AP-C WEP 1. Wok a. Calculate the wok done by a specified constant foce on an object that undegoes a specified displacement. b. Relate the wok done by a foce to the aea unde a gaph of foce as a function of position, and calculate this wok in the case whee the foce is a linea function of position. c. Use integation to calculate the wok pefomed by a foce F(x) on an object that undegoes a specified displacement in one dimension. d. Use the scala poduct opeation to calculate the wok pefomed by a specified constant foce F on an object that undegoes a displacement in a plane. 2. Wok-Enegy Theoem a. Calculate the change in kinetic enegy o speed that esults fom pefoming a specified amount of wok on an object. b. Calculate the wok pefomed by the net foce, o by each of the foces that make up the net foce, on an object that undegoes a specified change in speed o kinetic enegy. c. Apply the theoem to detemine the change in an object s kinetic enegy and speed that esults fom the application of specified foces, o to detemine the foce that is equied in ode to bing an object to est in a specified distance. 3. Consevative Foces a. State altenative definitions of consevative foce and explain why these definitions ae equivalent. b. Descibe examples of consevative foces and non-consevative foces. 4. Potential Enegy a. State the geneal elation between foce and potential enegy, and explain why potential enegy can be associated only with consevative foces. b. Calculate a potential enegy function associated with a specified one-dimensional foce F(x). c. Calculate the magnitude and diection of a one-dimensional foce when given the potential enegy function U(x) fo the foce. d. Wite an expession fo the foce exeted by an ideal sping and fo the potential enegy of a stetched o compessed sping. e. Calculate the potential enegy of one o moe objects in a unifom gavitational field. 5. Consevation of Enegy a. State and apply the elation between the wok pefomed on an object by nonconsevative foces and the change in an object s mechanical enegy. b. Descibe and identify situations in which mechanical enegy is conveted to othe foms of enegy. c. Analyze situations in which an object s mechanical enegy is changed by fiction o by a specified extenally applied foce. d. Identify situations in which mechanical enegy is o is not conseved. e. Apply consevation of enegy in analyzing the motion of systems of connected objects, such as an Atwood s machine. f. Apply consevation of enegy in analyzing the motion of objects that move unde the influence of spings. g. Apply consevation of enegy in analyzing the motion of objects that move unde the influence of othe nonconstant one-dimensional foces. h. Students should be able to ecognize and solve poblems that call fo application both of consevation of enegy and Newton s Laws. 6. Powe a. Calculate the powe equied to maintain the motion of an object with constant acceleation (e.g., to move an object along a level suface, to aise an object at a constant ate, o to ovecome fiction fo an object that is moving at a constant speed). b. Calculate the wok pefomed by a foce that supplies constant powe, o the aveage powe supplied by a foce that pefoms a specified amount of wok. -1-
Wok 1. Wok a. Calculate the wok done by a specified constant foce on an object that undegoes a specified displacement. b. Relate the wok done by a foce to the aea unde a gaph of foce as a function of position, and calculate this wok in the case whee the foce is a linea function of position. c. Use integation to calculate the wok pefomed by a foce F(x) on an object that undegoes a specified displacement in one dimension. d. Use the scala poduct opeation to calculate the wok pefomed by a specified constant foce F on an object that undegoes a displacement in a plane. When a foce acts on an object to cause a displacement, the foce has done wok on the object. Fo a constant foce: W = F cosθδx = F i Δx Fo a non-constant foce, the wok done is the aea unde the foce vs. displacement