At 11:00 a.m. on September 7, 2001, more than one million...

At 11:00 a.m. on September 7, 2001, more than one million British schoolchildren jumped up and down for one minute to simulate an earthquake. (a) Find the energy stored in the children’s bodies that was converted into internal energy in the ground and their bodies and propagated into the ground by seismic waves during the experiment. Assume 1 050 000 children of average mass 36.0 kg jumped 12 times each, raising their centers of mass by 25.0 cm each time and briefly resting between one jump and the next. (b) Of the energy that propagated into the ground, most produced high-frequency “microtremor” vibrations that were rapidly damped and did not travel far. Assume 0.01% of the total energy was carried away by long-range seismic waves. The magnitude of an earthquake on the Richter scale is given by $$ M=\frac{\log E-4.8}{1.5} $$ where $E$ is the seismic wave energy in joules. According to this model, what was the magnitude of the demonstration quake?

At 11:00 a.m. on September 7, 2001, more than one million British schoolchildren jumped up and down for one minute to simulate an earthquake. (a) Find the energy stored in the children’s bodies that was converted into internal energy in the ground and their bodies and propagated into the ground by seismic waves during the experiment. Assume 1 050 000 children of average mass 36.0 kg jumped
12 times each, raising their centers of mass by 25.0 cm each time and briefly resting between one jump and the next. (b) Of the energy that propagated into the ground, most produced high-frequency “microtremor” vibrations that were rapidly damped and did not travel far. Assume 0.01% of the total energy was carried away by long-range seismic waves. The magnitude of an earthquake on the Richter
scale is given by $$ M=\frac{\log E-4.8}{1.5} $$ where $E$ is the seismic wave energy in joules. According to this model, what was the magnitude of the demonstration quake?

Key Concepts

Logarithmic Earthquake Magnitude Scales

Earthquake magnitude scales, like the Richter scale, use a logarithmic scale to relate the vast range of seismic energies to manageable numbers. The logarithmic relation means that each unit increase in magnitude represents an exponential increase in energy. In the given context, this concept is used to convert the calculated seismic wave energy (a small fraction of the total energy) into an earthquake magnitude, illustrating the non-linear relationship between energy release and perceived earthquake strength.

Seismic Waves and Energy Propagation

Seismic waves are energy waves that travel through the Earth, typically generated by events such as earthquakes or other disturbances. The problem illustrates how a fraction of the energy from the jumps is transmitted into the ground as seismic waves, with most of this energy manifesting as high-frequency microtremor vibrations that damp quickly, and only a small percentage propagating over long distances.

Energy Conversion and Dissipation

Energy conversion involves transforming one form of energy into another, while energy dissipation pertains to the loss of usable energy due to factors such as friction, sound, or internal heating. In scenarios like a group of people jumping, the gravitational potential energy is transformed into other forms, including thermal energy and vibrational energy in the ground, with most energy being rapidly damped or converted into less useful forms.

Gravitational Potential Energy

This concept refers to the energy an object possesses due to its position in a gravitational field. When an object's center of mass is raised, energy is stored according to the formula E = mgh, where m is mass, g is gravitational acceleration, and h is the height raised. This principle underlies the calculation of energy input when individuals jump, converting kinetic energy into gravitational potential energy.

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