Princeton phys

~:: J'J- ;:a , & , Vo/6~ YT;h7; )op ill /ulvq P' " i-({~d' ~lA v-,_~/, ; -' '1 MCAT PHYSICS TABLE OF CONTENTS ~ t9 t't? c;o ""' vr/ 1/1 ryvr; ;\./ 7/t' tf vv~ /;1 V';/v-;; f KINEMATICS 1.1 Units and Dimensions 1.2 Kinematics Displacement Velocity 11 Acceleration 14 1.3 Uniformly Accelerated Motion 18 Kinematics with Graphs 20 Free Fall ~ 24 Projectile Motion 26 1.4 Summary of Formulas 30 MECHANICS I .• 31 2.1 2.2 2.3 2.4 2.5 2.6 Mass, Force, and Newton's Laws 31 Newton's Law of Gravitation 36 Friction 40 Inclined Planes 44 Pulleys 46 Summary of Formulas 50 MECHANICS II • • 51 3.1 Center of Mass 51 Center of Gravity 57 3.2 Uniform Circular Motion 58 3.3 Torque 66 · 3.4 Equilibrium ~ 73 * 3.5 Rotational Inertia ~ 76 3.6 Summary of Formulas 78 MECHANICS Ill 79 4.1 Work 79 4.2 Power 83 4.3 Kinetic Energy 85 The Work-Energy Theorem 86 4.4 Potential Energy 90 Gravity Is a Conservative Force 92 Friction Is Not a Conservative Force 94 4.5 Total Mechanical Energy 95 Using the Energy Method When There Is Friction 100 A Note about Thermal Energy and Modes of Heat Transfer 102 4.6 Momentum 103 Impulse 04 Conservation of Total Momentum 106 Collisions 108 * 4.7 A Note on Angular Momentum 114 4.8 Summary of Formulas 116 FLUIDS AND ELASTICITY OF SOLIDS 117 5.1 Hydrostatics: Fluids at Rest 117 Density and Specific Gravity 117 Pressure 119 Buoyancy and Archimedes' Principle 123 Pascal's Law 128 Surface Tension 130 5.2 Hydrodynamics: Fluids in Motion 131 Flow Rate and the Continuity Equation 131 Bernoulli's Equation 133 The Bernoulli Effect 13~ 5.3 The Elasticity of Solids 140 Stress 140 Strain 141 Hooke's Law 141 5.4 Summary of Formulas 144 ELECTROSTATICS 145 6.1 Electric Charge 145 6.2 Electric Force and Coulomb's Law 147 The Principle of Superposition for Electric Forces 150 6.3 Electric Fields 153 The Principle of Superposition for Electric Fields ~ 160 Conductors, Insulators, and Polarization ~ 162 6.4 Electric Potential 163 The Principle of Superposition for Electric Potential 169 6.5 Summary of Formulas 174 ELECTRICITY AND MAGNETISM ·····"*····················•· 175 1· Electric Circuits 175 Current 175 Voitage 177 Resistance 178 Ohm's Law 180 Resistors 181 DC Circuits 185 7.2 Capacitors 201 Dielectrics 209 Dielectric Breakdown 214 Combinations of Capacitors 215 7.3 Alternating Current 217 7.4 Magnetic Fields and Forces ·219 Sources of Magnetic Fields 231 Magnets 238 7.5 Summary of Formulas 244 OSCILLATIONS AND WAVES 245 8.1 Simple Harmonic Motion (SHM) ~ 245 The Dynamics· of SHM 246 The Kinematics of SHM 248 Pendulums 249 8.2 Waves 250 Transverse Waves 250 Frequency and Period 250 Wavelength and Amplitude 250 Wave Speed · 251 Two Big Rules for Waves · 251 8.3 Interference of Waves 254 8.4 Standing Waves 255 8.5 Summary of Formulas 258 SOUND • 259 9.1 9.2 9.3 9.4 9.5 9.6 10 Sound Waves 259 Standing Sound Waves in Tubes 261 Beats 262 Intensity and Intensity-Level 262 The Doppler Effect 264 Summary of Formulas 268 LIGHT AND GEOMETRICAL OPTICS 269 10.1 Electromagnetic Waves 269 · Photons .270 10.2 Reflection and Refraction 271 Index of Refraction 272 Total Internal Reflection 275 10.3 Wave Effects 276 Diffraction 276 Polarization 276 Dispersion 276 10.4 Mirrors 277 Plane Mirrors 277 Curved Mirrors 2n The Mirror Equation ~ 279 The Magnification Equation 279 10.5 Lenses 282 Lens Power .284 The Basics of Eyesight Correction 285 10.6 Summary of Formulas 286 C31l. 14 for a given pair of surfaces [§2~3] coulomb The SI unit of electric charge, abbreviated C; the fundamental electric charge (the charge on a proton or the magnitude of the charge on an electron) is defined to be e = 1.6 x l0-19 C Therefore, one coulomb is equal to the total charge on 6.25 x 1018 protons [§6.1] Coulomb's law The law that gives the electric force between two charged objects: F =kq 1q2 /r 2• (§6.2] critical angle The angle at which an incident beam of light no longer refracts If n1 is the refractive index of the incident medium and n2 is the index of the refracting medium (and n1 > ~ ), then the critical angle, 8c, is defined by the equation sin 8c = n2 / n1 • [§10.2] 292 MCAT PHYSICAL SciENCES REVIEW fluid A substance that can flow, or more precisely, a substance that cannot withstand a ,shear stress Both liquids and gases are fluids [§5.1] focal length The distance from a mirror or lens to its focal point Concave mirrors and converging lenses have positive focal lengths; convex mirrors and diverging lenses have negative focal lengths [§10.4, 10.5] focal point (or focus) For a concave mirror or a converging lens, the