Chapter 1: Mathematical Concepts and Vectors

Transcription

1 Chapter 1: Mathematical Concepts and Vectors giga G 1 9 mega M 1 6 kilo k 1 3 centi c 1 - milli m 1-3 micro μ 1-6 nano n in =.54 cm 1 m = 1 cm = 3.81 t 1 mi = 58 t = 169 m 1 hr = 36 s 1 day = 86,4 s 1 year = days You must know how to express numbers in scientiic notation and with the correct number o signiicant igures. when converting between units, you multiply by conversion actors that equal 1 ex: m m 1mi 36 s mi 1. = 1. =.4 s s 169m 1hr hr Trig Review: Pythagorean theorem: a + b = c c b a SOH CAH TOA sin opposite = cos hypotenuse adjacent = tan hypotenuse opposite = adjacent Adding Vectors Graphically: 1) Draw the irst vector to the correct length and in the correct direction. ) Draw the second vector (correct length and direction) starting at the tip o the irst 3) The resultant vector starts at the tail o the irst vector and ends at the tip o the second Adding Vectors Analytically: 1) Break each vector up into x- and y-components. ) Add all x-components together to ind R x. Add all y-components together to ind R y. 3) Find the magnitude and direction o the resultant vector: R = R + R θ tan x y R 1 y = Rx 4) Check that the angle is in the correct quadrant (i not add 18 ) and that your magnitude seems reasonable.

2 Chapter : Kinematics in One Dimension Displacement: Δ x = x x total distance Average speed = elapsed time Δx x x Average velocity: vave = = Δt t t Δv v v Average acceleration: aave = = Δt t t Δx Instantaneous velocity: v = lim Δ t Δt Δv Instantaneous acceleration: a = lim Δ t Δt An object speeds up i the velocity and acceleration point in the same direction; it slows down i the velocity and acceleration point in opposite directions Equations o Constant Acceleration: v = v + at 1 x x = ( v + v ) t 1 x x = vt+ at v = v + a( x x ) reeall: gravity is the only orce acting on an object (no air resistance) Equations o Freeall: v = v + at 1 y y = ( v + v ) t 1 y y = vt+ at v = v + a( y y ) Notes or Freeall: a = 9.8 m/s downward or a = 9.8 m/s i up is deined as + v = m/s at the highest point reeall motion is symmetric

3 Chapter 3: Kinematics in Two Dimensions Displacement: Δ r = r r Δr r r Average velocity: vave = = Δt t t Δv v v Average acceleration: aave = = Δt t t Δr Instantaneous velocity: v = lim Δ t Δt Δv Instantaneous acceleration: a = lim Δ t Δt Note: In two dimensional motion, each component (x and y) can be treated separately. The x-component is independent o the y-component and vice versa. The two components are connected by time t. Equations o Constant Acceleration: x-component o motion y-component o motion v = v + a t x x x 1 x x = ( vx + vx ) t 1 x x = vxt+ axt v = v + a ( x x ) x x x v = v + a t y y y 1 y y = ( vy + vy ) t 1 y y = vyt+ ayt v = v + a ( y y ) y y y Projectile Motion: In projectile motion, we assume no air resistance so: a x = m/s a y = -9.8 m/s (assuming up is +) Since a x = m/s, the 4 equations or the horizontal component o projectile motion reduce down to two: vx = vx x x = v t x The equations or the vertical component o projectile motion are the same as the above equations or the y-component o motion with a y = -9.8 m/s

