SPH4U. Conservation of Energy. Review: Springs. More Spring Review. 1-D Variable Force Example: Spring. Page 1. For a spring we recall that F x = -kx.

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1 -D Variable Force Exaple: Spring SPH4U Conseration of Energ For a spring we recall that F x = -kx. F(x) x x x relaxe position -kx F = - k x the ass F = - k x Reiew: Springs Hooke s Law: The force exerte b a spring is proportional to the istance the spring is stretche or copresse fro its relaxe position. F X = -kx Where x is the isplaceent fro the equilibriu an k is the constant of proportionalit. relaxe position F X = 0 x ore Spring Reiew The work one b the spring W s uring a isplaceent fro x to x is the area uner the F(x) s x plot between x an x. F(x) x x W s x -kx In this exaple it is a negatie nuber. The spring oes negatie work on the ass W s x x F( x) x x ( kx) x x kx x x W k x x s Page

2 Proble: Spring pulls on ass. A spring (constant k) is stretche a istance, an a ass is hooke to its en. The ass is release (fro rest). What is the spee of the ass when it returns to the relaxe position if it slies without friction? relaxe position : Spring pulls on ass. First fin the net work one on the ass uring the otion fro x = to x = 0 (onl ue to the spring): Ws k x x k 0 k stretche position (at rest) after release back at relaxe position x x stretche position (at rest) relaxe position r r Proble: Spring pulls on ass. Now fin the change in kinetic energ of the ass: ΔK r 0 r : Spring pulls on ass. Now use work kinetic-energ theore: W net = W S = K. k r k r x x x x stretche position (at rest) stretche position (at rest) relaxe position relaxe position r r Page

3 Springs : Unerstaning A spring with spring constant 40 N/ has a relaxe length of. When the spring is stretche so that it is.5 long, what force is exerte on a block attache to the en of the spring? x = 0 k x = x = 0 k x =.5 Unerstaning Forces an otion A block of ass = 5. kg is supporte on a frictionless rap b a spring haing constant k = 5 N/. When the rap is horizontal the equilibriu position of the ass is at x = 0. When the angle of the rap is change to 30 o what is the new equilibriu position of the block x? (a) x = 0c (b) x = 5c (c) x = 30c F X = -kx (a) -0 N (b) 60 N (c) -60 N F X = - (40N/) (.5) k x = 0 q = 30 o F X = - 0 N Choose the x-axis to be along ownwar irection of rap. FBD: The total force on the block is zero since it s at rest. Consier x-irection: Force of grait on block is F x,g = g sin(q Force of spring on block is F x,s = -kx N Since the total force in the x-irection ust be 0: Μg sin θ g sinq kx 0 x k 5.kg 9.8 x 5 N s x q q g F x,g = g sinq x q Page 3

4 Work b ariable force in 3-D: Nice to know explanation Work W F of a force F acting through an infinitesial isplaceent r is: W = F. r The work of a big isplaceent through a ariable force will be the integral of a set of infinitesial isplaceents: F r W TOT = F. r Work/Kinetic Energ Theore for a Variable Force in 3D r W F r kinetic energ r Su up F.r along path That s the work integral That equals change in KE For conseratie forces, the work is path inepenent an epens onl on starting point an en point Work b ariable force in 3-D: Newton s Graitational Force Integrate W g to fin the total work one b grait in a big isplaceent: R R W g = W g = (-G / R ) R = G (/R - /R ) R R F g (R ) Work b ariable force in 3-D: Newton s Graitational Force Work one epens onl on R an R, not on the path taken. Wg G R R R F g (R ) R R R Page 4

