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This Week 7/12 Lecture – Chapter 8 7/13 Recitation – Bungee Problems: 8.4, 8.23, 8.44 7/14 Lab – Kinematics in 1-D Homework #4 Due @ 5pm 7/15 Lecture – Chapter 9 7/16 Recitation – ??? Problems: 7/12/04 ??? 1 Chapter 8 Potential Energy and Conservation of Energy 7/12/04 2 Review: Kinetic Energy: Work (const force): Work (variable force): Work-KE theorem 7/12/04 K mv 1 2 2 W F r Fr cos W F dl K W 3 Potential Energy What happens to the energy I use when I do work on an object? know about kinetic energy… In certain circumstances, it is stored as potential energy Already U W In the case of non-conservative forces, the energy can be lost (usually as heat) 7/12/04 4 Non-Conservative Forces Non-conservative if the force does not reverse the energy transfer when the path is reversed Path dependent! Examples: Friction Air resistance John 7/12/04 Kerry ???? 5 Conservative Forces Conservative force does reverse the energy transfer when the path is reversed Path independent, work depends only upon position Examples: Gravity Springs 7/12/04 6 Gravitational Potential Energy Recall the work done lifting an object against gravity: Wgrav Fgrav d mgh Lifting: Dropping: Wgrav Fgrav d mgh Ugrav = -W = +mgh (if lifting) Ugrav = -W = -mgh (if dropping) h d Note: We get the same energy out as we put in (conservative) 7/12/04 7 Gravitational Potential Energy An easier way to get the sign right… U mgy U 2 U1 mgy2 mgy1 So the gravitational potential energy can be written as: U mgy But isn’t the choice of y = 0 arbitrary? 7/12/04 8 Potential Energy from Spring Energy associated with distortion of spring from equilibrium length F x=0 x=xf xf Wspring F ( x)dx 0 7/12/04 xf 0 1 2 (kx)dx kx f 2 1 2 U spring Wspring kx f 2 9 Important Points About Potential Energy 1. U depends on the position/arrangement of objects U between two arrangements does not depend on path 2. Only U has physical meaning -- the numerical value of U itself is arbitrary This means you decide for yourself where U = 0 (like you decide where x = 0) 7/12/04 10 Important Points about Potential Energy 3. U is only defined for "conservative" forces, which do not dissipate energy No U for frictional forces 4. Can rewrite conservation of energy: Old: Kinitial + Wtot = Kfinal New: Kfinal – Kinitial = Wtot = Wc + Wnc But: Wc = -ΔU = Uinitial – Ufinal Kinitial + Uinitial + Wnc = Kfinal + Ufinal Define E = K + U Einitial + Wnc = Efinal 7/12/04 11 Conservation of Mechanical Energy If there are no non-conservative forces: Kinitial + Uinitial + Wnc = Kfinal + Ufinal This gives us conservation of mechanical energy: Kinitial + Uinitial = Kfinal + Ufinal Einitial = Efinal 7/12/04 12 Example: Ball Thrown Upwards (neglect air resistance) E = K + U = ½mv2 + mgy = constant Set U = 0 at floor vmax ymax y = ymax E = U = mgymax y = 0 E = K = ½mvmax2 Energy E U K vmax = 2gymax 7/12/04 y 13 Example: Pig on a Curved Track No friction Starts from rest How fast is it going at the bottom? h Kinitial + Uinitial = Kfinal + Ufinal Kfinal = Uinitial - Ufinal = - U Kfinal = mgh ½mvf2 = mgh 7/12/04 Only endpoints matter, curve of track doesn't! vf = 2gh 14 