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Environmental Physics Chapter 1: Forces of Nature Copyright © 2007 by DBS Concepts • • • • • • • Many natural systems contain motion, forces and momentum that can be described by Newton’s laws Frictional forces are important in any real-world motion, dissipating energy Gravity acts between any two bodies, dependent on their mass and distance apart Rotational motion can be described by laws and equations analogous to those for straight line motion, particuarly relevent to climate and orbits Different types of wave observed in environmental systems have certain properties in common Electricity and magnetism and inextricably linked, as one induces the other The Earth’s magnetic field provides a tool to investigate the geological history of the Earth and a subsurface surveying technique Question What comes next? 1, 2, … 1, 2, 3… Predictable? 1, 2, 3, 2, 1, 2, 3… Predictable? Physical events are predictable and quantifiable More complex the pattern, the longer one must observe and the greater the need for accurate record keeping Newtonian Mechanics • Why are blue whales bigger than elephants • Why don’t we just shoot CO2 into space? • Why are skyscapers taller than trees? Buoyancy of the surrounding ocean water supports the weight of the whale's body tissues Use more energy disposing of the gas than is gained from producing it Forces acting on a building are reduced by use of appropriate materials and design Newton’s Laws 3 universal laws act in the universe The First Law is just a special case of the Second Law for which the net external force is zero Newtonian Mechanics 1st law • Law of inertia: – An object that is not moving will not move until a net force acts upon it. – An object that is in motion will not change its velocity (accelerate) until a net force acts upon it. e.g. hockey puck…should continue to move forever why does it stop? There are no perfect demonstrations! 2-16 Newtonian Mechanics Momentum and Inertia • 1st law is a statement of principle of ‘conservation of momentum’, Momentum is the tendancy to continue moving once doing so u = mv Where u = momentum (kg ms-1), m = mass (kg), v = velocity (m s-1) • Heavier something is, or faster it is moving, the more momemtum it has Objects tend to "keep on doing what they're doing" (unless acted upon by a force) Newtonian Mechanics Momentum and Inertia • Total momentum after collision is conserved u1 = u2 m1v1 = m2v2 – Elastic – bounce off each other - kinetic energy remains the same – Inelastic – objects coalesce - Kinetic energy is converted into heat or sound e.g. sticky reaction: m1v1 + m2v2 = (m1+m2)vf Question Calculate the recoil speed of a 8 lb Winchester .308 rifle which launches a bullet of mass 10 g with a speed of 2820 ft s-1. [Conversions: 1 ft = 30.5 cm = 0.305 m] Momentum is concerved: mgvg = -mbvb mbvb = ub = 0.01 kg x 2820 ft s-1 x 0.305 m / ft = 8.6 kg m s-1 This gun has a mass = 3.8 kg Momentum is concerved: mgvg = -mbvb vg = -mbvb = - ub/mg = - 8.6 kg m s-1 / 3.8 kg = -2.3 m s-1 Answer: 2.3 m s-1 Newtonian Mechanics Forces • 2nd Law: rate of change of momentum (u) is proportional to the applied force • Heavier something is and the faster it is moving the more difficult to stop it, cf. horse vs. cheetah f = Δu / Δt f = Δ(mv) / Δt = m Δv / Δt = m dv/dt = ma • Acceleration produced by a force is proportional to the magnitude of the force and inversely proportional to the mass of the object f = m a (units kg m s-2 or N) Effects of an identical force F acting on two different masses… Quote Quote 1-15 Law 3: Action and reaction are equal and Quote opposite 2-22 Newtonian Mechanics Motion • For constant velocity v, Velocity = distance / time v=s/t , v= u+ v 2 • For uniform motion in a straight line at constant acceleration s, v, and a are linked by 3 simple equations v = u + at s = ut + ½ at2 v2 = u2 + 2as Since a = dv/dt a = (v – u)/t v = u + at Question Derive s = ut + ½ at2 from the first 3 equations. s = v t = [(u + v)/2] x t Substitute: v = u + at s = [(u + u + at)/2] x t = ](2u + at)/2] x t s = ut + ½ at2 Question A car is travelling at 30 m s-1 and takes 10 seconds to acceleration to a new speed of 35 m s-1. What is its acceleration? Use v = u + at 35 m s-1 = 30 m s-1 + a x 10 s 10 a = 35 - 30 = 5 m s-1 a = 0.5 m s-2 Note About Homework • Mistake in question 1, units of acceleration are m s-2 • Use 10 for g (m s-2) • Note for HW you must show full working in problems!!! End • Review Newtonian Mechanics