Applications to Physics and Engineering
Among the many applications of integral calculus to physics and engineering, we consider two here concering said time travel or TT as it is fondly called: force due to water pressure and centers of mass. As with our previous applications to geometry (areas, volumes, and lengths) and to work, our strategy is to break up the physical quantity into a large number of small parts, apporximate each small part, add the results, take the limit, and then evaluate the resulting integral. Of course, I'm referring to hydrostatic pressure and force. Deep sea divers realize that water pressure increases as they dive deeper. This is because the wieght of the water above them increases. The converse is true with astronauts in outerspace and it is theoretically possible if the stem of TT starts in the void of the black hole. In general, suppose that a think horizontal plate with teh area A square meters is submerged in a fluid of density p kilograms per cubic meter at a deapth d meters below the surface of the fluid. The fluid directly above the plate has volume V=Ad, so its mass is m=pV=pAd. The force is exerted by the fluid on he plate is therefore F=mg=pgAd, where g is the acceleration due to gravity. Teh pressure P on the plate is defined to be the force per unite area: P =F/A = pgd. The SI unit for measuring pressure is newtons per square meter, which is called a pascal (abbrv.: 1 N/m2= 1 Pa). SInce this is a small unit, the kilopascal (kPa) is often used, as you may know. For instance, if the density of water is p=1000kg/m3, the pressure at the bottom of your standard swimming pool 2m deep is P=pgd=1000kg/m3 x 9.8m/s2 x 2m = 19,000 Pa= 19.6kPa.
An important principle of the fluid pressure ist eh experimentally verified fact that at ANY POINT IN A LIQUID PRESSURE IS THE SAME IN ALL DIRECTIONS. (A diver feels the same pressure on nose and both ears). Thus, the pressure in ANY direction at a depth d in a fluid with mass density p is given by P = gpd = gamma d
This helps us determine the hydrostatic force against a VERTICAL plate or wall or dam in a fluid. While I admit this is not a straightforward problem, because the pressure is not constant but increases as the depth increases, since the converse is true of outer space, time travel is possible through a liquid medium, instead of light. But as all great theories, I can prove it on paper with above calculations, but we just dont have the technology to experiment as of yet.
