( ) ( ) Review of Force. Part 1, Topic 1 Force Fields. Dr. Sven Achenbach - based on a script by Dr. Eric Salt - Outline. Review of Force. F r.

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1 S. Achenbach: PHYS 55 (Pat, Topic ) Handouts p. Review of Foce Univesity of Saskatchewan Undegaduate Couse Phys 55 Intoduction to Electicity and Magnetism Basics F o F v denotes foce it is a thee dimensional vecto e.g. in ectangula coodinates F = ( 7,3,) Pat, Topic Foce Fields units of foce (deived in the Système Intenational d Unités, SI) ae Newtons N D. Sven Achenbach - based on a scipt by D. Eic Salt - S. Achenbach S. Achenbach Phys55 -: Foce Fields 3 Outline Review of Foce slide # eview of foce 3 - special vectos - dot poduct - magnitude - angle between foce vectos foce fields 3 wok (tansfe of enegy) 9 - foce not in diection of movement - potential enegy - consevative foce fields - potential enegy evisited - wok done by a changing foce - wok as aea unde a cuve - pependicula component - enegy contou maps - elationship enegy contous & foce lines - estimating foce fom a contou map S. Achenbach Phys55 -: Foce Fields ( ) Special Vectos iˆ =,0,0 unit vecto in the x-diection ˆ j = ( 0,,0 ) unit vecto in the y-diection k ˆ = 0,0, unit vecto in the z-diection ( ) unit vecto means: length of unity () in the espective diection x, y o z with scalas F x, F y, F z : F = ( Fx, Fy, Fz ) = F iˆ x + F ˆ y j + F kˆ z times length (±) of foce diection (x-component) component in x-diection (unit vecto with length =) e.g. F = 7 iˆ + 3 ˆj + kˆ = scala means: a eal numbe without diection ( 7,3,) S. Achenbach Phys55 -: Foce Fields 4

2 S. Achenbach: PHYS 55 (Pat, Topic ) Handouts p. Review of Foce Review of Foce Dot Poduct the dot poduct opeato is a dot the dot poduct function (opeation) poduces a scala fom two aguments (the aguments themselves ae vectos) F G F G + F G + F G x x y y the symbol (sometimes also witten as := ) eads is defined as z z Magnitude the magnitude (i.e. length) of a vecto is a scala always positive denoted F o F defined as F F F = F + F + F x y z S. Achenbach Phys55 -: Foce Fields 5 S. Achenbach Phys55 -: Foce Fields 7 Review of Foce Review of Foce Example ( ) ( ) What is the dot poduct fo F =,,3 and G = 4,5,6? Example (,3, 3) F What is the magnitude of the vecto =? S. Achenbach Phys55 -: Foce Fields 6 S. Achenbach Phys55 -: Foce Fields 8

3 S. Achenbach: PHYS 55 (Pat, Topic ) Handouts p. 3 Review of Foce Review of Foce Example 3 Let F = 30 N, F x = 0 N, F y = 0 N. Find F z. Angle between Foce Vectos geometic intepetation of the angle θ between two vectos F and F : 3-dimensional view of vectos -dimensional view of the plane containing the two 3-dimensional vectos S. Achenbach Phys55 -: Foce Fields 9 S. Achenbach Phys55 -: Foce Fields Review of Foce Review of Foce Angle between Foce Vectos the angle θ between any two vectos is given by F F F F cos θ = = F F F F Example 4 ( ) ( ) Find the angle between the two vectos F = and F =,,3 4,5,6. with dot poduct opeation poducing a scala and the magnitude of a vecto F F = F F + F + F F F = F x F x + F y F y + F z F z x y z cos θ = F F x x F + F x y + F + F y z F y F + F x z + F F y z + F z S. Achenbach Phys55 -: Foce Fields 0 S. Achenbach Phys55 -: Foce Fields

