Tuesday, July 28; 9:35AM 10:50AM in 273 Willard 20 Mul=ple Choice Ques=ons See Folder in Exam Resources Midterm 2 Informa=on You are responsible for recording your 9 digit PSU Student ID on your scantron form bringing a #2 pencil bringing a calculator you cannot use a phone s calculator marking your sec=on on your exam booklet
Mandlebrot Set (image courtesy of Wikipedia) Fluids
Density One important quan=ty that allows us to dis=nguish one material from another is the mass to volume ra=o: the density. p = p0 + gd m = V This ra=o is iden=cal for any par=cular material regardless of its shape r size. p= p +o gd 0 p = p0 + gd m = V p = p0 + gd m = V water = 1.0 103 kg m3 3 kg copper = 9.0 10 3 m kg gold = 1.9 104 3 m water = 1.0 103 kg m3 m = V kg m3 3 kg copper = 9.0 10 3 m kg gold = 1.9 104 3 m water = 1.0 103
Pressure Pressure in general is defined to be a force applied over some area p = F A Pressure is not a vector quan=ty, despite arising from a force A ~N = mg
Concept Ques=on: You push down on a nail with a force of 10 N, and also on a brick with 10 N. By Newton s 3 rd Law, each object pushes back on you with the same force. Which object exerts a greater pressure on your hand? A) The nail B) The brick C) Pressure is the same D) Not enough informa=on to tell p = F A F = 10 N F = 10 N
Concept Ques=on: The oddly- shaped container below is open to the atmosphere and is filled with water (an incompressible fluid). How does the pressure of the fluid at Point A compare to the pressure of the fluid at Point B? A) P A > P B B) P A < P B C) P A = P B D) Depends on the density of the fluid p = p 0 + gd A B Same ques=on, different container. Note that one side is sealed. A) P A > P B B) P A < P B C) P A = P B D) This situa=on is not physically possible A B
Pascal s Principle ~F = mg A mass is added to the top of the liquid, and exerts a pressure due to its weight on the en=re surface of the liquid. p surface = 1 atm + mg A The added pressure is transmimed throughout the liquid A p A = p surface + gd Pascal s Principle: A change in pressure at any point in an incompressible fluid is transmimed everywhere throughout the fluid.
Example: Car Lin A hydraulic lin consists of an incompressible fluid in a sealed container. The cap on each side is free to slide ver=cally. For the lin shown here, what weight can be lined by the larger piston if a 250 N force is applied to the smaller piston? Assume each piston is circular, and that R 1 = 0.01 m and R 2 = 0.10 m. p A = p B + gd = p B p What about work done? F A x A = F B ( x B ) F A = F B A A A B AB =) F B = F A A A 0.10 2 = 250 0.01 2 = 25000 N
Archimedes Principle F net = F 2 F 1 =(P 2 P 1 )A ~F 1 ~F 2 P 2 = P 1 + gd =) P 2 P 1 = gd ) What is net horizontal force? F net = gda = g V = mg = Weight of fluid Archimedes Principle: The upward buoyant force on an object equals the weight of the fluid the object displaces.
Concept Ques=on: You drop a spherical piece of copper into a lake. Does the copper float or sink? A) Floats B) Sinks C) Depends on the volume of the copper D) Depends on the mass of the copper copper =9.0 10 3 kg m 3 water =1.0 10 3 kg m 3 What direc=on does the buoyant force of the water on the copper point? A) Up B) Down C) At some angle D) Cannot say with given informa=on
Concept Ques=on: You place iden=cally- sized blocks of three different materials into a tank of water. Rank the density of each material from least to greatest. A) A < B < C B) A < B = C C) C < B < A D) C = B < A E) A = B = C A B C F net = F buoy F g = water Vg mat Vg = gv ( water mat ) If F net = 0, water = mat
Concept Ques=on: You place iden=cally- sized blocks of three different materials into a tank of water. Rank the density of each material from least to greatest. A) A < B < C B) A < B = C C) C < B < A D) C = B < A E) A = B = C V t A V s B C F buoy = F g water V s g = mat V t g V s V t = mat water ice 0.9 water
Example: Density Scale What is the tension in the string if this block is suspended in water? Assume the block is 100 cm 3 of Al with ρ Al = 2700 kg/m 3. y X ~F = m~a =0 X X ~T ~ FB ~F G x T + F B mg =0 T = mg F B =( Al V )g water Vg = Vg( Al water ) 100 cm 3 3 1m =0.0001 m 3 100 cm = (100 cm 3 )(9.8 m/s 2 )(2700 1000)kg/m 3 =1.67 N Watch units in density problems
y y 2 y 1 1 Bernoulli s Equa=on v 1 p 2 v 2 p Another conserva=on law! p 1 + 1 2 v2 1 + gy 1 = p 2 + 1 2 v2 2 + gy 2 = Constant Pressure ( Work ) Speed ( Kine=c energy ) Height ( Gravita=onal poten=al energy ) Bernoulli s Equa=on: Relates the quan==es (pressure, speed, and height) at any two points within a moving fluid.
Concept Ques=on: Air (which is a fluid) flows around the object shown below. Because of the curve, air flows faster across the top than it does the bomom. What direc=on is the net force on this object? Neglect the buoyant force. A) Down B) Up C) Net force is zero D) Depends on density of object p 1 + 1 2 v2 1 + gy 1 = p 2 + 1 2 v2 2 + gy 2 = Constant v top >v bottom =) ) p top <p bottom
Example: Leaky Tub A rectangular tub, 2.0 m long, 0.60 m wide, and 0.45 m deep, is completely full of water. One end of the tub has a small drain plug right at the bomom edge. When you pull the plug, at what speed does water emerge from the hole? p t + 1 2 v2 t + gy t = p b + 1 2 v2 b + gy b p p p t + gy t = p b + 1 2 v2 b gy t = 1 2 v2 b p t = p b = 1 atm Drop a mass m from height y: =) v b = p 2gy = 2.97 m/s U g = mgy = K 1 2 mv2 Bernoulli s equa=on is just energy conserva=on applied to fluids! =) v = p 2gy
Reminders Reading: Chapter 14.1 14.5 Pre- lecture problem: KJF 14.P.020 HW 07 due Sunday at 10 PM Office hours Thursday at 1:00 (Osmond 116A) Midterm 2 tomorrow in 273 Willard