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Physics 121B - Mechanics Lecture 12 (T&M: 7.1-3) Mass-Energy & Momentum April 28, 2010 John G. Cramer Professor Emeritus of Physics B451 PAB jcramer@uw.edu Announcements Homework Assignment #5 should be submitted on the Tycho system by 11:59 PM on Friday, April 30. On Friday, May 7, we will have Exam 2 covering T&M Chapters 5-8. Sections will be Lecture multiple-choice (55 pts), Laboratory (25 pts), and Tutorial (20 pts). There will be assigned seating. Send E-mail if you have a new seating preference. As of 9:10 AM today, 153/175 Physics 121B students have registered their clickers. If you have not already done so, register your clicker at: http://courses.washington.edu/p121bs10/clicker.htm April 28, 2010 Physics 121B - Lecture 12 2/28 Lecture Schedule (Part 2) Physics 121B - Prof. John G. Cramer - 11:30-12:20 PM MWF - A118 PAB Textbook: Physics for Scientists and Engineers, 6th Edition, Paul A. Tipler and Gene Mosca, W. H. Freeman & Co. (2008) Week Date L# Lecture Topic Text Reading Pages Slides Hwk Tutorial Lab Forces Free Fall & Projectiles 16-Apr-10 E1 EXAM 1 - Chapters 1, 2, 3, and 4 HW#3 19-Apr-10 8 Friction, Circular Motion Ch. 5.1,2,3 5 29 4 21-Apr-10 9 Center of Mass; Work Ch. 5.5, 6.1-2 19 32 Newton's 2nd & 3rd 1-D Dynamics 23-Apr-10 10 Energy;Potential Energy Ch. 6.3-5, 7.1 10 19 HW#4 26-Apr-10 11 Conservation of Energy Ch. 7.2 to 7.3 12 32 5 28-Apr-10 12 Mass-Energy; Momentum Ch. 7.4 to 8.1 26 28 Work and KE Newton's Laws, Tension 30-Apr-10 13 Kinetic En; Impulse & Colliison Chs. 8.2 to 8.5 16 24 HW#5 3-May-10 14 Explosions & Rockets Ch. 8.4 to 8.5 19 21 We are here! 6 5-May-10 15 Review 2 22 Cons. of Energy Work & Energy 7-May-10 E2 EXAM 2 - Chapters 5, 6, 7, and 8 HW#6 Cons. of Momentum Momentum & Collisions 10-May-10 16 Rotational Kinematics & Energy Ch. 9.1,2,3 6 April 28, 2010 Physics 121B - Lecture 12 3/28 Potential Energy and Force dU ( x) F ( x) dx April 28, 2010 Physics 121B - Lecture 12 4/28 Clicker Question 1 A particle moves along the x-axis with the potential energy shown. What is the x component of the force on the particle when it is at x=4 m ? (a) 2 N (b) 1 N (c) 4 N (d) 2 N (e) 1 N April 28, 2010 Physics 121B - Lecture 12 5/28 Stable and Unstable Equilibrium Condition for Stable Equilibrium: In stable equilibrium, a small displacement in any direction results in a restoring force that accelerates the particle back to its equilibrium position. d2U/dx2 > 0. Condition for Unstable Equilibrium: In unstable equilibrium, a small displacement in any direction results in a force that accelerates the particle away from its equilibrium position. d2U/dx2 < 0. Condition for Neutral Equilibrium: In neutral equilibrium, a small displacement in any direction results in a zero force and the particle remains in equilibrium. d2U/dx2 = 0. April 28, 2010 Physics 121B - Lecture 12 6/28 Example: Force and the Potential-Energy Function In the region –a < x < a the force on a particle is represented by the potential energy function: U(x) = -b[1/(a+x) +1/(a-x)], where a and b are positive constants. (a) Find the force Fx in the region –a < x < a. (b) At what value of x is the force 0? (c) At that location, is the equilibrium stable or unstable? dU d 1 1 1 1 Fx b b 2 dx dx (a x) (a x) (a x ) (a x ) 2 At x 0, Fx 0. d 2U d2 1 1 1 1 2 b 2b 3 dx 2 dx (a x) (a x) (a x ) (a x ) 3 d 2U At x 0, 4b / a 0, so the equilibrium is unstable. dx 2 April 28, 2010 Physics 