4 S. Achenbach: PHYS 55 (Pat, Topic ) Handouts p. 4 Foce Fields eview of foce - special vectos - dot poduct - magnitude - angle between foce vectos foce fields wok (tansfe of enegy) - foce not in diection of movement - potential enegy - consevative foce fields - potential enegy evisited - wok done by a changing foce - wok as aea unde a cuve - pependicula component - enegy contou maps - elationship enegy contous & foce lines - estimating foce fom a contou map S. Achenbach Phys55 -: Foce Fields 3 Basics example of a gaphical epesentation of a foce field vectos (aows) show the foce on a standad (test) object located at the tail end of the foce aow point indicates (sample) position of object aow indicates diection of the foce.3 numbe indicates magnitude of the foce S. Achenbach Phys55 -: Foce Fields 5 Foce Fields Foce Fields Basics when an object expeiences a foce at evey point in space, it is in a foce field foce fields ae thee-dimensional thee is a 3-dimensional foce vecto associated with each point in 3-dimensional space Example 5 The chat shows the foce vecto acting on a kg mass at a few diffeent positions. The units of foce ae Newtons N. What is the foce on a kg mass located at x = 5 metes, y = 3 metes? fo pupose of illustation and simplicity, often only a plane in the field (-dimensional epesentation) is dawn on pape S. Achenbach Phys55 -: Foce Fields 4 S. Achenbach Phys55 -: Foce Fields 6

5 S. Achenbach: PHYS 55 (Pat, Topic ) Handouts p. 5 Foce Fields Basics altenative gaphical epesentation of a foce field diection of the foce is a tangent cuve magnitude is given at cetain points S. Achenbach Phys55 -: Foce Fields 7 eview of foce - special vectos - dot poduct - magnitude - angle between foce vectos foce fields wok (tansfe of enegy) - foce not in diection of movement - potential enegy - consevative foce fields - potential enegy evisited - wok done by a changing foce - wok as aea unde a cuve - pependicula component - enegy contou maps - elationship enegy contous & foce lines - estimating foce fom a contou map S. Achenbach Phys55 -: Foce Fields 9 Foce Fields Example 6 Estimate the foce vecto at x=4m, y=m. Basics enegy is a scala quantity (eal numbe) associated with an object units of enegy ae Newtons metes, called Joules J enegy can be tansfeed to o fom an object (e.g. while it is moving): wok is the enegy tansfeed to o fom an object by a foce if the foce is pushing o pulling the object - in the diection the object is moving, then the foce is doing positive wok on the object - opposite to the diection negative wok definition: with W = F d - magnitude of foce F applied to the object in the diection of the movement - distance d the object is moved - wok W done by foce F on the object as the object is moved a distance d S. Achenbach Phys55 -: Foce Fields 8 S. Achenbach Phys55 -: Foce Fields 0

6 S. Achenbach: PHYS 55 (Pat, Topic ) Handouts p. 6 Example 7 Foce not in Diection of Movement A foce with a magnitude of 3 N is applied to an object to slide it acoss a table. How much wok is done by the moving foce, if the object is moved 0.7 m? fee body diagam enegy is tansfeed fom the tacto to the box ca via the tow ope analysis of foces (and late enegy) using a fee body diagam foce applied by the ope has a component in the diection of movement does wok on the box ca othe foces (on wheels) act pependicula to the diection of movement do not contibute any wok (do not tansfe enegy to the box ca) S. Achenbach Phys55 -: Foce Fields S. Achenbach Phys55 -: Foce Fields 3 Example 8 A foce, F, with a magnitude of 0 N is applied to a block to slide it acoss a table. A second foce, F, which has the opposite diection of F is applied to the opposite side of the block. The magnitude of F is 6 N. The block is moved. m. ) How much wok is done by foce F? ) How much wok is done by foce F? 3) How much enegy is tansfeed fom F to F? Foce not in Diection of Movement equation fo wok done on an object moved in a staight line is W = F d = F d cos θ = ( F cos θ ) d with diffeence between end position p end and the stat position p stat of the object vecto d = p end p stat angle θ between vectos F and d 4) If the suface of the table is fictionless, how much enegy is used to acceleate the block (i.e. how much enegy is tansfeed fom F to the block)? S. Achenbach Phys55 -: Foce Fields S. Achenbach Phys55 -: Foce Fields 4

