One newton meter is approximately 0.7376 pound foot.

joule (J)  the SI unit of work or energy, defined to be the work done by a force of one newton acting to move an object through a distance of one meter in the direction in which the force is applied. Equivalently, since kinetic energy is one half the mass times the square of the velocity, one joule is the kinetic energy of a mass of two kilograms moving at a velocity of 1 m/sec. This is the same as 107 ergs in the CGS system, or approximately 0.737 562 foot-pound in the traditional English system. In other energy units, one joule equals about 9.478 170 x 10-4 Btu, 0.238 846 (small) calories, or 2.777 778 x 10-4 watt hour. The joule is named for the British physicist James Prescott Joule (1818-1889), who demonstrated the equivalence of mechanical and thermal energy in a famous experiment in 1843. Although Joule pronounced his name "jowl", the unit is usually pronounced "jew'l".

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If we assume that the wooden block slips at about 5 newtons force, then if the displacement is x cm or x/100 meters, then the energy is  5*x/100, or x/20 joules.

The magnitude of an earthquake, M, is given by

M = (log E - 4.8)/1.5  where E is measured in joules.

Substituting x/20 for E we have:

M = (log x/20 - 4.8)/1.5 = (log x - log20 -4.8)/1.5

M = (log x - 1.3 - 4.8)/1.5 = (log x - 6.1)/1.5

Or, solving for x, log x = 1.5 M + 6.1

x = 10**(1.5 M + 6.1)     (meaning, 10 raised to the power (1.5M + 6.1)

Using this equation, we can compute the value of x for a series of values of M:

M            x (cm)
-5.0            .039
-4.5            .2
-4.0            1.3
-3.5            7.1
-3            39.8
-2.5            223.9

These limits could be used to assign each slip event to a specific magnitude range.  The number of events in each bin could then be plotted versus the magnitude to see the magnitude distribution of the slip events.  This distribution can be compared with the distribution observed for the world's earthquakes.