5 Potential Energ For an conseratie force F we can efine a potential energ function U in the following wa: W = The work one b a conseratie force is equal an opposite to the change in the potential energ function. This can be written as: F. r = -U U = U - U = -W = - r F. r r r U r U Graitational Potential Energ So we see that the change in U near the Earth s surface is: U = -W g = g = g( - ). So U = g + U 0 where U 0 is an arbitrar constant. Haing an arbitrar constant U 0 is equialent to saing that we can choose the location where U = 0 to be anwhere we want to. Floor leel of 400 Hazel St Science Office (potential is zero here, for sure!) W g = -g Conseratie Forces: We hae seen that the work one b grait oes not epen on the path taken. Unerstaning Work & Energ A rock is roppe fro a istance R E aboe the surface of the earth, an is obsere to hae kinetic energ K when it hits the groun. An ientical rock is roppe fro twice the height (R E ) aboe the earth s surface an has kinetic energ K when it hits. R E is the raius of the earth. R R Wg G R R What is K / K? (a) The easiest wa to sole this proble is to use the W=K propert. h W g = -gh (b) (c) R E R E R E Be careful! Page 5

6 Since energ is consere, K = W G. R E R E R E W G =G R R Do not use gh forula as this onl works when h is er sall. ΔK=c R R For the first rock: For the secon rock: Where c = G is the sae for both rocks K =c c R E R E R E K =c c R E 3R E 3 R E K = 3 K 4 3 Conseratie Forces: In general, if the work one oes not epen on the path taken (onl epens the initial an final istances between objects), the force inole is sai to be conseratie. Grait is a conseratie force: Grait near the Earth s surface: Wg G R R Wg g A spring prouces a conseratie force: Ws k x x Conseratie Forces: A force that offers the opportunit of two-wa conersion between kinetic an potential energies is calle a conseratie force. The work one b a conseratie force alwas has these properties: It can alwas be expresse as the ifference between the initial an final alues of potential energ function. It is reersible. It is inepenent of the path of the bo an epens onl on the starting an ening points. When the starting an ening points are the sae, the total work is zero. Conseratie Forces: When the onl forces that o work are conseratie forces, then the total echanical energ is E = K + U Conseratie forces hae the nice propert of being able to be efine in ters of a potential energ. The usual efinition of potential energ is through the work-energ theore as for kinetic energ, i.e. W = U i - U f. W U U total total W K K U U K K U K U K Page 6

7 NonConseratie Forces: Not all forces are conseratie. Consier the friction force applie to a crate, the total work one b friction force when sliing the crate up a rap an back own is not zero. (when the irection of the otion reerses so oes the friction force, an the friction oes negatie work in both irections.) Conseratie Forces: We hae seen that the work one b a conseratie force oes not epen on the path taken. W = W Therefore the work one b a conseratie force in a close path is 0. W W W W NET = W - W = W - W = 0 W Potential energ change fro one point to another oes not epen on path Unerstaning Conseratie Forces The pictures below show force ectors at ifferent points in space for two forces. Which one is conseratie? (a) (b) (c) both Consier the work one b force when oing along ifferent paths in each case: No work is one when going perpenicular to force. W A = W B W A > W B x () x () () () Page 7

8 In fact, ou coul ake one on tpe () if it eer existe: Work one b this force in a roun trip is > 0! Free kinetic energ!! W NET = 0 J = K W = 0 Potential Energ Recap: For an conseratie force we can efine a potential energ function U such that: S U = U - U = -W = - F. r S The potential energ function U is alwas efine onl up to an aitie constant. W = 5 J W = -5 J You can choose the location where U = 0 to be anwhere conenient. W = 0 Conseratie Forces & Potential Energies (stuff ou shoul know): Force F Work (one b force) W Change in P.E U = U - U P.E. function U Unerstaning Potential Energ All springs an asses are ientical. (Grait acts own). Which of the sstes below has the ost potential energ store in its spring(s), relatie to the relaxe position? F g = -g -g( - ) g( - ) g + C G F g = G G G C R R R R R R F s = -kx k x x k x x kx (R is the center-to-center istance, x is the spring stretch) C () () (a) (b) (c) sae Page 8