SI Units • Units must be consistent! • Answers must have units! • What is the unit of force? kg ms-2 = N (Newtons) Newtonian Mechanics SI Units • The Reynolds number is used to characterize fluid flow (laminar or turbulent) it is the ratio of inertial (ρv) to viscous (μ/l) forces Re = ρvl μ Where ρ = density (kg m-3), v = velocity (m s-1), l = length (m) μ = viscosity (kg m-1 s-1) • Show that Re has dimensionless units Newtonian Mechanics Scalars and Vectors • Vectors: direction and magnitude – Force, velocity and acceleration • Scalar: magnitude but no direction – Mass, time, length, charge • Scalars can be added, vectors cannot…why is the wind speed not 35 km h-1? 15 km h-1 20 km h-1 Newtonian Mechanics Scalars and Vectors 15 km h-1 (i) Find apparent wind velocity for cyclilst 20 km h-1 v a2 = v h2 + v s 2 (152 + 202)1/2 = 25 km h-1 (ii) Resolve forces on kite to find wind speed, v v = fr x cosθ θ Newtonian Mechanics Friction and Air Resistance • Friction – force between two objects due to roughness of their touching surfaces • Static or dynamic – Depends on surface roughness – Affected by lubricants (oil, ball bearings, polymers etc.) • Friction between solid objects and fluids = drag – Depends on shape, size, surface characteristics and speed – Important for wind pollination, wind stress resistance and erosion What is friction in an electrical circuit called? Newtonian Mechanics Friction and Air Resistance • Skin drag – Important for small particles (e.g. PM) or objects in viscous fluid – Fd ~ radius x viscosity x velocity Fd = 6πrμv • Form drag – Important for larger objects in air – Fd ~ radius2 x velocity2 x air density x drag coefficient (dependant on objects shape) Fd = ½ ρaC πr2v2 Where ρa = air density (1.2 kg m-3), C = drag coefficient Form drag >>> Skin drag at high velocity (v2) Freefall Demo • Which of two objects will strike the floor first if dropped • All objects fall at the same rate when air resistance can be ignored • If the mass is too low compared to surface area (as in a feather) air resistance becomes important http://nssdc.gsfc.nasa.gov/planetary/image/featherdrop_sound.mov Apollo 15 Movie (1971) 1-14 Newtonian Mechanics Gravity • Newtonian Gravity – Anything falling under gravity falls at same rate g = 9.8 m s-2 f = mg • Calculate time of fall and velocity s = ut + ½at2 v = u + at Let a = g = 9.8 m s-2 t = 0.9 s, v = 8.9 m s-1 Newtonian Mechanics Mass, Weight and Density • • • Mass – Measurement of the amount of matter something contains Weight – Measurement of the pull of gravity on an object, w = mg Mass of an object – doesn't change with location – Weight does e.g. A 10 kg mass weights 10 x 9.8 = 98 N Moon gravity is 1/6 Earth, gm = 9.8 / 6 = 1.6 m s-2 10 kg mass is now, w = mgm = 16 N Question The moon’s gravitational pull is 1/6th as great as the Out of 168 people taking a Earth’s attraction quiz, 48 missed the question. If a pen is dropped on afloat moon, willbecause it: e.g. "It will away A) Float away the gravitational force is less than B) Float where is the Earth where it would hereit on C) Fall to thefall. surface of the moon I think it will float away because of what I have seen of the space rooms NASA uses to get astronauts ready for flight." Newtonian Mechanics Landslides • • Wind, water and glaciers move large amounts of material So does gravity • Landslide when component of gravity fs > frictional forces supporting it • Steeper slope has > fs Newton decided that g was somehow related to orbital motion… Newtonian Mechanics The Universal Force of Gravity • Any two objects in the Universe exert gravitational attraction on each other, with the force having a universal form: f = GMm r2 Force of gravity ~ M x m and ~ 1 / r2 Where M and m are two masses, r is the distance between them and G is Universal gravitational constant, The more massive two objects are the greater the force between them G = 6.67 x 10-11 Nm2 kg-2 The farther apart they are, the less the force will be Newtonian Mechanics Relationship Between Big G and Little g • Now notice also that at the Earth’s surface it is true by Newton's 2nd Law that: f = -GMEm1 = m1a = m1g rE 2 • Let ME = 6.02 x 1024 kg, rE = 6.40 x 106 m and solve for g: g = -GME rE 2 g = 9.8 m s-2 This says that the acceleration you feel due to the planet is independent of your mass. That's just what Galileo showed in his famous freefall experiments Newtonian Mechanics Terminal Velocity and Settling Velocity • Objects accelerate on falling until: drag