7 S. Achenbach: PHYS 55 (Pat, Topic ) Handouts p. 7 Foce not in Diection of Movement an intepetation of W ( F cos θ ) d is that is the component of foce =F F cos θ in the diection the object is moved Foce not in Diection of Movement Suppose the tacto pulls the box ca a distance of 00 m with a steady ope foce of 4000 N. The angle between the ope and the ails is 30 degees. wok done by tacto W = F ope d cos θ with - tensile foce in the ope F ope - distance box ca is moved d - angle θ between F ope and d d is in diection of ails angle between foce and distance vectos is θ = 30 = π / 6 wok done by the tacto on the box ca is W = F ope d cos θ = (4000 N) (00 m) cos ( π / 6 ) = kj S. Achenbach Phys55 -: Foce Fields 5 S. Achenbach Phys55 -: Foce Fields 7 Foce not in Diection of Movement Example 9 A wooden object is dagged m acoss a floo using a ope. fee body diagam component of foce of the tacto pulling the box ca in the diection of movement d has a magnitude F ope cos θ Recall that the diection of the foce exeted by a ope is always in line with the ope. If the magnitude of the foce exeted by Obelix on the ope is 0 N, how much wok is done by him (i.e. how much wok is done on the wooden object by the foce exeted by the ope)? S. Achenbach Phys55 -: Foce Fields 6 S. Achenbach Phys55 -: Foce Fields 8

8 S. Achenbach: PHYS 55 (Pat, Topic ) Handouts p. 8 Example 0 A sail boat is moved fom position (x,y) = (0m, 0m) to (00m, 30m) by a wind that exets a foce of (0 N, -0 N) [i.e. a foce of (0i -0j) N ] on the sail boat. What is the wok done by the wind on the sail boat? Example A 0 kg mass is hoisted by Obelix via a system of fictionless opes and pulleys. ) What is the gavitational foce on the mass? ) How much foce must Obelix apply to the ope to lift the mass? 3) How much wok is done by Obelix if the mass is hoisted m? 4) How much wok is done by the foce due to the gavitational field (gavitational foce)? 5) How much enegy is tansfeed fom Obelix to the mass? 6) Whee is the enegy that was added to the mass? S. Achenbach Phys55 -: Foce Fields 9 S. Achenbach Phys55 -: Foce Fields 3 Potential Enegy enegy can be stoed and etieved at a late time (usually) such enegy is called potential enegy fom latin potentia [foce, ability, powe] Example 3) How much wok is done by Obelix if the mass is hoisted m? e.g. enegy tansfeed by an ache to a bow (late, enegy can be etieved fom the bow and be tansfeed to an aow) fee body diagam enegy in a gavitational field tansfeed by lifting a mass up a vetical distance (late, wok done can be etieved by loweing the mass) Note: when a mass is elevated, we say the mass has potential enegy even though the enegy is stoed in the gavitational field S. Achenbach Phys55 -: Foce Fields 30 Note: Result would be the same if Obelix was pulling the mass vetically up, without pulley. But with pulley with additional tavelling wheel: same enegy distibuted ove double the distance ( half the foce) easie (but twice as long) to pull S. Achenbach Phys55 -: Foce Fields 3

9 S. Achenbach: PHYS 55 (Pat, Topic ) Handouts p. 9 Consevative Foce Fields Consevative Foce Fields Deivation time pemitting a field is called consevative if and only if the enegy equied to move a mass between two points in the field only depends on the stat and end points (and does not depend on the path taken) e.g., the gavitational field is a consevative field moving a mass staight fom A to C takes same enegy as moving the mass fom A to B fist and then fom B to C wok equied to move a mass fom A via B to C is wok to move it fom A to B plus wok to move it fom B to C foce equied to move a mass fom A to B is F with infinitesimal foce ΔF a = Fg + ΔFAB AB 0 in diection fom A to B to get the mass moving, but not doing wok F a F g as obvious fom fee body diagam wok equied to move a mass fom A to B is WAB = Fg d AB, and with = ( ) iˆ + ( y y ) ˆj W AB d AB 0 0 = [ 0 iˆ ( m g) ˆj ] ( 0) iˆ + ( y y0 ) = ()() ( m g) ( y y0 ) = mg ( ) y y 0 [ ˆj ] S. Achenbach Phys55 -: Foce Fields 33 S. Achenbach Phys55 -: Foce Fields 35 Consevative Foce Fields Deivation time pemitting foce equied to move a mass fom A to C is F with infinitesimal foce ΔF a = Fg + ΔFAC AC 0 in diection fom A to C to get the mass moving, but not doing wok F a F g as obvious fom fee body diagam with gavitational constant on eath g=9.8 m / s, and mass m : F g = m g Fa = Fg = ( 0 iˆ ( m g) ˆj ) wok equied to move a mass fom A to C is W = Fa d = ( Fa cos θ ) d = F g d, and with d = x x iˆ + y y ˆ ( 0 ) ( 0 ) j iˆ ( m g) ˆj x x iˆ 0 + y ()( 0 x x0 ) + ( m g)( y y0 ) W = mg ( ) = = [ ] [( ) ( y ) ˆj ] 0 0 W = AC y y 0 S. Achenbach Phys55 -: Foce Fields 34 Consevative Foce Fields wok equied to move a mass fom A via B to C is wok to move it fom A to B plus wok to move it fom B to C foce equied to move a mass fom B to C is F with infinitesimal foce ΔF a = Fg + ΔFBC BC 0 in diection fom B to C to get the mass moving, but not doing wok F a F g as obvious fom fee body diagam wok equied to move a mass fom B to C is WBC = Fg d BC, and with = ( x x ) iˆ ( 0) ˆj W AB d BC 0 + [ 0 iˆ ( m g) ˆj ] ( x x ) iˆ 0 + ( 0) ()( x x ) + ( m ) ( 0) = = g = [ ˆj ] Deivation time pemitting S. Achenbach Phys55 -: Foce Fields 36