121B - Lecture 12 7/23 Suspension and Equilibrium By choosing several pivot points, one can locate the CM as the cross point of the several An object hangs so that the CM is verticals below the pivot point. from the pivot points. April 28, 2010 Physics 121B - Lecture 12 8/28 The Conservation of Energy The total energy of the universe is constant. Energy can be converted from one form to another or transmitted from one region to another, but energy can never be created or destroyed. Ein Eout Esys Esys Emech Etherm Echem Eother The Work-Energy Theorem Wext Esys Emech Etherm Echem Eother April 28, 2010 Physics 121B - Lecture 12 9/28 Clicker Question 2 4.0 m/s A spring-loaded gun shoots a plastic ball with a speed of 4.0 m/s. If the spring is compressed twice as far, what is the ball’s speed? a) 2.0 m/s b) 4.0 m/s c) 8.0 m/s d) 16.0 m/s e) 32.0 m/s April 28, 2010 Physics 121B - Lecture 12 10/28 Example: Falling Clay A ball of modeling clay with mass m is released from rest from a height h and falls to the perfectly rigid floor (thud). Discuss the application of the law of conservation of energy to (a) the system consisting of the clay ball alone, and Wext mgh; Emech 0; Wext Emech Etherm Etherm mgh (b) the system consisting of Earth, the floor, and the clay ball. Wext 0; Wext Emech Etherm 0 Emechi mgh; Emech f 0 Emech 0 mgh mgh Etherm Emech mgh April 28, 2010 Physics 121B - Lecture 12 11/28 Problems Involving Kinetic Friction Wext 0 Emech Etherm Emech Kblock Kboard 1 2 mv 2 1 mvi2 f 2 1 2 MV f2 0 fk max ; fk x max x; 2ax x v2 vi2 f f k x 1 m v2 vi2 1 mv2 1 mvi2 2 f 2 f 2 f k X MAx X 1 M V f2 Vi 2 1 MV f2 0 2 2 fk x X 1 mv2 1 mvi2 1 MVf2 2 f 2 2 f k srel Emech ; f k srel Etherm Wext Emech Etherm Emech f k srel April 28, 2010 Physics 121B - Lecture 12 12/28 Example: Pushing a Box A 4.0-kg box is initially at rest on a horizontal tabletop. You push the box a distance of 3.0 m along the tabletop with a horizontal force of 25 N. The coefficient of kinetic friction between the box and tabletop is 0.35. Find (a) the external work done on the block–table system, (b) the energy dissipated by friction, (c) the final kinetic energy of the box, and (d) the speed of the box. Wext Wby you on block Wby gravity on block Wby gravity on table Wby floor on table Fpush x 0 0 0 25 N 3.0 m 75 J Etherm f k x k Fn x k mg x 0.35 4.0 kg 9.81 m/s2 3.0 m 41 J K f Wext Etherm 75 J 41 J 34 J v f 2K f / m 2 34 J / 4.0 kg 4.1 m/s April 28, 2010 Physics 121B - Lecture 12 13/28 Example: A Playground Slide A child of mass 40 kg goes down an 8.0-m- long slide inclined at 30° with the horizontal. The coefficient of kinetic friction between the child and the slide is 0.35. If the child starts from rest at the top of the slide, how fast is he traveling when he reaches the bottom? Wext Emech Etherm (U K ) f k srel K K f 0 1 mv2 ; Wext 0; U mgh 2 Fn mg cos 0; fk k Fn k mg cos ; h s sin 0 mgh 1 mv2 fk s mgs sin 1 mv2 k mg cos s 2 f 2 f v f 2 gs sin k cos 2 9.81 (4.0) sin 30 0.35cos30 5.6 m/s April 28, 2010 Physics 121B - Lecture 12 14/28 Friction and Chemical Energy Fk srel Etherm Wext Emech Etherm Emech Fk srel Echem (mgh Etherm ) April 28, 2010 Physics 121B - Lecture 12 15/28 Mass & Energy E mc2 E M 2 c April 28, 2010 Physics 121B - Lecture 12 16/28 Example: Nuclear Binding Energy A hydrogen atom consisting of a proton and an electron has a binding energy of 13.6 eV. By what percentage is the mass of a proton plus the mass of an electron greater than that