force = force of gravity • • Higher terminal velocity (vt) means large particles settle more quickly For a small sphere (PM) when viscous forces dominate (skin drag) Fg = mg = Fd mg = 6πrμv 4/3 πr3ρsg = 6πrμv vt = 2g ρs r2 9μ (m = ρs x 4/3 π r3) Where ρs is the density of the sphere. Stokes’ law: settling velocity is highest for a larger, dense object in a less viscous medium Question Derive an equation for vt for larger objects in less viscous fluids. Drag force predominates: mg = Fd 4/3 πr2ρsg = ½ ρaC πr 2 v 2 vt = 8 ρsrg 3 ρaC Question Find vt for a human being… vt = (8 ρsrg / 3 ρaC)1/2 = 8 x 1000 kg m-3 x 0.5 m x 10 m s-2 3 x 1.2 kg m-3 x 1.0 = 105 m s-1 Newtonian Mechanics Settling Chambers • For a chamber of height h and length l and gas velocity u, gas takes l /u seconds to traverse • Particle settles in h/vs seconds • Particles with large enough diameters such that h< l vs u settle out vs < hu l l h Newtonian Mechanics Settling Chambers vs < hu l Substitute into Stoke’s law (for small particles in air) and find the size of particle removed, r vt = 2g ρs r2 9μ Question Substitute into Stoke’s law and find r. r= 9μuh 2ρsgl Cheap method of removing large particles, not so good for fine! Important Note • Note: since the force due to bouyancy is not included these calculations assume large particulate density, ρs >>> ρa • If bouyancy were included: Fg = Fd + Fb mg = 6πrμv + Fb 4/3 πr3ρsg - 4/3 πr3ρag = 6πrμv 4/3 πr3g (ρs – ρa) = 6πrμv vt = 2r2g (ρs - ρa) 9μ End • Review Rotational Dynamics Moments of Inertia • Angular momentum: the tendency for something that is spinning to continue to spin • Conserved – contunes to spin unless external forces act Linear motion and forces: Rotational motion and forces: distance, s velocity, v mass, m momentum, u = mv angle, angular velocity, ω moment of inertia, I angular momentum, L = I ω Rotational Dynamics Moments of Inertia • Angular momentum: depends on speed of spin (ω) and mass distribution (I) L=Iω L = angular momentum (torque), I = moment of inertia, ω = angular velocity large I To conserve L ω increases Smaller I Rotational Dynamics Central Forces • Central Forces – Objects velocity is at a tangent to spin direction (ω = v / r) – Change in direction = change in velocity = acceleration fc = mv2 = mr ω2 r v2 / r = r ω2 is accn. component These are wrong way around in text! Demo http://www.ac.wwu.edu/~vawter/PhysicsNet/QTMovies/Rotations/Centrif ugalH2OMain.html Rotational Dynamics Coriolis Force • Force felt by any body moving relative to something rotating fc = 2mωv • Anything moving on the Earth’s surface is subjected to CF • Movement of air towards axis reduces inertia (I) so by law of conservation of momentum angular velocity (ω) increases – CF increases towards poles • CF = 0 at equator Rotational Dynamics Cyclone Separators • Air pollution control – Centrifugal force amplifies gravitational settling – Outer and inner vortex Rotational Dynamics The Vortex • • Vorticity is a measure of the rotation of a fluid about an axis Fluid speeds up as it is drawn towards center (conservation of angular momentum) e.g. Tornadoes, cyclonic winds and hurricanes Vortices form in turbulent flow http://en.wikipedia.org/wiki/Vortex Rotational Dynamics Orbits • • • Remote Sensing: geostationary or polar orbits/Low Earth Orbits (LEO) For LEO (< 2000 km) centripetal force = gravitational force mv2 = mg R Where m = mass of satellite, v = velocity of satellite, R = Earth’s radius (6400 km) and g = 10 m s-2 v = √ (gR) = 8000 m s-1 • • Time period T = 2πR/v = 83 mins Low orbit satellite travels at 8 km s-1 or 28,800 km hr-1 > 8 km s-1 escape velocity Question For geostationary orbit need to find r. Given T = 2π/ω equate the formulas for gravitational force and centripetal force and solve for r. r fc = fg msat rω2 = GmE msat / r2 rω2 = GmE / r2 r = (GmE / ω2)1/3 T = 2π/ω ω = 2π/86164 s Orbital radius r = 42164 km from center of earth. Subtracting Earth’s radius gives an altitude of 36,000 km Use v = ωr to find velocity Waves Wave Characteristics • Wavelength and frequency • • v = fλ Where v = speed (m s-1), f = frequency (cyces s-1), λ = wavelength (m) Speed varies according to medium waves travel in Waves Transverse and Longitudinal Waves • • • • Transverse: wave travels perpendicular to vibration e.g. light, EM waves Longitudinal: vibrations travel in same direction as wave e.g. sound Shape: natural waves often sine waves I = Imax sin 2π/λ (vt-x) Where I = intensity, t = time, x = distance Maybe