TUTORIAL SOLUTION SHEET 3 – Pressure A. 1. Concepts: – Pressure forces act inwards from all directions – Pressure and depth Relevant Equations: = ! + P:
Pressure !: Pressure at surface :
Density of fluid d: depth ” =
Diagrams: Solution: The pressure you experience when submerged in a fluid (including the atmosphere, remember tute 1!) causes forces that press inwards on you from all directions. Furthermore, pressures add. This way, deep down in the ocean the pressure is much greater than that of sea level. This is because the atmospheric pressure, !, is increased by the weight of the water for a distance above you (this is the part of the pressure equation, see Figure 1). Human lungs are mainly designed to withstand the pressure differences caused by your diaphragm, allowing air to be drawn in and pushed out from them. When your diaphragm pulls down to expand your lungs (Figure 2), you now have a larger area for the same amount of air, and so the pressure, #, goes down inside your lungs. This draws in air from the pressure difference (# < !) created between your lungs and the air outside, and vice versa for breathing out. As a diver goes further down in the ocean, the added pressure from the water above causes a large pressure difference between the inside of their lungs and its surrounds. To be able to breath at all, divers are accompanied with an oxygen tank and pressure regulator that keeps the pressure in their lungs at a safe level, $. If they go too deep, however, their lungs will eventually collapse under the added pressure from the ocean! It is usually recommended to try to avoid this effect. The pressure forces also compress the air in your blood to much smaller volumes. As a diver rises to the surface of the ocean, the air in your blood begins to expand as the pressure decreases. If you rise too fast, and don’t allow enough time for this extra volume of air to escape your blood stream through your lungs, bubbles form in your blood. This can be extremely painful and even cause death if the air bub
bles reach your heart. When diving, one must surface slowly to avoid getting the bends. Figure 1: Breathing in Figure 2: Diving pressures Figure 3: Pressure Regulator B. Concept: – Pressure at points of equal height in a fluid at rest are equal Relevant Equation: = ! + Solutions i): Water is less dense than mercury and cannot mix with it. We learned in last week’s tutorial that the buoyant force caused by a volume of liquid increases with the liquids density (remember that we compared air to water with the submerged scales question). In this way, the mercury exerts more upward force on the water than the water would on the mercury, and so the water floats and the mercury stays at the bottom. Diagrams ii): Solutions ii): The key to this problem is that the pressure at every point at a given horizontal level in a single body of fluid at rest must be the same. The mercury in this question is a single body of fluid, so the pressure at the height of the lowest mercury surface is equal to the pressure at the same height in the other arm of the U-tube. In each of the arms, the pressure is caused by the above fluid (including the air, which is at atmospheric pressure, !, and so is equal for both sides of the tube): %&'( = )*+,( [ ]-+ + ! = [ ]-!! + ! -+ = -!! -!. = -!. -!.-+ = 1.00 × 10/ kg.m0/ × 0.050 m13.55 × 10/ kg.m0/ = 0.0037 m = 0.37 cm Diagrams iii): Solutions iii): We
use the exact same method as above in ii) to find 123425: %&'( =
)*+,( [ ]%*67*8 + ! = [ ]-+ + [ ]-!. + ! [ ]%*67*8 =
[ ]-+ + [ ]-!! %*67*8 = -+ + -!. -!. %*67*8 = 13.55 × 10/ kg.m0/ ×
9.4 × 100/ m + 1.00 × 10/ kg.m0/ × 5.0 × 1009 m3.0 × 1009 m = 5.9 × 10/ kg . (2 sig figs) Comparing with the two possible elements, the liquid metal is gallium. There is an easier way to determine the unknown metal between gallium and caesium as well. Caesium is an extremely volatile element that would explode when put in contact with the water and even spontaneously combusts in air! Unfortunately, however, explosions are no longer an accepted method of diagnosis in the University of Sydney.