Chapter 9 Fluid Mechanics

Fluids “A nonsolid state of matter in which the atoms or molecules are free to move past each other, as in a gas or liquid.” (p. 318) Solids: definite shape and volume. Liquids: definite volume but no definite shape. Gases: no definite shape or volume; has volume and shape of container.

Mass Density “The mass per unit volume of a substance.” (p. 319) Mass density = mass / volume ρ = m / V For mass, we will use grams (g) or kilograms (kg) and for volume we will use cm 3, m 3, liters (L) or milliliters (mL). 1 m 3 = 1 x 10 6 cm 3 1 L = 1000 mL 1 cm 3 = 1 mL Common units of density are g / cm 3, g / mL, and kg / m 3 Solids and liquids are almost incompressible, which means their densities do not change. Gases are compressible; so their densities depend on temperature and pressure.

Buoyant Force “A force that acts upward on an object submerged in a liquid or floating on a liquid’s surface.” (p. 319) Archimedes Principle: “any object submerged in a fluid experiences an upward buoyant force equal in magnitude to the weight of the fluid displaced by the object.” (p. 320) Buoyant force = weight of displaced fluid F B = F g (displaced fluid) = m f g (m f = mass of fluid displaced)

Other buoyancy formulas…. An object submerged in a fluid has an “apparent weight” that is less then it’s normal weight: F net = F B – F g (object) = m f g – m o g (m o = mass of submerged object) = m f g – m o g (m o = mass of submerged object) And since ρ = m / V, m = ρV. So, the above formula becomes F net = ρ f V f g– ρ o V o g F net = ρ f V f g– ρ o V o g

Floating Objects If an object floats, then its density is less than the density of the fluid (ρ o < ρ f ) For floating objects, the buoyant force equals the weight of the floating object: F B = F g (object) = m o g The density of an object determines the depth of submersion: F net = ρ f V f g– ρ o V o g = 0 ρ f V f – ρ o V o = 0 (divide by g) ρ f V f – ρ o V o = 0 (divide by g) ρ f V f = ρ o V o (add ρ o V o to both sides) ρ f V f = ρ o V o (add ρ o V o to both sides) ρ f / ρ o = V o / V f (divide by ρ o V f )

Sinking Objects… If an object sinks, then its density is greater than the density of the fluid (ρ o > ρ f ). If an object sinks below the surface of a fluid, then the volume of the fluid displaced equals the volume of the object. So, V f = V o and we can replace both with just V: F net = ρ f Vg– ρ o Vg = F B – F g (object) A simple relationship results from the above equation: F g (object) / F B = ρ o / ρ f F g (object) / F B = ρ o / ρ f