4 Chapter 4: Forces and Newton s Laws Newton s Laws: Newton s 1 st law: An object at rest will remain at rest, an object in motion will remain in motion in a straight line at a constant speed unless acted upon by an unbalanced orce. Newton s nd law: F = ma Newton s 3 rd law: For every action there is an equal but opposite reaction Note that F = ma is really two equations, one or the x-component and one or the y-component: x x F = ma and F = ma The unit o orce is the Newton (N). 1 N = 1 kg m/s y y You must draw ree-body diagrams or all Newton s laws problems. Some Particular Forces: Weight: w = mg Tension: you must solve or tension (T) using Newton s nd law Normal Force: you must solve or the normal orce (F N ) using Newton s nd law Static Friction: < s < s,max where s,max = μ s F N ( s = s,max only when the object is on the verge o slipping) Kinetic Friction: k = μ k F N y Inclined Planes: For inclined planes, we usually deine the x-axis parallel to the incline and the y-axis perpendicular to the incline. θ x The weight then needs to be broken up into a component parallel to the incline (w x ) and a component perpendicular to the incline (w y ) w x = mg sinθ w y = mg cosθ

5 Chapter 5: Dynamics o Uniorm Circular Motion An object in uniorm circular motion is accelerating because its direction is constantly changing. Period (T): time or one complete revolution Speed: π r v = T Centripetal Acceleration: v magnitude is given by ac = r direction is towards the center o the circular path. Centripetal Force: Centripetal orce is the name given to the net orce required to keep an object moving in uniorm circular motion. Newton s nd law or circular motion: mv mv Fc = mac = F = r r Note: centripetal orce is not a new kind o orce. It is usually a orce or combination o orces such as tension, weight, normal orce, riction, Note: when doing uniorm circular motion problems, it is customary to deine positive in the direction o acceleration (towards the center o the circle)

6 Chapter 6: Work and Energy Work: W = Fscosθ The unit o work is the Joule (J). 1 J = 1 Nm = 1 kg m /s Work can be +, -, or. Kinetic Energy: K = 1 mv Work-Kinetic Energy Theorem: WT = Δ KE = KE KE 1 W 1 T = mv mv * W T is the total work done by all orces Gravitational Potential Energy: PE = mgh Work done by gravity: Wg = mgh mgh = ΔPE Total Mechanical Energy: 1 E = KE+ PE = mv + mgh Law o Conservation o Energy: energy can not be created or destroyed; it may be transerred rom one orm to another but the total amount o energy remains constant Conservation o Mechanical Energy (applies i only conservative orces do work): E = E KE + PE = KE + PE 1 mv 1 + mgh = mv + mgh Note: i non-conservative orces do work on an object, the total mechanical energy o the object changes. W = E E nc W = ( KE + PE ) ( KE + PE ) nc (W nc = the total work done by all non-conservative orces) Average Power: P = W t P = Δ E t

7 Chapter 7: Impulse and Momentum Linear Momentum: p = mv Impulse: Note this is really two equations: J = F Δt ave p = mv and p = mv x x y y Impulse-Momentum Theorem: ( ave ) J =Δ p = mδv F Δ t =Δ p = mδv Conservation o Momentum: Conservation o Momentum: in the absence o a net external orce ( F ext = ) momentum o a system remains constant. i Fext = P = Pi P = p + p + p +... = sum o p o all objects in system 1 3, the Collisions: Elastic Collision: collision in which kinetic energy is conserved Inelastic Collision: collision in which kinetic energy is not conserved Completely Inelastic Collision: an inelastic collision in which the objects stick together in a collision between two objects: p = pi mv + mv = mv + mv i the collision is completely inelastic: ( m1+ m) v = mv mv

8 Chapter 1: Simple Harmonic Motion Hooke s law: F = kx ([k] = N/m) the sign indicates that the direction o the orce is opposite the displacement Frequency: = 1 T ([] = Hz) Period: T = 1 ([T] = s) Simple Harmonic Motion: or an object oscillating in simple harmonic motion: π ω = π = T x = Acosωt xmax = A v = Aω sinωt v = Aω a = Aω ωt a = Aω max cos max Note: v = at x = +A; v = v max at x = a = when x = ; a=a max at x = +A or a mass m oscillating on a spring with spring constant k: k 1 k ω = = T = π m π m m k Elastic Potential Energy: PE = 1 kx Conservation o Energy: 1 mv + mgh + 1 kx = 1 mv + mgh + 1 kx Simple Pendulum: or a simple pendulum (or small angles): g 1 g ω = = T = π L π L L g

