Chapter 10. Energy - Physics & Astronomy

1 Chapter 10. EnergyThis pole vaulter can lift herself nearly 6 m (20 ft) off the ground by transforming the kinetic Energy of her run into gravitational potential Goal: To introduce the ideas of kinetic and potential Energy and to learn a new problem-solving strategy based on conservation of : A Natural Money Called Energy Kinetic Energy and Gravitational Potential Energy A Closer Look at Gravitational Potential Chapter 10. Energy A Closer Look at Gravitational Potential Energy Restoring Forces and Hooke s Law Elastic Potential Energy Elastic Collisions Energy Diagrams Money- Energy AnalogyFrom the Parable of the Lost PennyMoney Energy AnalogyFrom the law of conservation of energyKinetic and Potential EnergyThere are two basic forms of Energy . Kinetic Energy is an Energy of motionGravitational potential Energy is an Energy of positionpositionThe sum K+ Ugis not changed when an object is in freefall. Its initial and final values are equalKinetic and Potential EnergyThis is the conservation law for free fall motion: the quantityFree-Fall motion 222 ()fyiyifvvg yy = = = = 221122fyfiyivgyvgy+=++=++=++=+7motion: the quantityhas the same value before and after the motion.

2 212yvgy++++Free-Fall MotionThenivrrrr222 ()fyiyifvvg yy = = = = fxixvv====222211112222fxfyfixiyivvgyvvgy ++=++++=++++=++++=++8fvrrrr2222221122ffi ivgyvgy+=++=++=++=+ conservation law: the quantityhas the same value before and after the motion. 212vgy++++Frictionless surface: acceleration yiyfy sinag ====arrrrs222fivvas = = = =Motion with constant acceleration: sinifyys ====yy 9222 sin22siniffiiyyvvggygy == == == == 221122ffiivgyvgy+=++=++=++=+ conservation law: the quantityhas the same value before and after the motion. 212vgy++++yiyfyConservation law: the quantityhas the same value at the initial and the final points 212vgy++++10In all cases the velocity at the final point is the same (FRICTIONLESS MOTION)212vgy++++212mvmgy++++Conservatio n law:or212 Kmv====Kinetic Energy Energy of motion gUmgy====Gravitational Potential Energy Energy of position 11 The units of Energy is Joule: 22mJkgs====mechgEK U=+=+=+=+Mechanical Energy conservation law of mechanical Energy (without friction).

3 ConstantmechEK U=+==+==+==+=1213 Zero of potential energyyy0,1iy,2iyy, ,1,1, ,1ig ifg fKUKU+=++=++=++=+, ,2,2, ,2ig ifg fKUKU+=++=++=++=+,1, ,1, ,1()fig ig fKKUU=+ =+ =+ =+ 140,2fy,1fy,2, ,2, ,2()fig ig fKKUU=+ =+ =+ =+ ,1, ,2, ,2,2,2,1,1, ,1, ,1,2()()() ()gg ig fififg ig fgUUUmg yymg yyUUU = = = = = = = = = = = == = = = = = = = = = = = Only the change of potential Energy has the physical meaning ,,ig ifg fKUKU+=++=++=++=+2012iKmv====,0g iUmgy====,10g fUmgy========15,10g fUmgy========2112fKmv====221001122mvmvmg y=+=+=+=+210024 /vvgym s=+=+ =+=+ =+=+ =+=+ Restoring Force: Hooke s Law16spFk s= = = = spring constant Hooke s Law:Restoring Force: Hooke s LawspFk s= = = = The sign of a restoring force is always opposite to the sign of displacement 17 Restoring Force: Elastic Potential EnergyspFk s= = = = 21()2sUk s= = = = 18 The elastic potential Energy is always positive Restoring Force: Elastic Potential Energy21()2sUk s= = = = 2211()constant22sK Umvk s+=+ =+=+ =+=+ =+=+ =19 Restoring Force: Elastic Potential Energy21()2sUk s= = = = 202211()constant22sK Umvk s+=+ =+=+ =+=+ =+=+ =Law of conservation of Mechanical Energy2211()constant22gsK UUmvmgyk s++=++ =++=++ =++=++ =++=++ =21(((())))2,22,21,11,1fxixfxixm vm vm vm v = = = = 1,12,21,12,2ixixfxfxm vm vm vm v+=++=++=++=+,1,2,1,2ixixfxfxpppp+=++=++ =++=+22,1,2,1,2ixixfxfxpppp+=++=++=++=+, ,ix totalfx totalpp====The law of conservation of momentumElastic CollisionsPerfectly Elastic Collisions AA,A ivrrrrvrrrr,A fvrrrrDuring the collision the kinetic Energy will be transformed into elastic Energy and then elastic Energy will transformed back into kinetic Energy Perfectly Elastic Collision.