9 The isplaceent of () fro equilibriu will be half of that of () (each spring exerts half of the force neee to balance g) () () 0 The potential energ store in () is: k k The potential energ store in () is: k k The spring P.E. is twice as big in ()! Conseration of echanical Energ If onl conseratie forces are present, the total kinetic plus potential energ of is consere, i.e. the total echanical energ is consere (ef. of E). (note: E=Eechanical throughout this iscussion) E = K + U E = K + U = W + U = W + (-W) = 0 using K = W using U = -W E = K + U is constant!!! Both K an U can change, but E = K + U reains constant. But we ll see that if issipatie forces act, then energ can be lost to other oes (theral, soun, etc) changing E echanical an external forces can change E echanical Exaple: The siple penulu Suppose we release a ass fro rest a istance h aboe its lowest possible point. What is the axiu spee of the ass an where oes this happen? To what height h oes it rise on the other sie? Exaple: The siple penulu Kinetic+potential energ is consere since grait is a conseratie force (E = K + U is constant) Choose = 0 at the botto of the swing, an U = 0 at = 0 (arbitrar choice) E = / + g h h = 0 h h Page 9

10 Exaple: The siple penulu E = / + g. Initiall, = h an = 0, so E = gh. Since E = gh initiall, E = gh alwas since energ is consere. Exaple: The siple penulu / will be axiu at the botto of the swing. So at = 0 / = gh = gh gh = h = 0 = 0 h Exaple: The siple penulu Since E = gh = / + g it is clear that the axiu height on the other sie will be at = h = h an = 0. The ball returns to its original height. Exaple: The siple penulu The ball will oscillate back an forth. The liits on its height an spee are a consequence of the sharing of energ between K an U. E = / + g = K + U = constant. = h = h = 0 Page 0

11 Exaple: Airtrack & Glier A glier of ass is initiall at rest on a horizontal frictionless track. A ass is attache to it with a assless string hung oer a assless pulle as shown. What is the spee of after has fallen a istance? Exaple: Airtrack & Glier Kinetic+potential energ is consere since all forces are conseratie. Choose initial configuration to hae U=0. K = -U g g Proble: Hotwheel A to car slies on the frictionless track shown below. It starts at rest, rops a istance, oes horizontall at spee, rises a istance h, an ens up oing horizontall with spee. Fin an. Proble: Hotwheel... K+U energ is consere, so E = 0 K = - U oing own a istance, U = -g, K = / Soling for the spee: g h h Page

12 Proble: Hotwheel... At the en, we are a istance - h below our starting point. U = -g( - h), K = / Soling for the spee: g h Hooke s Law (reiew) The agnitue of the force exerte b the spring is irectl proportional to the istance the spring has oe fro its equilibriu. Fx kx Force is opposite to the irection spring is oe - h h Fx kx This is the Force applie to the spring Exaple A kg ass is hung fro a ertical spring that is allowe to stretch slowl fro its unstretche equilibriu position until it coes to its new equilibriu position 0.0 below its initial one. a) Deterine the force constant of the spring? b) If the ball is returne to the spring s initial unstreche equilibriu position an then allowe to fall, what is the Net Force on the ass when it has roppe 0.08? c) Deterine the acceleration of the ass at position b) F=-kx F 0 g kx 0 kx g g k x N 0.085kg 9. 8 kg 0.0 F=g N 4.65 a) Deterine the force constant of the spring? Therefore k= 4. N/ Page

13 F=-kx F=g N F g kx b) If the ball is returne to the spring s initial unstreche equilibriu position an then allowe to fall, what is the Net Force on the ass when it has roppe 0.08? N N kg 0.495N 0.085kg Therefore F= 0.49 N F=-kx F=g F a 0.495N a c) Deterine the acceleration of the ass at position b) Therefore a= 5.8 /s own 0.495N a kg 5.78 s Elastic Potential Energ (reiew) The energ store in objects that are stretche, copresse, bent, or twiste. Us kx Unerstaning A 0.0 kg ass is hung fro a ertical spring (k=9.6 N/). The ass is hel so that the spring is at its unstretche equilibriu position. The ass is then allowe to fall. Neglect the ass of the spring. a) How uch elastic potential energ is store in the spring when the ass has fallen c? b) What is the spee of the ass when it has fallen c? Page 3

14 x=0 c x= c U s kx N J a) How uch elastic potential energ is store in the spring when the ass has fallen c? E E i gh kx gh kx f f f x=0 c x= c f gh kx 0kg N s 0. 0kg.0 s b) What is the spee of the ass when it has fallen c? Flash Page 4