progressive (moving) or standing (stationary) Waves Wave Properties (a) Incident wave (b) Reflection (c) Transmission and absorption (d) Refraction (a) Diffraction (b) Scattering (diffraction+reflection) (c) Interference • Polarization Waves Seismic Waves • • • Body waves consist of ‘P’ and ‘S’ waves – P travel in solid and liquid – S in solid only Surface waves cause more damage Resonance P waves are refracted End • Review Question How does the electrical force differ from the gravitational force? Gravity always pulls together, electrical force acts both ways (attracts and repels) Electrical force is many times more powerful than gravity (action of a comb on paper) Electromagnetism Electric Charge and Current • • • • Electric charge (Coulomb = 1 amp / s) Current (A) is how much charge is moving Voltage (V) is difference in electrical potential energy (provides push) Ohm’s Law V=IxR • Where R = resistance (ohms) AC and DC current Electromagnetism Electric Fields • • • Objects carrying a static charge emit an electric field in all directions that becomes weaker at greater distances Like chartges repel, unlike attract Field strength: E=V/d • Where V = voltage (v), d = distance (m) Electromagnetism Electrostatic Precipitators • • • Used to control dust and particulate matter (aerosols) High voltage Flow of e- charges particles Electromagnetism Magnets • A magnet is a metallic object (Fe, Co, Ni, alloys) that attracts another metallic object – Every magnet has at least two poles, N and S (dipole) – Like magnetic poles repel each other, while unlike poles attract – Lines of force extend from N to S create a field – Loss of magnetic field at Curie point – Presence of iron in the earth creates a field Demo • • Strength is measured in Teslas (T) Flux density 1 T = 1 N A-1 m-1 Electromagnetism The Earth’s Magnetic Field • • • • Core is above Curie point, induced by fluid and electrical currents in outer liquid iron core - Geomagnetic dynamo Influenced by solar flux Reverses periodically – 300x during last 200 million yrs Mid-ocean ridge shows magnetic reversals Electromagnetism The Earth’s Magnetic Field Computer model Source: NASA http://science.nasa.gov/headlines/y2003/29dec_magneticfield.htm Electricity and Magnetism Induction • Electrical current creates magnetism, changing magnetic fields produce current – The Electromagnet – Electromagnetic induction • Electricity and magnetism combine to produce force and motion 20-12 Electromagnet Electromagnetic Induction Demo 20-12 Electromagnetism In Animals and Plants • • • • First noticed in Robins, seen in all Migratory birds Organisms as diverse as hamsters, salamanders, sparrows, rainbow trout, spiny lobsters, and bacteria Whale beachings Bacteria contain magnetite crystals so they can tell ‘up’ from ’down’ Electromagnetism Transmission Lines • Transmission Lines – harmful to humans? • Hypothesized EMF link to childhood leukemia • Proposed: – Direct effect of electric field disrupting electrical activity in the body – Indirect effect of polarized water vapor which dissolves ionized gaseous pollutants from the air allowing them to be more readily absorbed • No conclusive evidence • Magnetic fields still an issue Demos 19-01 19-02 19-06 19-19 19-20 19-24 Further Reading • • http://www.phys.unsw.edu.au/~jw/demo/projectiles.html The monkey and the hunter http://physics.bu.edu/~duffy/semester1/c04_monkeyhunter.html Monkey-hunter Journals • Fews, A.P., Henshaw, D.L., Wilding, R.J., and Keitch, P.A., (1999) Corona ions from powerlines and increased exposure to pollutant aerosols. International Journal of Radiation Biology, Vol. 75, No. 12, pp. 1523-1531. • Fews, AP, Henshaw, D.L., Keitch, P.A., Close, J.J. and Wilding, R.J. (1999) Increased exposure to pollutant aerosols under high voltage powerlines. International Journal of Radiation Biology, Vol. 75 No. 12, pp. 1505-1521. • Gailitis, A., Lielausis, O., Dement’ev, S. et al. (1999) Detection of a flow induced magnetic field eigenmode in the Riga dynamo facility. Physical Review Letters, Vol. 84, No. 19, pp. 4365-4368. • UK Childhood Cancer Study Investigators (1999) Exposure to power-frequency magnetic fields and the risk of childhood cancer. The Lancet, Vol. 354, No. 9194, pp. 1925-1931. Books • Cutnell, J.D., and Johnson, K.W. (2000) Physics (5th Edition). John Wiley, New York. • Warren, P. (1988) Physics for Life. John Murray, London. Movies • Magnetic Storm http://www.pbs.org/wgbh/nova/magnetic/ Movies • Harmful Effects of Electromagnetism