AP Physics “B” – Unit 4/Chapter 8 Notes Rotational Equilibrium & Rotational Dynamics

Torque -  (tau) – SI derived unit is N·m - net torque produces rotation →torque – a quantity that measures the ability of a to rotate an object about some axis - torque depends on a force and a lever arm →how easily an object rotates depends not only on how much force is applied but also on where the force is applied →lever arm – the ┴ distance from the axis of rotation to a line drawn along the direction of the force - torque also depends on the angle between a force and a lever arm → do not have to be ┴ to cause rotation

   Fr(sin)    Fd

- two equal but opposite forces can produce a rotational acceleration if they do not act along the same line - if torques are equal and opposite, there will be no rotational acceleration →seesaw – momentary torque produced by pushing with legs

Rotational Equilibrium - equilibrium requires zero net force and zero net torque →if the net force on an object is zero, the object is in translational equilibrium →if the net torque on an object is zero, the object is in rotational equilibrium - the dependence of equilibrium on the absence of net torque is called the second condition of equilibrium →the resultant torque acting on an object in rotational equilibrium is independent of where the axis is placed - an unknown force that acts along a line passing through this axis of rotation will not produce any torque - beginning a diagram by arbitrarily setting an axis where a force acts can eliminate an unknown in a problem - conditions for equilibrium Type of Equation Symbolic Eq. Meaning translational Fnet=0 Fnet on an object must be zero rotational net=0 net on an object must be zero - unstable equilibrium – center of mass is above support point - stable equilibrium – center of mass is below support point

AP Physics “B” – Unit 4/Chapter 9 Notes – Yockers Fluids

Defining a Fluid – a nonsolid state of matter in which the atoms or molecules are free to move past each other, as in a gas or liquid - liquids have a definite volume - gases do not have a definite volume

Density - mass density is mass per unit volume of a substance - mass density =  (rho) - SI unit = kg/m3 m   V - note: because they are compressible, there are no standard densities for gases - specific gravity of a substance is the ratio of its density to the density of water at 4°C (1.0 x 103 kg/m3) →specific gravity is a dimensionless quantity →for example, if the specific gravity of a substance is 3.0, its density is 3 3 3 3 3.0(1.0 x 10 kg/m ) = 3.0 x 10 kg/m

Pressure – force per unit area (the average P in a fluid at the level to which an object has been submerged is defined as the ratio of force to area) F P  A - SI unit is Pa (pascal) = N/m2 →pressure of atmosphere at sea level is about 105 Pa - 105 Pa ≈ 1 atm (atmosphere) →total air pressure in a typical automobile tire ≈ 3 x 105 Pa (3 atm) - a fluid exerts pressure →hydraulic press F F P  1  2 A1 A2 - pressure varies with depth in a fluid →”columns” - air - water F ma Va Aha P   g  g  g  ha , so A A A A g →fluid pressure as a function of depth (Po = )

P  Po  hag →the pressure P at a depth h below the surface of a liquid open to the atmosphere is greater than atmospheric pressure by the amount hag

Atmospheric Pressure – is pressure from above - the force on our bodies (surface area of 2m2) is 200,000 N (40,000 lb) - survival? - fluids and gases within our bodies push outward with a pressure equal to that of the athmosphere (what would happen to our bodies in the vacuum of space?) - mercury barometer measures the pressure of the atmosphere

Buoyant Force – a force that acts upward on an object submerged in a liquid or floating on the liquid’s surface - because the buoyant force acts in a direction opposite the force of gravity, objects submerged in a fluid such as water have a net force on them that is smaller than their →apparent weight - Archimedes’ principle – any object completely or partially submerged in a fluid is buoyed up by a force whose magnitude is equal to the weight of the fluid displaced by the object →buoyant force = FB →mass of displaced fluid = mf

FB  m f ag  Fg (displaced fluid ) - float or sink? →mass of submerged object = mo

Fnet  FB  Fg (object )

Fnet  m f ag  mo ag and m  V, so

Fnet   f V f  oVo ag →a floating object cannot be denser than the fluid in which it floats

