# Spring Force and Power

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1 Lecture 13 Chapter 9 Sprng Force and Power Yeah, energy s better than orces. What s net? Course webste:

2 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

3 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.

4 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

5 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

6 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

7

8 I want power!!! Power

9 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

10 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 kg(0m / 10s s) 0000Watt 0kW

11 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.

12 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

13 Lecture 14 Chapter 10 I have lots o potental Potental Energy Conservaton o Energy Course webste:

14 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 (don t read t. Only you have a strong desre) Sprng Potental Energy: Secton 10.3 Conservaton o Mechancal Energy: Secton 10.4

15 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

16 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

17 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)

18 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.

19 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

20 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

21 Thank you

22 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

23 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?

24 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

25 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: 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

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