Lecture 17: Fluids II l Archimedes’ Principle (continued) l Continuity Equation l Bernoulli's Equation

Archimedes’ Principle l Buoyant Force (F B ) è weight of fluid displaced è F B =  fluid V displaced g è F g = mg =  object V object g è object sinks if  object >  fluid è object floats if  object <  fluid l If object floats… è F B = F g è Therefore:  fluid g V displ. =  object g V object è Therefore: V displ. /V object =  object /  fluid

Continuity of Fluid Flow Fluid moving from narrow tube to wide tube:  A is cross-sectional area of each region.  v is velocity of fluid in each region. Mass of the fluid is conserved:  Continuity Equation:  1 A 1 v 1 =  2 A 2 v 2  If density does not change: A 1 v 1 = A 2 v 2 Flow rates:  Mass Flow Rate:  A v (units: kg / s)  Volume Flow Rate: A v (units: m 3 / s) A1A1 A2A2 v2v2 v1v1

Fluid moving with change in height and velocity:  Bernoulli's Equation: P 1 +  1 g y 1 + ½  1 v 2 = P 1 +  1 g y 1 + ½  2 v 2  This equation shows us that the pressure in a fluid depedns on the height of the fluid (the higher you go the lower the pressure) and the velocity of the fluid (the faster the fluid is moving the lower the pressure). y1y1 y2y2 v2v2 v1v1 Bernoulli’s Equation P1P1 P2P2

Fluid Flow Summary Mass flow rate:  Av (kg/s) Volume flow rate: Av (m 3 /s) Continuity: A 1 v 1 = A 2 v 2 Bernoulli: P / 2  v  gh 1 = P / 2  v  gh 2 A 1 P 1 A 2 P 2 v1v1 v2v2 

Example l A garden hose, with diameter 2 cm, carries water at 1.0 m/s. To spray your friend, you place your thumb over the nozzle giving an effective opening diameter of 0.5 cm. What is the speed of the water exiting the hose? What is the pressure difference between inside the hose and outside? è To answer the first question we will use: »Continuity Equation è To answer the second question we will use: »Bernoulli’s Equation

Example l A garden hose, with diameter 2 cm, carries water at 1.0 m/s. To spray your friend, you place your thumb over the nozzle giving an effective opening diameter of 0.5 cm. What is the speed of the water exiting the hose? What is the pressure difference between inside the hose and outside? è Continuity Equation: »A 1 v 1 = A 2 v 2 »v 2 = v 1 ( A 1 / A 2 ) »v 2 = v 1 ( r 1 2 / r 2 2 ) »v 2 = 16 m/s

Example l A garden hose, with diameter 2 cm, carries water at 1.0 m/s. To spray your friend, you place your thumb over the nozzle giving an effective opening diameter of 0.5 cm. What is the speed of the water exiting the hose? What is the pressure difference between inside the hose and outside? è Bernoulli's Equation: »P 1 +  g y 1 + ½  v 1 2 = P 2 +  g y 2 + ½  v 2 2 »P 1 – P 2 = ½  ( v v 1 2 )(note: y 1 = y 2 ) »P 1 – P 2 = x 10 5 Pa Note: this difference is a little more than atmospheric pressure.