9 Chapter 16: Waves and Sound Waves: Transverse Wave: the disturbance occurs perpendicular to the direction o travel o the wave Longitudinal Wave: the disturbance occurs parallel to the direction o travel o the wave 1 1 = T = T λ v = v = λ T i the requency is increased, the wavelength is decreased but wave speed doesn t change Speed o waves on a string: v = F ( m/ L) Sound: Intensity: I = E P ta = P I = or spherically uniorm radiation A 4π r (1 db) log I β = I where 1 I = 1. 1 W m Doppler Eect: = v 1± v s v 1 s v Numerator: + observer moving toward source - observer moving away rom source Denominator: - source moving toward observer + source moving away rom observer

10 Chapter 17: Linear Superposition and Intererence Linear Superposition: The Principle o Linear Superposition: when two or more waves are present at the same place at the same time, the resultant disturbance is the sum o the disturbances rom the individual waves or wave sources vibrating in phase: Beats: Δ L= nλ n =,1,,... constructive intererence nλ Δ L= n = 1,3,5,... destructive intererence = beats 1 Standing Waves: or standing waves on a string ixed at both ends: L v λ = n = n n = 1,,3,... n L or standing waves in a tube open at both ends: L v λ = n = n n = 1,,3,... n L or standing waves in a tube open at only one end: 4L v λ = n = n n = 1,3,5,... n 4L

11 Chapter 11: Fluids Density: ρ = m V ρ = 1. 1 water 3 kg m 3 Pressure: Pressure: F P = P atm = Pa = 1 atm A Pressure in a Static Fluid: P = P1+ ρ gh Pascal s Principle: Any change in the pressure applied to a completely enclosed luid is transmitted undiminished to all parts o the luid and the enclosing walls Buoyancy: Archimede s Principle: the magnitude o the buoyant orce on an object partially or completely immersed in a luid equals the weight o the luid displaced Buoyant Force: FB = ρ luidvsubg i an object is completely submerged, V sub = V obj i an object is loating, F B = w = mg Fluids in Motion: Equation o Continuity: ρ1av 1 1 = ρav i the luid is incompressible (ρ 1 =ρ ): Av 1 1= Av Bernoulli s Equation: P1+ 1 ρv ρgy1 = P + ρv + ρgy Bernoulli s Principle: where the speed o a luid increases, the pressure in the luid decreases

12 Chapter 1 and 13: Temperature and Heat; Transer o Heat Temperature: the temperature o an object depends upon the average translational KE o the atoms and molecules in the object Heat: ice point steam point Celsius scale C 1 C Kelvin scale 73K 373K Fahrenheit scale 3 F 1 F 1C 9 = F T = T K C heat (Q): energy transerred between objects because o a temperature dierence Q > i object absorbs heat Q < i object loses heat Q = mcδ T where c = speciic heat capacity rom conservation o energy (i no heat is lost to the surroundings): Q lost by hotter object = Q gained by cooler object during a phase change, heat is absorbed or lost but the temperature doesn t change solid to liquid / liquid to solid Q = +ml (L = latent heat o usion) liquid to gas / gas to liquid Q = +ml v (L v = latent heat o vaporization) heat is transerred by three processes: convection, conduction, and radiation conduction: radiation: ( kaδt ) t Q = L Q = σ 4 e T At Q = eσ T At 4 (radiation emitted) (radiation absorbed) net power radiated: P net = P emitted - P absorbed

13 Chapter 14: The Ideal Gas Law Molecular Mass, the Mole, and Avogadro s Number: Atomic mass unit: 1 u = kg the mass per mole (g/mol) o a substance has the same numerical value as the atomic or molecular mass o the substance (in u) n # o moles N # o particles N A mol -1 m mass o sample m particle mass o particle n = N N A m = m particle N Mass per mole = m particle N A n = m Mass per mole Ideal Gas Law: PV = nrt R = 8.31 J molik PV = NkT k = J K Boyle s law (applies i n and T are constant): PV i i = PV Charles law (applies i n and P are constant): V V i = T T i