Pressure and Fluids § 12.1–12.3

Density Relating “how big” to “how much” § 12.1

Density Mass per volume  = dm/dV; m =  dV If  is constant, m =  V SI units kg/m 3 ; conventionally g/cm 3 Intensive quantity –Depends on substance, not amount

Densities Substancekg/m 3 g/cm 3 Air Plain water1, Sea water1, Lead11,

Pressure One type of stress § 12.2

Pressure Force per unit area p = F  /A F  is component perpendicular to surface Scalar quantity

Units of Pressure SI Units: N/m 2 = pascal = Pa Bar = 100,000 Pa; mbar = hPa = 100 Pa Atmosphere = atm = mean sea level pressure = 101,325 Pa = 1.01 bar Torr: Height in mm of a mercury barometer (760 torr = 1 atm) lb/in 2 = psi ( psi = 1 atm)

Gauge and Absolute Pressure Gauge: increase above surrounding atmosphere –e.g. tire pressure 32 psig Absolute: total

Poll Question When a molecule bounces elastically off a surface, what is the direction of its momentum change? A. B. C. D.

Poll Question When a molecule bounces elastically off a surface, what is the direction of the force on the surface? A. B. C. D.

Collisions Collision exerts a force into surface Pressure from force averaged over time and area Force

Pressure in a fluid Pascal’s Principle: pressure in a fluid is exerted uniformly in all directions throughout the fluid.

Hydraulics just another simple machine

Quick Question If a force F 1 is applied to the small piston with area A 1, what is the pressure increase in the fluid? A.  p = F 1 A 1. B.  p = F 1 + A 1. C.  p = A 1 /F 1. D.  p = F 1 /A 1. 12

Poll Question If a force F 1 is applied to the narrow piston with area A 1, How does the magnitude of the force F 2 exerted by the fluid on the wide piston compare to F 1 ? A. F 2 = F 1. B. F 2 < F 1. C. F 2 > F 1. 12

Hydraulics Pistons have different areas Pressure p = F/A for both pistons F1F1 A1A1 F2F2 A2A2 = F1F1 F2F2 A1A1 A2A2 = rearranges to

Poll Question If the narrow piston with area A 1 is moved a distance  x 1, How does the distance  x 2 that the wide piston moves compare to  x 1 ? A.  x 2 =  x 1. B.  x 2 <  x 1. C.  x 2 >  x x1x1 x2x2 ?

Hydraulics are Simple Machines Volume changes  V are opposites  V = A 1  x 1, so  x 1 =  V/A 1 ;  x 2 = –  V/A 2 F1F1 A1A1 F2F2 A2A2 = F 1  x 1 = –F 2  x 2 input, output work equal and opposite = VV A1A1 F1F1 VV A2A2 F2F2

Example Problem In an auto shop a hydraulic jack is used to lift vehicles for service. If the radius of the cylinder below piston 1 is r 1 = 0.02 m, and the radius of the cylinder below the piston holding the car is r 2 = 0.1 m, what force F 1 must be exerted on the small piston to lift a 1500-kg car?

Pressure with Depth

Pressure beneath a fluid Force comes from weight of fluid above Pressure increases with depth Supports weight above

Pressure in a liquid p = F  /A =  gh weight of fluid above cross-section = mass  g A density  volume  g A = A  Ahg = h p = pressure here

Depth Pressure Formula p = p 0 +  gh p 0 = pressure at depth 0 p = pressure at depth h  = constant density of liquid h = depth under top of liquid h p = pressure here p 0 = pressure here

Buoyancy How do things float? § 12.3

What forces are present?

Pressure in a fluid Pressure increases with depth Greater pressure at bottom than top of an immersed object Results in upward buoyancy force that is the (vector) sum of all pA forces F =  p dA A

Buoyancy from U g Force = –dU g /dh U g = U g obj + U g displaced fluid  U g = mg(  h) +  Vg(–  h) ∑F = –dU g /dh = –mg +  Vg –mg = gravity force  Vg = buoyancy force hh

Buoyancy Force Buoyancy force = weight of fluid displaced F =  Vg   = constant density of fluid  V = volume of fluid displaced = volume of object submerged  g = 9.8 N/kg

Quick example The density of fresh water  = 1000 kg/m 3. a.What is the buoyancy force on a 1-m 3 parcel of water? b.What is the buoyancy force on a 1-m 3 rock submerged under water? c.What is the buoyancy force on a 1-m 3 chunk of Styrofoam submerged under water?

Quick Questions All blocks are cubes with 1-m sides. Which ones float? Which one makes the water level change the most? Which one makes the water level change the least?

Buoyancy and Density If an object is more dense than the surrounding fluid, it sinks If an object is less dense than the surrounding fluid, it rises A floating object displaces exactly enough fluid to support its weight

Quick Question Which boat (with cargo) has the greatest volume? A. The high boat (left). B. The low boat (right). C. Both have the same volume.

Quick Question Which boat (with cargo) has the greatest volume under the water line? A. The high boat (left). B. The low boat (right). C. Both have the same volume under water.

Quick Question Which boat (with cargo) experiences the greatest buoyancy force? A. The high boat (left). B. The low boat (right). C. Both experience the same buoyancy force.

Quick Question Which boat (with cargo) has the greatest mass? A. The high boat (left). B. The low boat (right). C. Both have the same mass.

Poll Question After a boat sinks, it displaces A.more water than when it floated. B.less water than when it floated. C.the same volume of water as when it floated.