10 S. Achenbach: PHYS 55 (Pat, Topic ) Handouts p. 0 Consevative Foce Fields wok equied to move a mass fom A via B to C is wok to move it fom A to B plus wok to move it fom B to C W ABC = WAB + W ABC = mg ( y y0 ) + 0 as expected, W ABC = W AC path does not matte WBC Example A peson with a mass of 70 kg snowboads down a mountain in a zig-zag fashion. How much wok is done by the gavitational field on the peson if the peson stats at a point on the mountain that is 500 m above the day lodge (vetical distance) & ends at the day lodge? with W AC W AB W BC = mg = mg = 0 ( y y 0 ) ( ) y y 0 S. Achenbach Phys55 -: Foce Fields 37 S. Achenbach Phys55 -: Foce Fields 39 Consevative Foce Fields by the way: wok done to move an object in the gavitational field is given by W = mg ( ) y y 0 wok only depends on vaiables mass of the object m vetical distance of movement y -y 0 (it also depends on g, but this is not a vaiable on the suface of the eath) Potential Enegy evisited potential enegy is associated with an object and its location in a field is measued fom a efeence point in space is the enegy that could be etieved if the object was moved fom its pesent location back to the efeence point of an object in a gavitational field is denoted U and is given by U = m g h with - U potential enegy of the object - m mass of the object - g gavitational constant (on the eath suface: g = 9.8 m / s ) - h vetical distance fom the efeence point to the cente of gavity of the object Note: if the object is located below the efeence point, then U and h ae negative S. Achenbach Phys55 -: Foce Fields 38 S. Achenbach Phys55 -: Foce Fields 40

11 S. Achenbach: PHYS 55 (Pat, Topic ) Handouts p. Example 3 A tuck mounted cane is used to lift a 500 kg wecking ball. The cane is mounted to the bed (deck) of the tuck. What is the potential enegy of the wecking ball with espect to the bed of the tuck if the cente of gavity of the ball is 0 m above the bed of the tuck? What is the potential enegy of the ball with espect to the bed of the tuck if the ball is loweed into a hole so that the cente of gavity is m below the bed of the tuck? Wok done by a changing Foce an object is moved a distance D fom S 0 along cuve S the foce acting on the object is a vecto that is a function of its position x on the cuve: decomposition of foces (fo values of x) F(x) is foce acting on the object when it is located a distance x fom S 0 along the cuve S Note: x is not the hoizontal distance fom S 0, but the distance taveled along the cuve F(x) can be boken down in components F(x) = F s (x) + F (x) with F s (x) diection is tangent to the cuve at point x (a distance x fom S 0 along the cuve) F (x) diection is pependicula to the cuve at point x S. Achenbach Phys55 -: Foce Fields 4 S. Achenbach Phys55 -: Foce Fields 43 Wok done by a changing Foce an object is moved a distance D fom S 0 along cuve S the foce acting on the object is a vecto that is a function of its position x on the cuve: F(x) is foce acting on the object when it is located a distance x fom S 0 along the cuve S Note: x is not the hoizontal distance fom S 0, but the distance taveled along the cuve Wok done by a changing Foce theoy to find wok done on an object as it is moved along the cuve (path): ) subdivide the path along the cuve between S and S into N segments of length Δx = (D D ) / N (segments should be small enough to be appoximated by staight lines) S. Achenbach Phys55 -: Foce Fields 4 S. Achenbach Phys55 -: Foce Fields 44