gaph. You can use geomety fo simple gaphs, and integation fo moe complex gaphs. In moe than one dimension, you have to add up all the little bits of wok done by the foce fo each little displacement. dw = F i d This is known as a line integal W = dw = F i d 2 Geneal Fom W = F i d 1 Example: Enegy Stoes in a Sping A sping obeys Hooke s Law, whee F(x)=-kx. How much wok is done in compessing a sping fom equilibium to some point x? x d =dx x x x2 W = F i d W = F i dx W = kx dx W = k x dx = k = 12 kx 2 0 0 2 0 This is the potential enegy now stoed in the sping, since exactly that much wok was done to compess the sping. -2-
Dot Poduct Dot Poduct (aka Scala Poduct) The dot poduct is a mathematical opeation which takes two vectos and povides a scala poduct. A i B = A B cosθ = ABcosθ A i B = Ax Bx + Ay By + Az Bz +... The dot poduct of pependicula vectos is zeo. You can think of the dot poduct as the length of the pojection of vecto A onto the unit vecto of B. Example: Find the Dot Poduct If vecto A = <-7,4> and vecto B = <-2,9>, what is A B? A B= 7-2 + 4 9 = 50 What is the angle between A and B? A i B = A B cosθ ( 65)( 85)cosθ = 50 θ = 47.7 Example: Find Component fo Pependicula Vectos If A = <-2,4> and B = <6,By>, find the value of By such that A and B ae pependicula. A i B = 0 < 2,4 > i < 6, By >= 0 ( 2 6) + (4 By ) = 0 Hint: Two vectos ae pependicula if thei dot poduct is zeo. 4By 12 = 0 4By = 12 By = 3-3-
Wok-Enegy Theoem 1. Wok-Enegy Theoem a. Calculate the change in kinetic enegy o speed that esults fom pefoming a specified amount of wok on an object. b. Calculate the wok pefomed by the net foce, o by each of the foces that make up the net foce, on an object that undegoes a specified change in speed o kinetic enegy. c. Apply the theoem to detemine the change in an object s kinetic enegy and speed that esults fom the application of specified foces, o to detemine the foce that is equied in ode to bing an object to est in a specified distance. When you do wok on an object, you give it enegy. When an object tansfes enegy to anothe object, it does wok. Deivation of the Wok-Enegy Theoem in 1-D B B vb F =ma=m dv D dt WAB = F i d 1 W = Fx dx W = dx=vdt mv dv A W= v mv 2 B 2 v A A va = 12 m(v B2 v 2A ) = ΔKE Example: Foce Requied to Stop a Tain A tain ca with a mass of 200kg is taveling at 20 m/s. How much foce must the bakes exet in ode to stop the tain ca in a distance of 10 metes? Wnet = ΔKE = KE f KEi = 0 12 (200kg)(20 m s )2 = 40,000J Wnet = F d F = Wnet 40,000J = = 4,000N d 10m -4- You can veify this using kinematics and Newton s 2nd Law
Consevative Foces 1. Consevative Foces a. State altenative definitions of consevative foce and explain why these definitions ae equivalent. b. Descibe examples of consevative foces and non-consevative foces. Consevative Foce Definitions A foce in which the wok done on an object is independent of the path taken is known as a consevative foce. A foce in which the wok done moving along a closed path is zeo. A foce in which the wok done is diectly elated to a negative change in potential enegy. (W = ΔU ) Non-Consevative Foce Definitions A foce in which the wok done on an object is dependent upon the path taken is known as a non-consevative foce. A foce in which the wok done moving along a closed path is not zeo. A foce in which the wok done is not necessaily elated to a negative change in potential enegy. Consevative Foces Non-Consevative Foces Gavity Fiction Elastic Dag Coulombic Ai Resistance -5-