focal point (or focus) is the point to which rays of light that are initially parallel to the optical axis are focused after contact with the mirror or lens For a convex mirror or a diverging lens, the focal point is the point from which rays of light that are initially parallel to the optical axis are diverged after contact with the mirror or lens [§10.4, 10.5] force Intuitively, a push or pull exerted by one object on another that causes it to accelerate [§2.1] frequency The number of oscillations (or cycles) per second [§8.1, 8.2, 9.1, 10.1] friction The friction force is the parallel component of the contact force exerted by a surface on an object [§2.3] fundamental The lowest permissible frequency-or longest permi$sible wavelength-of a standing wave; also referred to as the first harmonic [§8.4] gravitational acceleration The acceleration produced by the gravitational pull of a body, directed toward the center of the body The magnitude of the gravitational acceleration, g, produced by the earth is approximately 10 m/ s2 near the surface [§1.3] gravitational force In Newton's Theory of Gravitation, every object exerts a force-a gravitational pull-on every other object If the masses of the two objects are m1 and m2, and if they are separated by a distance r, then the magnitude of the gravitational force between them is given by Fgra~ = Gm 1m2 /r 2• [§2.2] hertz The SI unit of frequency; hertz= 1Hz= cycle (or oscillation) per second [§8.1] Hooke's Law The magnitude of the force exerted by a stretched or compressed object or spring is proportional to the distance by which it is stretched or' compressed: F = -kx [§8.1] MCAT PHYSICS - 293 GLOSSARY hydrostatic gauge pressure The pressure at a point below the surface of a fluid at rest, due to the weight of the fluid above it: Pgauge = Pnw~D, where D is the depth [§5.1] Impulse The product of force and the time during which it acts: impulse= J =Ft [§4.6] impulse-momentum theorem The total impulse delivered to an object is equal to its change in momentum: I total= Ap [§4.6] index of refraction The index of refraction for a medium is equal to the ratio of the speed of light in vacuum to the speed of light through the medium: n = c/v [§10.2] inelastic collision A collision in which total momentum is conserved, but total kinetic energy is not [§4.6] inertia Resistance to acceleration; an object's inertia is measured by its mass and is the ratio of the net force on an object to its acceleration: inertia = m = Fne/ a [§2.1] Rotational inertia (also known as moment of inertia) ·is resistance to rotational acceleration [§3.5] insulator A material with a very high resistivity, and therefore does not permit charge to flow through it easily Glass and wood are examples of insulators [§6.3] intensity The intensity of a wave is the power it transmits per unit area; the units of I are therefore W /m2 • Intensity is related directly to the wave's amplitude, and diminishes with the square of the distance from the source [§9.4] interference The combination of two or more waves When the waves are in phase (crest meets crest, trough meets trough), this is constructive interference, and the amplitude of the resultant wave is equal to the sum of the individual amplitudes; when the waves are out of phase (crest meets trough,· trough meets crest), this is destructive interference, and the amplitude of the resultant wave is equal to the difference between the individual amplitudes [§8.3] joule The SI unit of work and energy; joule= J = N-m = kg-m2 /s2 • [§4.1] kinetic energy The energy due to motion; for an object of mass m and speed v, the kinetic energy is KE = tmv 2• [§4.3] 294 MCAT PHYSICAL SCIENCES REVIEW kinetic friction Also known as sliding friction, it is the friction that results when there is relative motion between the two surfaces-that is, when one surface slides across the other If FN is the magnitude of the normal force and ~ is the coefficient of kinetic friction between the two surfaces, then the force of kinetic friction is directed opposite to the direction of the sliding and its magnitude is Ffric =~FN [§2.3] Kirchhoff's Laws 1) The total amount of current entering a junction in a circuit must be equal to the total amount of current leaving the junction 2) The sum of the voltages around a closed loop in a circuit must be zero [§7.1] lens A thin piece of glass or plastic that forms an image by refracting light [§10.5] lever arm Denoted by l, it is the perpendicular distance from the pivot (reference) point to the line of action of a force [§3.3] longitudinal wave A wave in which the oscillations of the medium are parallel to the direction of propagation of the wave Sound waves are longitudinal [§9.1] magnetic field The force field created by a moving electric charge [§7.4] magnetic force The force exerted by a magnetic field on a moving charge If a charge q moves with velocity v through a magnetic field B, then the magnetic force on q is given by F ::= q(v x B) The magnitude ofF is lqlvBsin8 (where 6is the angle between v and B), and the direction ofF is given by the right-hand rule if q is positive and by the left-hand rule if q is negative [§7.4] magnification The ratio of the height of the imageto the height of the object; a negative value for the magnification means that the image is inverted relative to the object For a mirror or lens, the magnification is given by the equation m =-i/o, where i and o are the distances from the mirror or lens to the image and the object, respectively [§10.4, 10.5] mass The quantitative measure of an object's inertia; intuitively, we think of mass as measuring the amount of matter in an object In SI units, mass is exprest?ed in kilograms (kg) and is the ratio of the net force on an object to its acceleration: mass =m = Fne/ a [§2.1] mirror A surface that forms an image by reflecting light [§10.4] MCAT PHYSICS - GLOSSARY 295 moment of inertia Also known as rotational inertia, an object's moment of inertia measures its resistance to rotational acceleration Gust as an object's mass measures its resistance to translational acceleration) [§3.5] momentum The product of an object's mass and velocity: momentum= p = mv [§4.6] net force The sum of all the forces that act on an object [§2.1] neutron A subatomic particle with zero electric charge that is a constituent of atomic nuclei [§6.1] newton The SI unit of force: newton= N = kg-m/s 2• [§2.1] Newton's Laws of Motion 1) If Fnet = 0, then the object's velocity will not change 2) Fnet = ma 3) If Object exerts a force, F 1·on·l' on Object 2, then Object exerts a force, F 2·on-l' on Object These forces-known as an action-reaction pair-have the same magnitude but point in opposite directions, so F 2.on-l = -Fl-on-2, and act on different objects [§2.1] node A point where a standing wave has zero amplitude [§8.4] normal As an adjective, it means perpendicular As a noun, a normal is a line that's perpendicular to a surface [§2.3] normal force For an object in contact with a surface, the normal force is the component of the force exerted by the surface that is perpendicular to the surface [§2~3] north pole of magnet The pole from which the magnetic field lines emerge from a magnet [§7.4] ohm The unit of resistance: ohm = = volt per amp = VI A [§7.1] Ohm's Law A material is said to obey Ohm's Law if its resistance remains constant as the voltage across it varies; thus, for such a material, V = IR, where R is a constant [§7.1] 296 MCAT PHYSICAL SCIENCES REVIEW parallel resistors Resistors in a circuit are said to be in parallel if they provide alternate routes for current to flow from one point in the circuit to another; parallel resistors always share the same voltage drop [§7.1] pascal The unit of pressure: pascal= Pa = newton per square meter= Nlm2 • [§5.1] Pascal's Law A confined fluid transmits an externally applied change in pressure to all parts of the fluid equally [§5.1] period The time required for one complete oscillation (or cycle) [§8.1, 8.2] photon Light travels as a wave, but interacts with matter as a stream of particles; these "particles," each an indivisible quantum of energy, is a photon Photons have no mass and move at the speed of light The energy carried by each photon is proportional to the frequency of the light: Ephoton = hf, where h is a constant of nature known as Planck's constant [§10.1] polarized A transverse wave is polarized if the direction of its oscillations is constant (or is confined to vary in a particular way) For a plane-polarized electromagnetic wave, the direction of polarization is the direction of oscillation of the electric field [§10.1, 10.3] potential energy The energy of an object (or system) due to its position or configuration There are different forms of potential energy, depending on the force involved; for the MCAT, the three most important forms are gravitational PE, electrical PE, and elastic PE [§4.4, 6.4, 8.1] power In mechanics, power is the rate at which work is done or energy is used Power is thus equal to work (or energy) divided by time, and its SI unit is the watt, where watt = W = JIs [§4.2] In