| Basic Thermodynamics ~ J. Pohl © | www.THERMOspokenhere.com (45-A325) |
In Massachusetts (1710) an accused witch was hanged in the public square. At the moment of her death, when that rope snapped taut, an experience occurred to her family and loved ones, they believed an earthquake had happened. At night, in full darkness, friends cut her down and a rumor got started. "When she died, God lifted Earth to receive her," whispered one bystander to the hanging. With a day minds changed; the entire village came to believed she was innocent.
The next Sunday, the Bishop explained; "She was guilty. The tremor was not God - it was gravity. The witch fell down; Earth fell up." He provided the solution shown below. What really happened? Is the Bishop's solution correct?
♦ The exaggerated sketch shows the "before" and "after" of the event. We will use the momentum equation with the system being both Earth and the woman. The momentum equation written with the discrete summation sign is:
| (1) 1 |
The velocities for this event have only K components. So when we vector multiply by "•K" the velocities become z-component velocities or "vz's." We make that multiplication, separate variables of the differential equations, then integrate between limits. This same simple process gets done again and again.
| (2) 2 |
Next we express "vz" as "d z( t ) / dt." Make that substitution, separate variables again, then integrate once more.
| (3)3 |
| (4) 4 |
The two masses, taken as a system, experience no surface forces during the event. Therefore the center of mass of the system does not move. It is sufficient for us to assume the elevation, Zc.m. and then set it as a zero datum for the event. Thus the elevations of the above equation are "relative" to the location (zero value of z). We enter the elevations with some algebra:
| (5) 5 |
The Bishop is correct that mutual gravity caused Earth to move up. However we see that the amount of movement is very small. The tremor many people felt must have been a minor Earthquake.
A Second Solution: They had better obtain the same answer.
In Massachusetts (1710) an accused witch was hanged in the public square. At the moment of her death, when that rope snapped taut, an experience occurred to her family and loved ones, they believed an earthquake had happened. At night, in full darkness, friends cut her down and a rumor got started. "When she died, God lifted Earth to receive her," whispered one bystander to the hanging. With a day minds changed; the entire village came to believed she was innocent.