Potential Enegy 1. Potential Enegy a. State the geneal elation between foce and potential enegy, and explain why potential enegy can be associated only with consevative foces. b. Calculate a potential enegy function associated with a specified one-dimensional foce F(x). c. Calculate the magnitude and diection of a one-dimensional foce when given the potential enegy function U(x) fo the foce. d. Wite an expession fo the foce exeted by an ideal sping and fo the potential enegy of a stetched o compessed sping. e. Calculate the potential enegy of one o moe objects in a unifom gavitational field. Potential enegy is enegy an object possesses due to its position o condition. Potential Enegy is often symbolized as PE o U. Wok Done By A Consevative Foce W = ΔU x2 ΔU = Wcons.F = F i dx Common Potential Enegy Fomulas ΔU g = mgh x1 In unifom gavitational field U s = 12 kx 2 Gavitational Potential Enegy in a Non-Unifom Gavitational Field Stat with Newton s Law of Univesal Gavitation Fg = Then: Gm1m2 2 d Gm1m2 d U = Gm m g 1 2 2 2 ΔU = WconsF = Wgav = 1 Gm m 1 2 U g = Gm1m2 = Foce fom Potential Enegy The change in the potential enegy is: du = dwf = F i dl = F cosθdl Imagine a small displacement dl of an object moving along a path unde the influence of a consevative foce F. Fcosθ is the component of F in the diection of dl, which can be witten as Fcosθ=Fl. Theefoe: F cosθ= F l du = F cosθdl du = Fl dl Fl = du dl This means that if you know how the potential enegy depends on a coodinate, you can find the component of the foce in the diection of the coodinate. Example: Deivation of Hooke s Law Conside the gaph of a sping s potential enegy: U (x) = 12 kx 2 Total Enegy = Etotal=K+U Foce at any given point on the cuve is the opposite of the slope. Deivation of Hooke s Law! -6-
Consevation of Enegy 1. Consevation of Enegy a. State and apply the elation between the wok pefomed on an object by nonconsevative foces and the change in an object s mechanical enegy. b. Descibe and identify situations in which mechanical enegy is conveted to othe foms of enegy. c. Analyze situations in which an object s mechanical enegy is changed by fiction o by a specified extenally applied foce. d. Identify situations in which mechanical enegy is o is not conseved. e. Apply consevation of enegy in analyzing the motion of systems of connected objects, such as an Atwood s machine. f. Apply consevation of enegy in analyzing the motion of objects that move unde the influence of spings. g. Apply consevation of enegy in analyzing the motion of objects that move unde the influence of othe non-constant onedimensional foces. h. Students should be able to ecognize and solve poblems that call fo application both of consevation of enegy and Newton s Laws. Conside a single consevative foce doing wok on a closed system: WF = ΔK Wok-Enegy Theoem WF = ΔU Consevative Foce WF = ΔK = ΔU ΔK + ΔU = 0 This means that the sum of the kinetic and potential enegies doesn t change. This quantity is called the total mechanical enegy (E). E = K +U If only consevative foces ae doing wok, Ei=Ef, theefoe Ki+Ui=Kf+Uf. This is the Law of Consevation of Mechanical Enegy. Non-consevative foces change the total mechanical enegy of a system, but not the total enegy of a system (wok done by a non-consevative foce is typically conveted to intenal enegy (heat). Etotal = K + U + Wnc Emech = K + U -7-
Powe 1. Powe a. Calculate the powe equied to maintain the motion of an object with constant acceleation (e.g., to move an object along a level suface, to aise an object at a constant ate, o to ovecome fiction fo an object that is moving at a constant speed). b. Calculate the wok pefomed by a foce that supplies constant powe, o the aveage powe supplied by a foce that pefoms a specified amount of wok. Powe Powe is the ate at which wok is done / the ate at which a foce does wok. Pavg = ΔW Δt ΔW Δt dw P= dt P = F iv Pavg = Units of powe ae J/s, o Watts. Fo instantaneous powe, look at aveage powe ove a vey small time inteval d dw dw = Fid F i d v = dt P= P= P = F iv dt dt Note: 1 hosepowe = 746 Watts Example: Powe Deliveed to a Moving Mass Find the powe deliveed by the net foce to a 10-kg mass at t=4s given the position of the mass is given by x(t)=4t3-2t. Fist, find v(t) and a(t): v(t)=12t2-2 a(t)=24t Next, find the net foce using Newton s 2nd Law: Fnet=ma=(10kg)(24t)=240t Then, find the powe deliveed: θ=0 P = F i v = Fv cosθ cosθ=1 P = (240t)(12t 2 2) = 2880t 3 480t t=4s P = (2880)(43 ) 480(4) = 182,400 Watts -8-