optics, lens power is a measure of the focusing strength of a lens By definition, lens power is equal to the reciprocal of the focal length of the lens: P = 1I f Iff is expressed in meters, theri lens power has units of diopters, where diopter = D = m-1• [§10.5] pressure A scalar quantity equal to the magnitude of the force per unit area [§5.1] projectile motion The motion of a particle moving under the influence of uniform (constant) acceleration; if the object's initial velocity is not purely horizontal or vertical, the path of the object will be a parabola [§1.3] proton A subatomic particle with a positive electric charge (e·qual to the elementary electric charge, e) that is a constituent of atomic nuclei [§6.1] MCAT PHYSICS - GLOSSARY 297 quantized A quantity is said to be quantized if it exists only in discrete amounts Examples: (1) Electric charge on an object can only be an integer multiple of the basic unit of electric charge, e (2) Electromagnetic radiation of frequency f can be absorbed only in whole number multiples of the photon energy, hf [§6.1] radiation Energy emitted or absorbed due to propagation of waves (electromagnetic waves, unless a different kind of wave is specifically mentioned) [§10.1] A mode of heat transfer via electromagnetic waves [§4.5] rarefaction A region where the local density and pressure is momentarily decreased from standard due to the passage of a sound wave [§9.1] real Image An image formed by a mirror or lens where light rays actually intersect Unlike a virtual image, a real image can be projected onto a screen [§10.4] reflection When waves or particles "bounce off" a surface on which they are incident, the return of these waves or particles is called reflection [§10.2] refraction The change in direction of a wave when it passes from one medium into another [§10.2] resistance The ratio of the voltage to current: R = VI I [§7.1] resistivity The intrinsic resistance of a material [§7.1] resistor A component of an electrical circuit that provides resistance to the flow of current [§7.1] restoring force For an object undergoing oscillation, the force on the object that is directed toward equilibrium [§8.1] series resistors Resistors in a circuit are said to be in series if current must flow through each of them, one after the other; series resistors always share the same current [§7.1] shear stress The magnitude of the shearing force exerted on an object divided by the area parallel to which it acts [§5.3] 298 MCAT PHYSICAL SCIENCES REVIEW simple harmonic motion Periodic (oscillatory) movement where the period and frequency of the oscillations not depend on the amplitude, caused by a restoring force that is proportional to the displacement from equilibrium [§8.1] Snell's ·Law The Law of Refraction in optics, n1 sin 81 = n sin 82, where n and n2 are the refractive indexes of the incident and refracting media (respectively), and 81 is the angle of incidence and 82 the angle of refraction [§10.2] sound-level A measurement, in decibels, of the intensity of a sound wave The sound-level for a wave of intensity I is given by the equation f3 = 10 log10 (I/ I0), where I0 is the threshold of hearing [§9.1) south pole of magnet The pole into which the magnetic field lines enter a magnet [§7.4) specific gravity The unitless ratio of the density of a substance to the density of water: sp gr = Psubstance/ Pwater· [§5.1) speed The magnitude of an object's velocity [§1.2) standing wave A wave caused by the superposition of two oppositely directed traveling waves, for which the resulting crests and troughs not travel [§8.4) static friction The friction that results when there is no relative motion between the two surfaces-that is, when neither surface slides across the other If FN is the magnitude of the normal force and p.5 is the coefficient of static friction between the two surfaces, then the force of static friction is directed opposite to the direction of the intended motion and its maximum magnitude is Ffric, max =P,5FN [§2.3] strain The ratio of the change in one of an object's dimensions to the original, caused by an applied stress For a compressive or tensile stress, the strain is equal to the (magnitude) of the change in the object's length divided by the original length For a shear stress, the strain is equal to the distance the object is bent divided by the length perpendicular to the direction of the bend [§5.3] stress The magnitude of the force acting on an object, divided by the area over which it acts [§5.3] superposition The addition of separate physical