4 Mechanical Energy is Conserved (no internal friction)22,,112211A A iB B im vm v+=+=+=+=24BB,B ivrrrr,B fvrrrr22,,1122A A fB B fm vm v=+=+=+=+Perfectly inelastic collision:A collision in which the two objects stick together and move with a common final velocity NO conservation OF MECHANICAL ENERGYP erfectly Elastic Collisions ABA,A ivrrrr,B ivrrrr,A fvrrrrvrrrrConservation of Momentum and conservation of Energy isolated system25B,B fv2222,,,,11112222A A iB B iA A fB B fm vm vm vm v+=++=++=++=+,,,,A A iB B iA A fB B fm vm vm vm v+=++=++=++=+rrrrrrrrrrrrrrrrNow we have enough equations to find the final velocities. Example: ,A iv,0B iv====,A fv,B fv222,,,111222A A iA A fB B fm vm vm v=+=+=+=+,,,A A iA A fB B fm vm vm v=+=+=+=+,,ABA fA iABmmvvmm ====++++26,,2AB fA iABmvvmm====++++is always positive, can be positive (if ) or negative (if ),B fv,A fvABmm>>>>ABmm<<<<,0A fv====,,B fA ivv====ifABmm==== Chapter 10.

5 Summary SlidesChapter 10. Summary SlidesChapter 10. Summary SlidesChapter 10. Summary SlidesGeneral PrinciplesGeneral PrinciplesImportant ConceptsImportant ConceptsImportant ConceptsApplicationsApplicationsChapter 10. Clicker QuestionsChapter 10. Clicker QuestionsChapter 10. Clicker QuestionsChapter 10. Clicker QuestionsRank in order, from largest to smallest, the gravitational potential energies of balls 1 to (Ug)1 > (Ug)2 = (Ug)4 > (Ug)3B. (Ug)4 > (Ug)3 > (Ug)2 > (Ug)1C. (Ug)1 > (Ug)2 > (Ug)3 > (Ug)4D. (Ug)4 = (Ug)2 > (Ug)3 > (Ug)1E. (Ug)3 > (Ug)2 = (Ug)4 > (Ug)1 Rank in order, from largest to smallest, the gravitational potential energies of balls 1 to (Ug)1 > (Ug)2 = (Ug)4 > (Ug)3B. (Ug)4 > (Ug)3 > (Ug)2 > (Ug)1C. (Ug)1 > (Ug)2 > (Ug)3 > (Ug)4D. (Ug)4 = (Ug)2 > (Ug)3 > (Ug)1E. (Ug)3 > (Ug)2 = (Ug)4 > (Ug)1A small child slides down the four frictionless slides A D. Each has the same height. Rank in order, from largest to smallest, her speeds vA tovDat the > vA= vB> > vB> vA> > vA> vB> vB= vC= > vA= vB> vCA small child slides down the four frictionless slides A D.

6 Each has the same height. Rank in order, from largest to smallest, her speeds vA tovDat the > vA= vB> > vB> vA> > vA> vB> vB= vC= > vA= vB> vCA box slides along the frictionless surface shown in the figure. It is released from rest at the position shown. Is the highest point the box reaches on the other side at level a, at level b, or level c?reaches on the other side at level a, at level b, or level c?A. At level aB. At level bC. At level cA box slides along the frictionless surface shown in the figure. It is released from rest at the position shown. Is the highest point the box reaches on the other side at level a, at level b, or level c?reaches on the other side at level a, at level b, or level c?A. At level aB. At level bC. At level cThe graph shows force versus displacement for three springs. Rank in order, from largest to smallest, the spring constants k1, k2, and > k3 > > k2 > = k3 > > k1 = > k2 > k3 The graph shows force versus displacement for three springs.

7 Rank in order, from largest to smallest, the spring constants k1, k2, and > k3 > > k2 > = k3 > > k1 = > k2 > k3A spring-loaded gun shoots a plastic ball with a speed of 4 m/s. If the spring is compressed twice as far, the ball s speed will beA. 1 2 4 8 spring-loaded gun shoots a plastic ball with a speed of 4 m/s. If the spring is compressed twice as far, the ball s speed will beA. 1 2 4 8 particle with the potential Energy shown in the graph is moving to the right. It has 1 J of kinetic Energy at x= 1 m. Where is the particle s turning point? = 1 2 2 = 5 6 mA particle with the potential Energy shown in the graph is moving to the right. It has 1 J of kinetic Energy at x= 1 m. Where is the particle s turning point? = 1 2 2 = 5 6 m