12 S. Achenbach: PHYS 55 (Pat, Topic ) Handouts p. Wok done by a changing Foce theoy to find wok done on an object as it is moved along the cuve (path): ) subdivide the path along the cuve between S and S into N segments of length Δx = (D D ) / N (segments should be small enough to be appoximated by staight lines) ) calculate appoximate wok done on the object as it is moved though each segment, teating the foce as constant ove each individual segment (but changing fom segment to segment) 3) sum wok done in all segments appoximate value fo the wok equied to move the object fom S to S Wok done by a changing Foce 3) total wok done by the foce on an object moved fom S to S is appoximately W ± [ Fs ( D Δx) Δx. segment F s ( D +. Δx) Δx + ( D + ( N + 0. ) Δx) Δx] L 5 F s appoximation becomes equality as N fo N (and Δx dx), limit of the sum is the integal W = x = D. segment F s x = D ( x) N. segment - integal of F s (x) fom x=d to x=d - adding up all values of F s (x) as a function with the sole vaiable x in&times small incements of dx between the stat point x=d and the end point x=d dx S. Achenbach Phys55 -: Foce Fields 45 S. Achenbach Phys55 -: Foce Fields 47 Wok done by a changing Foce ) calculate the wok done on an object as it is moved though one segment (segment i) segment i is the inteval D + i Δx D + i + Δx [ ( ) ), with its mid point X mid point wok done in segment i is Wsegment i ± Fs ( xmid point ) Δx = ± [ F s D + ( i ) Δx Δx F ( x) F ( x) S = S (wok may be pos. o neg. depending on whethe aids o opposes the movement of the object) ( i + 0. ) Δx = D + 5 ( ) ] ( point ) F s x mid S. Achenbach Phys55 -: Foce Fields 46 Wok done by a changing Foce 3) total wok done by the foce on an object moved fom S to S is appoximately W ± [ Fs ( D Δx) Δx. segment F s ( D +. Δx) Δx + ( D + ( N + 0. ) Δx) Δx] L 5 F s appoximation becomes equality as N fo N (and Δx dx), limit of the sum is the integal W = x = D. segment N. segment Note: integals fo many functions ae well known f0 e.g., one impotant function is F s ( x) = (f 0 is a constant) x F s x = D S. Achenbach Phys55 -: Foce Fields 48 ( x) dx f D dx = D x D D f f

13 S. Achenbach: PHYS 55 (Pat, Topic ) Handouts p. 3 Wok as Aea unde a Cuve fo a function F s (x) that epesents foce as a function of distance integal F s (x) dx has units foce times distance [Nm] aea unde the cuve F s (x) in a foce-vesus-distance gaph has units foce times distance [Nm] wok done [Nm = J] is gaphically epesented by aea unde the cuve F s (x) vesus x between x=d and x=d Gavitational Foce Field a familia consevative foce field is that of gavity magnitude of the foce due to gavitational attaction on the eath (fo an object located above the suface of the eath, i.e. x eath = m) mass 4 m m N F = x kg with - F magnitude of the foce on the object - m mass of the object - x distance fom the cente of the eath to the object units: if m [kg] and x [m] F [N] diection of the foce: towads the cente of the eath Note: stength of the foce due to gavitational attaction deceases quadatically with inceased distance S. Achenbach Phys55 -: Foce Fields 49 S. Achenbach Phys55 -: Foce Fields 5 Example 4 An object is moved a distance of m by a foce. The object is moved along a cuve stating at a point 0 m fom a efeence point S 0 to a point m fom S 0. The tangential component of the foce is F s ( x) = N (x is the distance fom S 0 in metes) x How much enegy is tansfeed fom the foce to the object, i.e. how much wok is done by the foce moving the object? Example 5 How much enegy is equied to move an object with the mass m fom the eath s suface to oute space? S. Achenbach Phys55 -: Foce Fields 50 S. Achenbach Phys55 -: Foce Fields 5