phenomena, such as electric forces, fields, potentials, and waves, where the resultant is simply equal to the sum of the individual values [§6.2, 6.3, 6.4, 8.3] MCAT PHYSICS - GLOSSARY 299 tension A type of force applied to a solid object that tends to increase its length Tension is also used to describe the pulling force e.xerted by a stretched string, rope, chain, or spring [§2.1, 2.5, 5.3] tesla The SI unit of magnetic field strength: tesla = T = newton per amp-meter= N/ A-m [§7.4] threshold of hearing The lowest intensity the human ear can detect; denoted 10 , it is defined as 10-12 W /m2 • [§9.4] torque A quantity associated with a force that measures how effective the force is at producing rotational acceleration If r is the vector from the pivot point to the point of application of a force F, and the angle between r and F is 8, then the torque of the force is defined to be 1' = rF sin [§3.3] Torrlcelll's result The equation giving the speed of efflux for a static fluid from a large open container: v = ~2gD, where Dis the depth of the hole below the surface of the fluid [§5.2] total Internal reflection When an incident beam of light strikes the surface of a medium with a lower index of refraction, the beam will experience total internal reflection (TIR) if the angle of incidence is greater than the critical angle In this case, none of the beam's energy is transmitted to the other medium; it is only reflected [§10.2] total mechanical energy The sum of an object's kinetic energy and potential energy: E = KE + PE [§4.5] transverse wave A wave in which the oscillations that make up the wave are perpendicular to the direction of the wave's propagation Waves on a rope and electromagnetic waves are transverse [§8.2] velocity The rate of change of an object's position: v =As/ llt An object's velocity gives both the speed and the direction of motion of the object [§1.2] virtual Image An image formed by a mirror or lens where light rays don't actually intersect Unlike a real image, a virtual image cannot be displayed on a screen Convex mirrors and diverging (concave) lenses form only virtual images [§10.4] viscosity The internal friction of a fluid; an ideal fluid is one whose viscosity is negligible [§5.2] 300 MCAT PHYSICAL SCIENCES REVIEW volt The SI unit of electric potential and voltage; volt= V = joule per coulomb= J/C [§6.4] voltage The difference in electric potential between two points [§6.4, 7.1] watt The SI unit of power; watt = W = joule per second = JIs [§4.2] wave A disturbance that carries energy and momentum from one position to another [§8.2] wavelength The distance (denoted by A.) between consecutive crests (or between consecutive troughs) of a wave [§8.2] weight The gravitational force exerted on an object: w = mg [§2.2] work The work done by a constant force F as it acts through a displacement d is given by the equation W = Fd cos 8, where is the angle between F and d Work is a scalar quantity, and its SI unit is the joule, where joule= J = kg-m2 /s2 • [§4~1] work-energy theorem The total amount of work done on an object is equal to the change in the object's kinetic energy: Wtotai = AKE [§4.3] MCAT PHYSICS FoRMULA SHEET 301 ,;' tle c!!rinceton MCAT Physics Formula Sheet Rev1ew Kinematics D_lnamics d =Vt =i(Vo +V}t 2) v =Vo +at 3) d = v t+iat 1) 4) d = vt-1.at v =v3 +2ad 5) horizontal vertical Newton's Laws 1) Fnet =0 ==> v =constant =Voxt Vx =Vox Bx =0 y =Voyt-igt2 2) Fnet Vy =Voy -gt By =-g 3) F2-on-1 = -F1-on-2 Projectile Motign Tbe Big Eive X IVox = VoC~S8o Gravity v0 y = v0 s1n80 "-If a is constant oo cosO {4/2 J3/2 -/2/2 sine {0/2 ~12 30° -12/2 Fgrav goo 120° 135° 150° 180° -/1/2 {0/2 /1/2 /2./2 -.J3/2 -{4/2 J3/2 {4/2 JS/2 -/2./2 ~/2 {0/2 60° 45° =rna J2 ~ 1.4, f3 ~ 1.7 =·w =mg Mm Fgrav = G -2- r M g =G2, r gearth ~to "; s Inclined Plane {8 =Incline angle to horizontal) Friction 11' static MAX =P.sF~ 11 kinetic = J4 FN Force due to gravity parallel to ramp = mg sin FN = magnitude of normal force Force due to gravity L to ramp = mg cos (= fN) Center of Mass(= Center of Gravity) XcM = m1X1 + + mnXn - W1X1 + +WnXn - - m1 +···+mn W1 + ·+Wn -XcG Uniform Circular Motion centripetal acceleration: Torque Momentum z = rFsin8 =iF p=mv r v2 centripetal force: Fe =mBa = mr hilpulse: J =Ft Impulse-Momentum theorem: J Be = ~ =6.p Conservation of Momentum: total Pi = total Pt Stress and Strain Work, Energy, Power work: W =Fd.cos8 kinetic energy: KE = f mv stress Work-Energy theorem: Wtotat = 6.KE power: P = ~, 6.L = FA ; stram = L P =Fv if Fllv potential energy: PEgrav = mgh (if h