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 force 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 →forces 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 pressure 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 2 - SI unit is Pa (pascal) = N/m 5 →pressure of atmosphere at sea level is about 10 Pa 5 - 10 Pa ≈ 1 atm (atmosphere) 5 →total air pressure in a typical automobile tire ≈ 3 x 10 Pa (3 atm) - a fluid exerts pressure →hydraulic press F1 F2 P 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 = atmospheric pressure) 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 weight →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 Ax 1 A1x1 2 A2 x2 x v t 1 A1v1t 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 (½v ), 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 P1<P2 →if A1>A2 then v1<v2 →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 - gas laws →when the density of a gas is sufficiently low, the pressure, volume, and temperature 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 temperatures and pressures - the ideal gas law 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 P1V1 P2V2 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 .