Conservation of Energy


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1 Lecture 3 Chapter 8 Physcs I Conservaton o Energy Course webste: Lecture Capture: , Fall 03, Lecture 3
2 Outlne Chapter 8 Conservatve & Nonconservatve orces Potental Energy Gravtatonal Pot. Energy Elastc Pot. Energy Conservaton o Mechancal Energy 95.4, Fall 03, Lecture 3
3 Conservatve Forces The work done by a conservatve orce n movng an object rom pont A to pont B depends only on the postons A and B, not the path or the velocty o the object The net work done by a conservatve orce or a round trp and returnng an object to ts ntal poston s zero Conservatve orces: gravty, sprng, electrostatc Nonconservatve orces: rcton, drag 95.4, Fall 03, Lecture 3
4 y Gravtatonal Potental Energy y mg K d y K WorkKnetc Energy Prncple x mg( y y) Consder a block sldng down on a rctonless surace under the nluence o gravty F G dl mg mg( ˆ) j dx( ˆ) dy( ˆ) j The work done by the gravtatonal orce: W W G G F y y G W G K W G dl mg( ˆ) j [ dx(ˆ) dy( ˆ)] j mgdy mg( y y) mv mv mv mv mv mgy mgy mv Rearrange t: 95.4, Fall 03, Lecture 3
5 Conservaton o Mechancal Energy!!! mv mgy mv mgy 95.4, Fall 03, Lecture 3 mgy represents a new orm o energy, potental energy Gravtatonal potental energy Total Mechancal Energy K U mgy U K U E K U E E so, we got E constant Conservaton o Mechancal Energy Energy s transormed between knetc and potental As the object alls, t reveals ts potental energy n orm o knetc one and can do work
6 General Potental Energy For a system, where only conservatve orces do work, we have: K U K U U U K K K W WorkKE Prncple U W K Relaton between potental energy and work: U W U W F G.d l In general, we dene the change n potental energy assocated wth a conservatve orce F as the negatve o the work done by that orce. 95.4, Fall 03, Lecture 3
7 Potental Energy Energy dened as the ablty to do work Knetc Energy: assocated wth energy o moton Knetc Energy K mv Other types o stored energy that can do work A compressed sprng An object at a heght that can roll or drop These systems have the potental to do work Call t a stored potental energy Potental energy can only be assocated wth conservatve orces Only changes o potental energy mportant, not absolute values Choose a sutable reerence U=0 or each problem , Fall 03, Lecture 3
8 Example: Roller coaster A 000kg roller coaster moves rom pont to ponts and 3. What s the potental energy o the roller coaster at ponts and 3 relatve to pont? What s the change n potental energy rom ponts to 3? U 0 U 0 Frst, choose a reerence level U 0 U U mg(y y ) U 3 U 3 y mg( y ) U 3 0 U U 98kJ U 3 U 47kJ 95.4, Fall 03, Lecture 3 U 3 U (47 98)kJ 45kJ Subtract them:
9 Example: Droppng ball An object o mass m s dropped v 0 rom a heght h above the ground. Fnd speed o the object as t hts the ground: Equatons o moton or constant acceleraton F mg From N. nd law we got ths knematc eqn: 0 v v gh v 95.4, Fall 03, Lecture 3 mv gh Energy conservaton K U mgy K mv mv mgh Thus, both approaches are equvalent 0 h Re. level U=0 y U mgy v 0 v? 0 h
10 Elastc/Sprng Potental Energy What s the potental energy o a sprng compressed rom equlbrum by a dstance x? F x kx Use a relaton between potental energy and work: U U U ( x) U (0) x kxdx kx 0 Choose U = 0 when x = 0 Potental energy o a sprng U W x F dl 0 S U sprng kx 95.4, Fall 03, Lecture 3
11 Example: Brck/sprng on a track (II) A kg mass, wth an ntal velocty o 5 m/s, sldes down the rctonless track shown below and nto a sprng wth sprng constant k=50 N/m. How ar s the sprng compressed? ntal v 5 m s y =m Re. level U=0 nal K Energy conservaton: U mv mgy mv mgy kx K U 0 U 0 sp So, the sprng compresson, x: x m k v gy x. 7m 95.4, Fall 03, Lecture 3
12 Force Potental Energy Force Potental Energy Gven a conservatve orce as a uncton o poston, the change n potental energy assocated wth ths (conservatve) orce s: U U(x)U(0) 0 x F x dx Potental Energy Force Gven a potental energy as a uncton o poston, the assocated conservatve orce s: F(x) du(x) dx 95.4, Fall 03, Lecture 3
13 Example: D Force Potental Energy Gven the potental energy: U(x) Ax Bx C nd the orce F as a uncton o x F(x) du dx d dx (Ax Bx C) Ax B 95.4, Fall 03, Lecture 3
14 Example: Potental Energy 3D Force In 3D F(x, y, z) U x 95.4, Fall 03, Lecture 3 î U y U(x, y, z) 3xy 4z ĵ U z Partal Dervatve: When takng dervatves wth respect to one varable, treat other varables as constants U x 3y U y 3x So F(x, y, z) 3yî U z 4 3xĵ 4 ˆk Potental energy s a scalar, orce s a vector ˆk
15 ConcepTest Paul and Kathleen start rom rest at the same tme on rctonless water sldes wth derent shapes. At the bottom, whose velocty s greater? Water Slde I A) Paul B) Kathleen C) both the same Conservaton o Energy: E E thereore: mgh mv v gh Because they both start rom the same heght, they have the same velocty at the bottom. Re. level U=0
16 ConcepTest Paul and Kathleen start rom rest at the same tme on rctonless water sldes wth derent shapes. Who makes t to the bottom rst? Water Slde II A) Paul B) Kathleen C) both the same Even though they both have the same nal velocty, Kathleen s at a lower heght than Paul or most o her rde. Thus, she always has a larger velocty durng her rde and thereore arrves earler! Re. level U=0
17 Example: Conservaton o Energy Martn s tghtywhteys 95.4, Fall 03, Lecture 3
18 (a) What s the sprng constant o Martn s tghtywhteys? N. nd law ky mg 0 F sp y=5 m k mg y (40 kg) (9.8 m/s 5 m ) M=40kg k 78 N/m mg 95.4, Fall 03, Lecture 3
19 (b) What s the potental sprng energy beore the cougar lets go? y = 5 m θ=37 M=40kg USp kx 5 78( ) sn 37 70J 95.4, Fall 03, Lecture 3
20 nal (c) What s Martn s launch speed? y=5m θ=37 U G =0 M=40kg ntal U Sp kx 70J K U G 0 0 U Sp K mv kx mgy kx mv mgy U G U Sp Energy conservaton: 0 v kx mgy m v 6 m s 95.4, Fall 03, Lecture 3
21 (d) What horzontal dstance does he travel? Projectle moton problem, not conservaton o energy. Use knematc equatons. V o =6 m/s θ=37 y 0 =5 m y=0 (ground) 95.4, Fall 03, Lecture 3
22 Thank you See you on Monday 95.4, Fall 03, Lecture 3
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