|THERMO Spoken Here! ~ J. Pohl © ~ 2019||www.ThermoSpokenHere.com (A6000-054)|
In Massachusetts (1710) an accused witch was taken to the public square to be hanged. As her family and loved ones cried, support beneath her was promptly removed. But everyone noticed, the instant she began to fall Earth seemed to tremble. It seemed to all an earthquake had occured.
Upon darkness as friends were cutting her down, they discussed, "Was there an earthquake?" All had felt something. About to leave, a bystander whispered, "T'was the Power of GOD we felt. When she fell, God lifted Earth to receive her." Nearly the next day the entire village believed there had been a Visitation of GOD because she was innocent.
Sunday, from the pulpit, the Bishop said, "Visitation? Nonsense. She was guilty. The tremor wasn't God; it was gravity. The witch fell down four feet; Earth fell up just a little." He provided the solution shown below. Is the Bishop's solution correct?
♦ Exaggerated sketches of the "before" and "after" are shown. We will use the momentum equation (Newton's 2nd Law) with the system being both Earth and the woman. The momentum equation written with the discrete summation sign is:
The sum of forces equals zero because there are NO forces. All interactions we might think of as forces are inter- ior to the system.
The velocities for this event have only K components. So when we vector multiply by "•K" (scalar multiplication) the velocities become z-component velocities or "vz's." We make that multiplication, separate variables of the differentials, then integrate between limits.
Next we express velocities, "vz," as "dz(t)/dt." Make that substitution, separate variables again, then integrate once more.
Perform the integrations:
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:
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: The answers had better be the same.