Energy and Energy Transer Chapter 7 Scalar Product (Dot) Work Done by a Constant Force F s constant over the dsplacement r 1
Denton o the scalar (dot) product o vectors Scalar product o unt vectors = 1 j = 0 jj = 1 jk = 0 k k = 1 k = 0 Scalar product a vector wth tsel: Queston 1 For the vectors A = (3 4j + 4k) and B = ( + 3j - 7k) o Express the scalar product A B o Express the magntude o each vector A and B o Fnd the angle between A and B (use the denton o scalar product)
Work - transer o energy [Nm = Joule J ] Work done on an object by a orce that s constant over the object s dsplacement Energy transerred n, Work + Energy transerred out, Work Note: The work done by k s always negatve snce the angle θ between k and Δr s always180 o the object always looses energy and slows Queston A orce F = (6 j) N acts on a partcle that undergoes a dsplacement Δr = (3 + j) m. Fnd the work done by F on the partcle Fnd the angle between F and Δr 3
Queston 3 A man pulls a 4.6 kg vacuum cleaner wth a orce o 50.0 N at an angle o 30.0 o above the loor. The coecent o knetc rcton s 0.00 Calculate the work done on the vacuum cleaner by the orces shown as the vacuum s dsplaced 3.00 m to the rght k The Sprng Force Physcal system (block and sprng) or whch orce vares wth poston 4
Hooke s Law descrbes the sprng orce F s (restorng orce) and ts relaton to dsplacement x k sprng constant [N/m] measure o the sprng stness. sgn F s and x are always n opposte drectons. The elastc lmt o a sprng Graph o F app vs. x showng behavor wthn and beyond the elastc lmt x 5
Work Done by a Varyng Force (lke F s ) F vares over the dsplacement dr Work done by a orce that vares over the object s dsplacement 6
Queston 4 A mass s subject to a net orce F x that vares wth poston, as shown. The mass starts movng at x = 0. Fnd the work done by F x when the mass moves rom x = 0 to x = 15m. Queston 5 A orce F = (4x + 3yj) N acts on an object as t moves n the x drecton rom x = 0 to x = 5 m and y = 0 to y = 3 m What s the work done on the object by the orce? 7
The work done by F s vares over the dsplacement x ò x Work Fs = (-kx) dx x Hooke s Law anmaton = - 1 k[x - x ] = - 1 kx + 1 kx Queston 6 An archer pulls her bowstrng back rom 0 to 0.400 m by exertng a orce that vares unormly rom 0 to 30 N (ncreases). Assume the bowstrng s a sprng. o What s the sprng constant o the bowstrng? o What s the work done by the archer n drawng the bow? 8
Knetc Energy and The Work-Knetc Energy Theorem Energy o moton Knetc Energy K [kg m /s = J] K 1 mv When the energy o the system changes K K K 1 1 mv mv 9
Work-Knetc Energy Theorem Work K K Dervaton Work ( F ) dx x x x x x dv maxdx x m dx x dt v dx v m dv mvdv v dt v 1 1 mv mv Queston 7 A 7.80x10-3 kg bullet ntally movng at 575 m/s strkes the hand o a superhero, causng the hand to move 5.50x10 - m n the drecton o the bullet s velocty beore stoppng. Fnd the average orce that stops the bullet. 10
Gravtatonal Potental Energy Elastc Potental Energy Energy stored n a system just watng to be let loose ether do Work or be converted to K Potental Energy U [ J ] - retrevable energy o Only conservatve orces store U n a system as they do work o When conservatve orces act on a system, any K lost (or ganed) by the system, s balanced by an equal gan (or loss) o U A orce s conservatve t has the ollowng propertes: The work that t does on a partcle movng between ponts s ndependent o the path taken by the partcle (only depends on the endponts, and ) That work that t does on a partcle movng through a closed path = 0 (endponts are the same, = ) 11
Gravtatonal Potental Energy U g As long as object s postoned at some dstance y wth respect to a chosen zero-reerence pont. U g mgy y I the potental energy o the system changes: U g U g mgy U g mgy Elastc Potental Energy U s As long as sprng s stretched/compressed at some dstance x wth respect to equlbrum U s 1 kx I the potental energy o the system changes: U s U 1 s kx U s 1 kx 1
Work done by a conservatve orce = ΔU 13