Pendulum: Qualitative View Turning Points energy E K U K can never be negative 7/12/04 Motion is bounded 15 Pendulum Problem L Lcosθ If I release the pendulum at rest from an angle , how fast is it going at the bottom? L-Lcosθ At bottom: y = 0 (set U = 0 there) At angle : y = L – Lcos Kinitial + Uinitial = Kfinal + Ufinal mg(L – Lcos ) = ½mv2 7/12/04 v = 2gL(1 – cos ) 16 Defenestration n. [Lat., de-,”out of”; fenstra, “window”.] An act of throwing something or someone out of a window. Traditional political custom: Prague 1419 A popular uprising led by the priest Jan Zelivsky included the throwing of the city councilors from the windows of the New Town Hall. 1618 The governors of Bohemia attempted to crush Czech Protestantism. They were thrown from the windows of the council room in Hradcany. This event helped precipitate the Thirty Years War. 7/12/04 17 Potential Energy Example If the royal councilors were given the heave-ho at 5 m/s who is going fastest when they hit the ground? a b c Ei=Ef Ei = Ki+Ui = ½mvi2 + mgh Ef = Kf + Uf = ½mvf2 7/12/04 ½mvi2 + mgh = ½mvf2 vf2 = vi2 + 2gh They all hit the ground with the same velocity! 18 Graphical Representation of Energy For a closed system: E = K + U = constant Can plot U(x) to see how system evolves Since K cannot be negative, motion is bounded by E U bounds E1 trapped in well E2 7/12/04 x 19 Force From Potential Energy -U Work done by force U Fx U F x In the limit of small x, we get: dU F dx Why the minus sign? 7/12/04 20 Force From Potential Energy dU F dx Force pushes in direction to decrease potential energy, i.e., "downhill" U F F F=0 x 7/12/04 dU/dx < 0 F>0 dU/dx > 0 F<0 21 How High to “Loop the Loop”? Ei = mgh = Ef = mg(2R) + ½mv2 Centripetal: v2/R = g (minimum v, barely loops) mgh = 2mgR + ½mRg v h R h= 2R + 0.5R = 2.5R (minimum) 7/12/04 22 Pig Sliding Up a Frictional Plane mk = 0.3 v0 = 4 m/s = 30 How high does he get? Set U = 0 at bottom Kinitial + Uinitial + Wfriction = Kfinal + Ufinal 7/12/04 23 Example (continued) d Kinitial + Wfriction = Ufinal h = 30 Kinitial = ½mv02 Wfriction = -|Ff|d = -mkNd = -mk (mgcos) d Ufinal = mgh = mgdsin ½mv02 – mk mgd cos = mgd sin 7/12/04 24 Example (continued) 1 2 v0 m k gd cos gd sin 2 2 v0 d ( m k g cos g sin ) 2 2 v0 d 2( m k g cos g sin ) d h = 30 1 (4 m / s) 2 d 2 (0.3)(9.8 m / s 2 ) cos 30 (9.8 m / s 2 ) sin 30 1.07 m 7/12/04 25 Constraint Forces The normal force and tension, in many situations, exert forces which are perpendicular to the motion Motion on a fixed surface (e.g. track) Tension from a rope with one end fixed (e.g. pendulum) If F•dx is always zero, no work can be done! 7/12/04 26 Constraint Forces No motion in direction of constraint No work No potential energy Example - pole vaulting Originally invented by Dutch farmers 1 Emech = mvi2 2 1 2 mvi = mgymax 2 vi2 ymax = 2g 7/12/04 27 Energy in the Pole Vault More detail: pole acts as a spring Energy Pole mgh Kinetic Time 7/12/04 28 Estimating the Record Fast sprinter travels ~10 m/s Vaulter running: Erun=½mv2 At top of motion: Etop=mgh ½mv2=mgh Record: 6.14 m Sergey Bubka, 1994. 7/12/04 h=v2/2g About 5 m. Remember-center of mass is about 1m up at start. 