Chapter 1: Mathematical Concepts and Vectors

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1 Chapte : Mathematical Concepts and Vectos giga G 9 mega M 6 kilo k 3 centi c - milli m -3 mico μ -6 nano n -9 in =.54 cm m = cm = 3.8 t mi = 58 t = 69 m h = 36 s da = 86,4 s ea = das You must know how to expess numbes in scientiic notation and with the coect numbe o signiicant igues. when conveting between units, ou multipl b convesion actos that equal ex: m m mi 36 s mi. =. =.4 s s 69m h h Tig Review: Pthagoean theoem: a + b = c c b a SOH CAH TOA sin opposite = cos hpotenuse adjacent = tan hpotenuse opposite = adjacent Adding Vectos Gaphicall: ) Daw the ist vecto to the coect length and in the coect diection. ) Daw the second vecto (coect length and diection) stating at the tip o the ist 3) The esultant vecto stats at the tail o the ist vecto and ends at the tip o the second Adding Vectos Analticall: ) Beak each vecto up into x- and -components. ) Add all x-components togethe to ind R x. Add all -components togethe to ind R. 3) Find the magnitude and diection o the esultant vecto: R = R + R θ tan x R = Rx 4) Check that the angle is in the coect quadant (i not add 8 ) and that ou magnitude seems easonable.

2 Chapte : Kinematics in One Dimension Displacement: Δ x = x x total distance Aveage speed = elapsed time Δx x x Aveage velocit: vave Δv v v Aveage acceleation: aave Δx Instantaneous velocit: v = lim Δ t Δt Δv Instantaneous acceleation: a = lim Δ t Δt An object speeds up i the velocit and acceleation point in the same diection; it slows down i the velocit and acceleation point in opposite diections Equations o Constant Acceleation: v = v + at x x = ( v + v ) t x x = vt+ at v = v + a( x x ) eeall: gavit is the onl oce acting on an object (no ai esistance) Equations o Feeall: v = v + at = ( v + v ) t = vt+ at v = v + a( ) Notes o Feeall: a = 9.8 m/s downwad o a = 9.8 m/s i up is deined as + v = m/s at the highest point eeall motion is smmetic

3 Chapte 3: Kinematics in Two Dimensions Displacement: Δ = Δ Aveage velocit: vave Δv v v Aveage acceleation: aave Δ Instantaneous velocit: v = lim Δ t Δt Δv Instantaneous acceleation: a = lim Δ t Δt Note: In two dimensional motion, each component (x and ) can be teated sepaatel. The x-component is independent o the -component and vice vesa. The two components ae connected b time t. Equations o Constant Acceleation: x-component o motion -component o motion v = v + a t x x x x x = ( vx + vx ) t x x = vxt+ axt v = v + a ( x x ) x x x v = v + a t = ( v + v ) t = vt+ at v = v + a ( ) Pojectile Motion: In pojectile motion, we assume no ai esistance so: a x = m/s a = -9.8 m/s (assuming up is +) Since a x = m/s, the 4 equations o the hoizontal component o pojectile motion educe down to two: vx = vx x x = v t x The equations o the vetical component o pojectile motion ae the same as the above equations o the -component o motion with a = -9.8 m/s

4 Chapte 4: Foces and Newton s Laws Newton s Laws: Newton s st law: An object at est will emain at est, an object in motion will emain in motion in a staight line at a constant speed unless acted upon b an unbalanced oce. Newton s nd law: F = ma Newton s 3 d law: Fo eve action thee is an equal but opposite eaction Note that F = ma is eall two equations, one o the x-component and one o the -component: x x F = ma and F = ma The unit o oce is the Newton (N). N = kg m/s You must daw ee-bod diagams o all Newton s laws poblems. Some Paticula Foces: Weight: w = mg Tension: ou must solve o tension (T) using Newton s nd law Nomal Foce: ou must solve o the nomal oce (F N ) using Newton s nd law Static Fiction: < s < s,max whee s,max = μ s F N ( s = s,max onl when the object is on the vege o slipping) Kinetic Fiction: k = μ k F N Inclined Planes: Fo inclined planes, we usuall deine the x-axis paallel to the incline and the -axis pependicula to the incline. θ x The weight then needs to be boken up into a component paallel to the incline (w x ) and a component pependicula to the incline (w ) w x = mg sinθ w = mg cosθ

5 Chapte 5: Dnamics o Uniom Cicula Motion An object in uniom cicula motion is acceleating because its diection is constantl changing. Peiod (T): time o one complete evolution Speed: π v = T Centipetal Acceleation: v magnitude is given b ac = diection is towads the cente o the cicula path. Centipetal Foce: Centipetal oce is the name given to the net oce equied to keep an object moving in uniom cicula motion. Newton s nd law o cicula motion: mv mv Fc = mac = F = Note: centipetal oce is not a new kind o oce. It is usuall a oce o combination o oces such as tension, weight, nomal oce, iction, Note: when doing uniom cicula motion poblems, it is customa to deine positive in the diection o acceleation (towads the cente o the cicle) Banked and Unbanked Cuves: In an unbanked cuve, static iction supplies the centipetal oce. max speed: v = μ g s In a banked (ictionless) cuve, a component o the nomal oce supplies the centipetal oce. angle o banked cuve: v tanθ = g

6 Chapte 6: Wok and Eneg Wok: W = Fscosθ The unit o wok is the Joule (J). J = Nm = kg m /s Wok can be +, -, o. Kinetic Eneg: K = mv Wok-Kinetic Eneg Theoem: WT = Δ KE = KE KE W T = mv mv * W T is the total wok done b all oces Gavitational Potential Eneg: PE = mgh Wok done b gavit: Wg = mgh mgh = ΔPE Total Mechanical Eneg: E = KE+ PE = mv + mgh Law o Consevation o Eneg: eneg can not be ceated o destoed; it ma be tanseed om one om to anothe but the total amount o eneg emains constant Consevation o Mechanical Eneg (applies i onl consevative oces do wok): E = E KE + PE = KE + PE mv + mgh = mv + mgh Note: i non-consevative oces do wok on an object, the total mechanical eneg o the object changes. W = E E nc W = ( KE + PE ) ( KE + PE ) nc (W nc = the total wok done b all non-consevative oces) Aveage Powe: P = W t P = Δ E t

7 Chapte 7: Impulse and Momentum Linea Momentum: p = mv Impulse: Note this is eall two equations: J = F Δt ave p = mv and p = mv x x Impulse-Momentum Theoem: ( ave ) J =Δ p = mδv F Δ t =Δ p = mδv Consevation o Momentum: Consevation o Momentum: in the absence o a net extenal oce ( F ext = ) momentum o a sstem emains constant. i Fext = P = Pi P = p + p + p +... = sum o p o all objects in sstem 3, the Collisions: Elastic Collision: collision in which kinetic eneg is conseved Inelastic Collision: collision in which kinetic eneg is not conseved Completel Inelastic Collision: an inelastic collision in which the objects stick togethe in a collision between two objects: p = pi mv + mv = mv + mv i the collision is completel inelastic: ( m+ m) v = mv + mv

Chapter 1: Mathematical Concepts and Vectors

Chapter 1: Mathematical Concepts and Vectors 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 1-9 1 in =.54 cm 1 m = 1 cm = 3.81 t 1 mi = 58 t = 169 m 1 hr = 36 s 1 day = 86,4

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