APPENDICES 717

Density and Specific Gravity Some Useful Geometry Formulas The density (ρ) of an object or material sample is equal to its mass (m) divided by its volume (V). Rectangle or Square = —m (Length = L, Height = H) ρ V The specific gravity (SG) of a material is equal to its density Area = LH divided by the density of pure water. Perimeter = 2(L+H)

The specific gravity is greater than one for materials that are Circle denser than water and less than one for materials that are less (Radius = R) dense than water. Area = pR2 Circumference = 2pR Hydrostatic Fluid Pressure The following very general formula can be used to calculate the change in pressure (Dp) associated with a specific change Triangle in depth (Dh) in any incompressible fluid having density (rfluid) (Base = B, Height = H) subjected to any constant acceleration (a). For example, this 1 Area = –BH formula can be used to calculate pressure changes under an 2 ocean of liquid water on Europa (one of Jupiter’s moons) or even under the sea of liquid methane thought to exist on Titan (one of Saturn’s moons). Rectangular Block or Cube (Length = L, Width = W, Height = H) Δp = ρfluid aΔh Area = 2(LW + LH + WH) In the specific case of water on or near earth’s surface, where Volume = LWH the freefall acceleration due to gravity is g = 9.8 m/s2, this becomes: Note that for a cube L=W=H, so... Area = 6L2 Δp = ρ gΔh 3 H2O Volume = L For additional information about calculating hydrostatic pressure on earth, including tables of pressure as a function of depth in both freshwater and seawater, please refer to Appendix III: Hydrostatic Pressure. Cylinder (Length = L, Radius = R) Absolute and Gauge Pressure Area of circle at each end = pR2 Under water, the total pressure (called the absolute pressure) Area without ends = 2pRL is the sum of two pressures: the pressure of the air pressing Total Area = 2pR2 + 2pRL down on the surface of the water, which is one atmosphere (1 Volume = pR2L atm) of pressure, and the additional pressure (Pgauge) caused by the weight of the water pressing down from above.

P absolute = P gauge + 1 atm Sphere Note that tire pressure gauges and most other pressure gauges (Radius = R) display gauge pressure, so they read 0 in air at sea level, even Area = 4pR2 4 though they are really exposed to 1 atmosphere of pressure Volume = – pR3 there. However, for calculating changes in gas volume 3 associated with changes in depth, you should always use the absolute pressure. Cone Hydrostatic Force (Height = H, Radius of base = R) When hydrostatic pressure acts on a surface, it applies a force Area of circular base = pR2 perpendicular to the surface. Hydrostatic forces can be huge Area without base = pRH and are one of greatest threats to underwater vehicles. The Total area = pR2 + pRH magnitude of the net hydrostatic force pushing inward (F ) in –1 2 on a section of pressure hull is equal to the pressure difference Volume = 3 pR H between inside and outside (pout–pin) multiplied by the surface area (A) exposed to the pressure difference.

Fin = (pout – pin)A