14 S. Achenbach: PHYS 55 (Pat, Topic ) Handouts p. 4 Fequently asked Question Why ae we always consideing the tangential component, e.g. F s (x), only? Answe: Because all the wok is done by the tangential component of the foce, and none by the pependicula component. And why is no wok done by the pependicula component? Answe: - with W = F d = F d cos θ and pependicula θ = 90 cos 90 = 0 wok pependicula to the motion = 0 - and how about motion pependicula to field lines? Well, look. Once upon a time, thee was a table... S. Achenbach Phys55 -: Foce Fields 53 Fequently asked Question Find the wok done if the puck is moved adially away fom the hole, fom 0 cm to 0 cm. F move is in the diection of motion wok done by extenal foce F move will be positive F move is exactly in the diection of motion θ = 0, cos θ = wok done is W = F move d cos θ = F move d = (9.8 N) (0 cm) = 0.98 Nm = 0.98 J In tems of the synthetic plana field, it would be said that the potential enegy of the puck inceased by 0.98 J. But eally, the potential enegy of the kg mass unde the table inceased as it was lifted 0 cm and gained mgh = 0.98 J. we could say the enegy is stoed in the synthetic plana foce field, but eally it is stoed in the gavitational field S. Achenbach Phys55 -: Foce Fields 55 Fequently asked Question conside the synthetic plana foce field set up on the suface of a table the,object is a weightless puck with a nail in it the field foce is ceated synthetically by a sting foce on the sting is povided by a kg mass dangling below the table the suface of the table is fictionless the foce on the puck - is always in the diection to the hole in the table - has a magnitude of mg = 9.8 N fo any position on the table Pependicula Comp. does no Wok component of the wok tangential to the motion (following the diection of field lines) what about the pependicula component? conside the wok done if the puck was moved in a semicicle (pependicula to field lines) ) use common sense to find the wok done potential enegy changes if wok is done potential enegy of the mass unde the table would have to change height of the mass unde the table would have to change height of the mass does not change no wok done while moving puck pependicula to natual movement along field lines S. Achenbach Phys55 -: Foce Fields 54 S. Achenbach Phys55 -: Foce Fields 56

15 S. Achenbach: PHYS 55 (Pat, Topic ) Handouts p. 5 Pependicula Comp. does no Wok component of the wok tangential to the motion (following the diection of field lines) what about the pependicula component? conside the wok done if the puck was moved in a semicicle (pependicula to field lines) ) find the wok done mathematically let x be the distance along the cicula path (x = 0 at stat position of the puck); (x = π = π 0 cm = 0.34 m at end position) fictionless movement on the cicle (mass not lifted): foce along that path F s (x) = 0 (constant fo all x) W = F d = (0 N) (0.34 m) = 0 J Enegy Contou Maps enegy contou maps show the potential enegy of an object as a function of position with espect to a efeence point e.g., - potential enegy of the puck at point B is J moe than when it is at point A (this is equivalent to saying potential enegy of the puck at point B with espect to point A is J ) - potential enegy of the puck at point C with espect to point A is - J - potential enegy of the puck at point B with espect to point C is 4 J no wok done while moving puck pependicula to natual movement along field lines S. Achenbach Phys55 -: Foce Fields 57 S. Achenbach Phys55 -: Foce Fields 59 Enegy Contou Maps enegy contous (o equipotential sufaces) ae sufaces ove which an object can be moved without changing its potential enegy all consevative foce fields have enegy contous e.g., an enegy contou fo the gavitational field of the eath is the suface of a sphee whose cente coesponds with the cente of the eath an object can be moved along an enegy contou with zeo foce (actually, with infinitesimal foce) because component of the field tangent to the enegy contou (pependicula to field line o line of foce ) is zeo Relationship En. Cont. Foce Lines enegy contous must be pependicula to lines of foce set up by the foce field (enegy contous and foce lines intesect at ight angles) in a plane of a foce field, enegy contous ae lines (equipotential lines) e.g., in the plana puck-on-a-sting field : enegy contous ae the cicumfeences of cicles (emembe: it takes no wok to move the puck in a cicumfeence of a cicle centeed on the hole in the table pot. enegy unchanged) S. Achenbach Phys55 -: Foce Fields 58 S. Achenbach Phys55 -: Foce Fields 60

