Lecture 13 Chapter 9 Sprng Force and Power Yeah, energy s better than orces. What s net? Course webste: http://aculty.uml.edu/andry_danylov/teachng/physcsi
IN THIS CHAPTER, you wll learn how to solve problems usng two new concepts: work and knetc energy nstead o orces. Today we are gong to dscuss: Chapter 9: Work done by a Sprng: Secton 9.4 Skp: Secton 9.5 Power: Secton 9.6
Work Done By a Varyng Force In the prevous class, we were ndng work done by constant orces But there are orces whch are not constant, let s look at one o those, sprng orce Object oscllatng on a sprng F k We need to gure out how to deal wth ths case.
Let s rst just ntroduce a sprng orce and then get work done by t. The Sprng Force varable orce The varable orce eerted by a sprng s gven by Hooke s Law: F sprng k k sprng constant (postve) Equlbrum Equlbrum Equlbrum 0 0 F S F S 0 F s 0 0 F s k 0 The orce s to the let 0 F s k( ) 0 The orce s to the rght The sprng orce returns the cart to the equlbrum. It s called a Restorng orce
Work done by a sprng Let s calculate work done a sprng orce Sprng orce: F sprng k Stretched Equlbrum d - dsplacement 0 F sp d Work done by F sp : W sp F sp d ( k ) d d k Ths s used to ntegrate: n n1 d n 1 k W sp k
Let s learn a useul trck o calculatng work Work s an area under a curve F-vs- Graphcal meanng o an ntegral s an area under a curve F-vs-. Proo: ( you want) Area = Work done by F Let s convnce ourselves: Work done by F over each : W 1 F1 1 ; W F ; etc Total work done by F s a sum: W W 1 W W 3... We must evaluate the ntegral ether geometrcally, by ndng the area under the curve, or by actually dong the ntegraton. W W 7 1 W lm 0 F By denton, ths s an ntegral: W 7 1 F d F
I want power!!! Power
Power Sometmes normaton about work done s not enough to descrbe some stuatons: 1) Derence between a ast and slow worker ) Derence between a sport and a regular car So we need to ntroduce a new quantty: The average Power s the work done dvded by the tme t takes to do the work. W work done by a orce P t tme taken to do ths work P dw dt Unts Watts = Joules/sec The nstantaneous Power s the rate at whch work s done s F ds dt Snce ds s nntesmally small, we can say that F s constant over ds F ds dt dw F ds dt s F ds F v
Eample Average Car Power A certan 1000 kg car can accelerate rom rest to a speed o 0 m/s n a tme o 10 s. What average power must the motor produce n order to cause ths acceleraton? v 0 0 v 0 m / s The work done by the motor n acceleratng the car can be ound rom the work-ke prncple: 0 W K K K mv 10 s s the tme taken or ths work By denton, the average Power s P W t work done tme taken to by do a orce ths work P W t 1 mv t 1 1000kg(0m / 10s s) 0000Watt 0kW
ConcepTest Tme or Work Mke perormed 5 J o work n 10 secs. Joe dd 3 J o work n 5 secs. Who produced the greater power? A) Mke produced more power B) Joe produced more power C) both produced the same amount o power Because power = work / tme, we see that Mke produced 0.5 W and Joe produced 0.6 W o power. Thus, even though Mke dd more work, he requred twce the tme to do the work, and thereore hs power output was lower.
Eample Average Runner Power How much power does t take a 50-kg runner to run up a 5 m hgh hll n 10 s? Assume acceleraton s zero. By denton, the average Power s P work done tme taken to by do a orce ths work
Lecture 14 Chapter 10 I have lots o potental Potental Energy Conservaton o Energy Course webste: http://aculty.uml.edu/andry_danylov/teachng/physcsi
IN THIS CHAPTER, we wll add a new very mportant player to our energy game team (KE, work): potental energy. Today we are gong to dscuss: Chapter 10: Potental Energy: Secton 10.1- (don t read t. Only you have a strong desre) Sprng Potental Energy: Secton 10.3 Conservaton o Mechancal Energy: Secton 10.4
Conservatve Forces (denton) 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 1 A F C W A W B W C B Work done by F s the same or any path Conservatve orces: gravty, sprng Non-conservatve orces: rcton
Gravtatonal Potental Energy y y 1 Consder a block sldng down on a rctonless surace under the nluence o gravty mg K 1 ds Reerence level y K F G mg mg( ˆ) j ds Work done by the gravtatonal orce: W G d( ˆ) dy( ˆ) j FG ds mg( ˆ) j [ d(ˆ) dy( ˆ)] j 1 W G y y 1 mgdy mg( y ) y1 The work done by gravty depends only on coordnates o the nal and ntal postons, so gravtatonal orce s conservatve 1 ˆ ( j ˆ) ˆj ˆ cos90 0 ˆ ( j ˆ) j ˆj ˆj cos0 1 You see there s eactly the same structure o both terms, mgy, so let s gve t a nce name and symbol Gravtatonal potental energy (a new orm o energy) U 0 W G U mgy ( U ) U1 Actually, n general t s W G U U 0 U Reerence pont mgy
Conservaton o Mechancal Energy!!! Combne W U Relaton between potental energy and work U U1) wth W K ( U K K 1 W K Work-KE Prncple K U K 1 U 1 So we got a new constructon K+U, so let s gve t a nce name and symbol also Total Mechancal Energy E K U E E 1 E constant Whch s Conservaton o Mechancal Energy Only changes o potental energy mportant, not absolute values Choose a sutable reerence U 0 =0 or each problem (lke a PE orgn)
Eample Roller coaster The roller-coaster car starts rom rest at the top o the hll. The heght o the hll s 40 m. Calculate a) the speed o the car at the bottom o the hll; b) at what heght t wll have hal ths speed.
ConcepTest Water Slde I Paul and Kathleen start rom rest at the A) Paul same tme on rctonless water sldes B) Kathleen wth derent shapes. At the bottom, C) both the same whose velocty s greater? Conservaton o Energy (or any o them): thereore: mgh 1 mv K E E U K U v gh because they both start rom the same heght (h), they have the same velocty at the bottom. Re. level U=0
Energy Energy s dened as the ablty to do work Knetc Energy: assocated wth energy o moton K 1 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
Thank you
Elastc/Sprng Potental Energy F sp k What s the potental energy o a sprng compressed rom equlbrum by a dstance? Use a relaton between potental energy and work: Work done by a sprng W (rom the prevous class) sp k k ( U U ) Let s combne them From here you can see that the PE o a sprng s U sprng 1 k Potental energy o a sprng Where s a dsplacement rom an equlbrum o a sprng End o Class
Eample Brck/sprng on a track 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?
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! http://phys3p.sl.psu.edu/phys_anm/mech/ramped.av Re. level U=0
Eample Droppng ball h y v 0 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: Now we are much more eperenced and We can apply two methods Re. level U=0 v? Knematc equatons From N. nd law we got ths knematc eq-n: 0 1 0 mv mgy v v gh v gh K Thus, both approaches are equvalent Energy conservaton U v 1 K mv 1 mv mgh gh U 0 h mgy