29 Another Cultural Moment… 7/12/04 30 Cow-a-pult 400 kg Say that a spring of constant k = 104 N/m is stretched 2 m to launch the cow. What is the max range of the cow if it is released 5m above the ground? Uspring,i+Ugrav,i+Ki = Uspring,f+Ugrav,f+Kf U spring,i 12 kx2 U grav, f mgh K f 12 mv2 7/12/04 31 Cow-a-pult (continued) 1 2 kx2 mgh 12 mv2 kx2 2mgh v m (10 4 N / m)( 2 m) 2 2(400 kg)(9.8 m / s 2 )(5 m) v 30 m / s 400 kg Maximum range when = 45: v0 sin 2 (30 m / s ) 2 R 92 m 2 g 9.8 m / s 2 7/12/04 32 Example: = 30 A 2 kg piglet on rough plane is compressing a spring by x = 0.1 meters and is released from rest on this plane (mk = 0.5). If v = 4 m/s after traveling 6m, what is k of the spring? 7/12/04 33 Example (continued) = 30 Kinitial + Uinitial + Wfriction = Kfinal + Ufinal Ugravity = 0 at start Uspring = ½kx2 at start Ugravity = mgd sin at end Uspring = 0 at end Wfriction = -|Ff|d = -mkNd = -mkmgd cos 7/12/04 Kinitial = 0 Kfinal = ½mv2 34 Example (continued) = 30 Kinitial + Uinitial + Wfriction = Kfinal + Ufinal 0 12 kx2 m k mgd cos 12 mv2 mgd sin 1 2 kx2 12 mv2 mgd sin m k mgd cos mv2 2mgd sin 2m k mgd cos 4 k 2 . 7 10 N /m 2 x 7/12/04 35 A More Complicated Potential The bow provides a non-constant force to the arrow F x What is the kinetic energy of the arrow? What is the final speed of the arrow? 7/12/04 36 Arrows of Outrageous Fortune… Force vs Distance Area = Uinitial = 50 J 200.00 Uinitial=Kfinal= ½mv2 180.00 160.00 An arrow has a mass ~ 0.03 kg… Force (N) 140.00 120.00 100.00 80.00 60.00 v 40.00 20.00 0.00 0.00 0.10 0.20 0.30 0.40 0.50 2 K final m 58 m / s Distance (m) 7/12/04 37 Work Due to Gravity Near the Earth Away from the Earth F mg m1 m2 F G 2 x xf xf xi xi W = F dx -mg dx xf xf m1 m2 W = F dx G 2 dx x xi xi xf W mgx x = mg ( x f -xi ) xf i 7/12/04 1 W = G m1m2 x xi 1 1 = G m1m2 x x f i 38 The End of the World as We Know It? 7/12/04 39 Extinction! 70 Million years ago Dinosaurs ruled the Earth They disappeared at the boundary between the Cretaceous and Tertiary periods (K-T boundary) Luis Alvarez (a Nobel Prize winner in Physics) suggested an asteroid impact might be responsible 7/12/04 40 The Impact Site Alvarez calculated the asteroid would need to be 10 km across and would leave a crater 150 km in diameter A huge crater off the Yucatan peninsula of Mexico has been identified as a possible impact site. Research on this crater has shown it is the result of an extra-terrestrial impact. 7/12/04 41 Assume an asteroid started at rest in the middle of the inner Oort cloud (~5000 RE-S) Assume it is acted on primarily by the Sun Assume mass ~1016 kg (10 km rock) 1 1 W = G ms ma x f xi = (6.672 10-11 N m 2 /kg 2 )(1.99 1030 kg)(1016 kg) = 8.9 10 24 2K v= = m 7/12/04 1 1 11 14 1.5 10 m 7.5 10 m J 2 (8.9 1024 J ) = 42,100 m/s 16 10 kg 1 Ton TNT =4109J Asteroid Impact =2109 MT TNT Over 80,000 MPH! 42 The Ball Race 1 2 d = h/sinθ h θ Ball 1 falls straight down, ball 2 rolls down a plane Which reaches the bottom first? Which is traveling fastest at the bottom? Ball 1: h=½gt2 t1= 2h/g Ball 2: a||=g sinθ t2= 2d/(gsinθ) = 2h/(gsin2θ) = t1/sinθ > t1 Balls 1,2: same Ki=0, Ui=mgh, Uf=0 same Kf same vf 7/12/04 43