16 S. Achenbach: PHYS 55 (Pat, Topic ) Handouts p. 6 Relationship En. Cont. Foce Lines an enegy contou can be found & dawn on a map that only has the lines of foce shown: daw a line pependicula to the foce line at the choosen point A extend this line so that it is pependicula to all foce lines Relationship En. Cont. Foce Lines e.g., conside a plane of the gavitational field of the eath (containing the cente of the eath, fo example though the equato) a satellite with an elliptical obit does not stay on a cicula enegy contou satellite expeiences a foce in (o opposite to) the diection of motion satellite peiodically changes speed thoughout the obit S. Achenbach Phys55 -: Foce Fields 6 S. Achenbach Phys55 -: Foce Fields 63 Relationship En. Cont. Foce Lines e.g., conside a plane of the gavitational field of the eath (containing the cente of the eath, fo example though the equato) a satellite with a cicula obit stays on a cicula enegy contou satellite expeiences no foce in the diection of motion satellite neve changes speed Relationship En. Cont. Foce Lines an enegy contou can be found & dawn on a map that only has the lines of foce shown: stat with a map showing the foce lines choose a efeence point stating at that point, daw an enegy contou pependicula (intesecting at 90 ) to all foce lines fist enegy countou fo moe contou lines: stat at intesection of the contou line and a foce line and find the distance (on the foce line) that the standad object must be moved fo (e.g.) J of wok (o 0 J o 0 J,...) stating at this new point, daw anothe contou line epeat until complete map is geneated S. Achenbach Phys55 -: Foce Fields 6 S. Achenbach Phys55 -: Foce Fields 64

17 S. Achenbach: PHYS 55 (Pat, Topic ) Handouts p. 7 Example 6 Geneate the enegy contous fo this foce field. The magnitude of the foce on the foce line labled S is constant at F/m = 4.0 N / kg. The efeence enegy contou should go though A. Othe enegy contous should be 0 J / kg apat. Estimating Foce fom a Contou Map enegy contou maps can be used to estimate the aveage foce exeted on an object that is moved between contou lines this foce is appoximately equal to the foce at a point half way between the contous suppose an object is moved along a cuve between points x = D and x = D with a tangential foce opposing the motion F s (x) x D wok done is W = = Fs ( x)dx x = D epesented by aea undeneath the cuve F s (x) between D and D ( ) wok also given by W = Fave D D with F ave = aveage of F s (x) on the inteval (D, D ) S. Achenbach Phys55 -: Foce Fields 65 if W and (D, D ) ae known, F ave can be obtained fom F ave W = D D S. Achenbach Phys55 -: Foce Fields 67 Example 7 Conside the gavitational foce field of the eath. Using the suface of a sphee with the adius i = m as a efeence, find the adius of the sphee whose suface is at an enegy of 0 6 J/kg above the efeence. I.e., at what adius would be a mass of kg if it is lifted fom the suface of the eath (efeence) outwad until a wok of 0 6 J is done? Estimating Foce fom a Contou Map e.g., finding the aveage foce fo a staight line path fom an enegy contou map aveage foce in the diection of movement fom points A to B (with a distance D AB ) F = ave W D AB (D AB can be measued with a ule) 0 J 0 J = = 33.3 N 0.3 m if points A and B ae on the same field line F ave is the aveage of ± F ( x) ove the inteval between points A and B as stated befoe, with x mid point half way between points A and B F ± F ave x mid ( ) point S. Achenbach Phys55 -: Foce Fields 66 S. Achenbach Phys55 -: Foce Fields 68

18 S. Achenbach: PHYS 55 (Pat, Topic ) Handouts p. 8 Example 8 Appoximating the foce at a point with an aveage foce: ) A kg object is moved in a staight line fom A to B, which ae at a distance of m. What is the aveage value of the component of foce that does the wok in moving the object? ) Estimate the foce (i.e. F = F iˆ x + F ˆ y j ) the field exets on a kg mass located at point D. Example 8 ) Estimate the foce (i.e. F = F iˆ x + F ˆ y j ) the field exets on a kg mass located at point D. Solution: Note: this esult is an appoximation because field foce at point D was appoximated with an aveage foce potential eos dawing the field line though point D potential angula eos dawing the un and ise potential measuement eos of the un and ise magnitude? diection / components? S. Achenbach Phys55 -: Foce Fields 69 S. Achenbach Phys55 -: Foce Fields 7 Example 8 ) Estimate the foce (i.e. F = F iˆ x + F ˆ y j ) the field exets on a kg mass located at point D. Solution: the diection of the foce exeted by the field is the same as the diection of the field foce line diection of the field foce line is always fom a point of highe enegy to a point of lowe enegy to sepaate the magnitude of foce F D = 0 N into x and y components, slope of the field line (ise ove un) must be measued (e.g. with a ule) hee: ise = un = ( / ) m F slope = - y = F y = - F F x x fom diection of foce line, F x is negative, F y is positive S. Achenbach Phys55 -: Foce Fields 70