Fnet  0   f V f  oVo ag  V f  o o V f →displaced volume of fluid can never be greater than the volume of the object →the object’s density can never be greater than the density of the fluid in which it floats →if densities are equal, the entire object is submerged - for floating objects, the buoyant force equals the object’s weight

FB  Fg (object)  moag

- the apparent weight of a submerged (V f  Vo ) object depends on density

Fnet   f V f  oVo ag V f  Vo , so

Fnet   f  o Va g

Fnet  FB  Fg (object ) F (object )  Va  g  o g  o FB  f Va g  f

Buoyant Forces in a Fluid

Pnet  Pbottom  Ptop

Pnet  Po  h2 ag  Po  h1ag   ag h2  h1 

Pnet  ag L

Fnet  Pnet A  ag LA  agV  m f ag

Fluid Flow - flow is laminar if every particle that passes a point follows the same smooth path traveled by earlier particles - path of flow is called a streamline - irregular flow is turbulent (very complex) - we’ll focus on laminar flow

Ideal Fluid - a fluid that has no internal friction or viscosity and is incompressible →density remains constant →not realistic, but explains many properties of real fluids →viscosity = amount of internal friction - ideal flow → the velocity, density, and pressure at each point in a fluid do not change in time - nonturbulent flow - we’ll assume we’re working with ideal fluids

Conservation Laws in Fluids - mass in bottom (one end) must equal mass out of top (out other end)

m1  m2 m  V 

1V1   2V2 V  Ax

1 A1x1   2 A2 x2 x  v  t

1 A1v1t   2 A2v2 t 1   2 & t  t

A1v1  A2v2

- continuity equation →based on the conservation of mass Flow rate  Av - conservation of energy in fluids → pressure + kinetic energy + potential energy is constant 1 P  v 2  a h  constant 2 g       F  s F  s F W  F  s      P V (A)(s) A  m  K  1 mv2  1  v 2  1 v 2 2 2  V  2  m  U g  mag h   ag h  ag h  V  →Bernoulli’s equation - comparing energy in a fluid at two different points 1 1 P  v 2  a h  P  v 2  a h 1 2 1 g 1 2 2 2 g 2 - Bernoulli’s equation states that the sum of the pressure (P), the kinetic energy 2 per unit volume (½v ), and the potential energy per unit volume (agh) has the same value at all points along a streamline

→Bernoulli’s equation -if fluid is not moving (v = 0), and h1 = 0 (zero level)

P1  P2  ag h2 the same as fluid depth pressure formula -if h is constant (horizontal) 1 1 P  v 2  P  v 2 1 2 1 2 2 2 →if v1>v2 then P1A2 then v1

→Bernoulli’s principle - swiftly moving fluids exert less pressure than do slowly moving fluids (P↓→v↑)

THE FOLLOWING IS NOT INCLUDED IN CHAPTER 9. IT IS COVERED LATER.

Kinetic Theory of Gases - gas particles act similarly to billiard balls constantly colliding - force per unit area is the gas pressure

Properties of Gases - →when the density of a gas is sufficiently low, the pressure, volume, and of the gas tend to be related to one another in a fairly simple way →this relationship is a good approximation for the behavior of many real gases over a wide range of and - the relates gas volume, pressure, and temperature

PV  Nk BT or PV  nRT →N = the number of gas particles -23 J →kB = Boltzmann’s constant = 1.23 x 10 /K →n = the number of moles J →R = 8.31 /mol·K - if a gas undergoes a change in volume, pressure, or temperature (or any combination of these), and - if the number of particles in the gas is constant, then N1=N2, so PV P V 1 1  2 2 T1 T2 - and considering the ideal gas law’s dependence on mass density, if we assume each particle in the gas has a mass, m, then the total mass of the gas is N x m = M - so we can also write the ideal gas law as Mk T PV  Nk T  B B m

Mk T  M  k T k T P  B    B  B mV  V  m m - a real gas can often be modeled as an ideal gas →no real gas obeys the ideal gas law exactly for all temperatures and pressures - at high pressures or low temperatures a gas nearly liquefies →remember, an ideal fluid is a gas or liquid that is assumed to be incompressible →but since we often confine gases to containers, it cannot flow, and, thus, must be compressed