of the hydrogen atom? April 28, 2010 Physics 121B - Lecture 12 17/28 Example: Nuclear Fusion In a typical nuclear fusion reaction, a triton (t) and a deuteron (d) fuse together to form an alpha particle (a) plus a neutron (n). The reaction is written d + t → a + n. How much energy is released per deuteron produced for this fusion reaction? April 28, 2010 Physics 121B - Lecture 12 18/28 Nonrelativistic (Newtonian) Mechanics and Relativity As the speed of a particle approaches a significant fraction of the speed of light, Newton’s second law breaks down, and we must modify Newtonian mechanics according to Einstein’s theory of relativity. The criterion for the validity of Newtonian mechanics can also be stated in terms of the energy of a particle. In nonrelativistic (Newtonian) mechanics, the kinetic energy of a particle moving with speed v is: where E0 = mc2 is the rest energy of the particle. Solving for v/c gives: Nonrelativistic mechanics is valid if the speed v of the particle is much less than c, the speed of light, or alternatively, if the kinetic energy K of a particle is much less than its rest energy E0. April 28, 2010 Physics 121B - Lecture 12 19/28 Quantization of Energy In microscopic mechanical systems (e.g., molecules), because of quantum mechanics, the system energy cannot change continuously, but rather can change only in discrete steps to well defined “state” values. The “ground state” E0 is the lowest of these energy values. En (n )hf , n 0,1, 2,3, 1 E0 1 hf 2 2 h Planck's constant 6.636 10-34 J s =4.136 10-15 eV s Erad E f Ei Ephoton hf April 28, 2010 Physics 121B - Lecture 12 20/28 Momentum Momentum: p mv Conservation of Linear Momentum Momentum: p mv dp d (mv ) dv dp m ma Fnet dt dt dt dt P mi vi pi Mvcm sys i i dPsys dvcm F i ext Fnet ext dt M dt Macm If Fext 0, then P mi vi Mvcm constant Conservation sys of Momentum April 28, 2010 Physics 121B - Lecture 12 22/28 Example: A Space Repair During repair of the Hubble Space Telescope, an astronaut replaces a damaged solar panel during a spacewalk. Pushing the detached panel away into space, she is propelled in the opposite direction. The astronaut’s mass is 60 kg, and the panel’s mass is 80 kg. Both astronaut and panel are initially at rest relative to the telescope, until the astronaut gives the panel a shove, giving it a velocity of 0.30 m/s relative to the telescope. Assuming her tether is slack, what is her velocity relative to the telescope? dPsys F ext 0 dt , so Psys constant mP vPf mAv Af mP vPi mAv Ai 0 mP vPf mAv Af mP (80 kg) ˆ ˆ v Af vPf (0.30 m/s)i (0.40 m/s)i mA (60 kg) April 28, 2010 Physics 121B - Lecture 12 23/28 Kinetic Energy of a System K K i 1 mi vi2 1 mi (vi vi ) 2 2 vi vCM ui i i i K 1 mi (vCM ui ) (vCM ui ) 1 mi (v CM u i2 2vCM ui ) 2 2 2 i i K 1 mi v CM 1 mu i2 vCM mi ui 2 i 2 2 i i mu i i i 0 K 1 mi vCM 1 mi ui2 1 MvCM K rel 2 2 2 2 2 i i In an isolated system, only the relative kinetic energy can change due to internal forces. I. e., no “space drives”. April 28, 2010 Physics 121B - Lecture 12 24/28 Example: Hubble Repair During repair of the Hubble Space Telescope, an astronaut replaces a damaged solar panel during a spacewalk. Pushing the detached panel away into space, she is propelled in the opposite direction. The astronaut’s mass is 60 kg and the panel’s mass is 80 kg. Both the astronaut and the panel initially are at rest relative to the telescope. The astronaut then gives the panel a shove. After the shove it is moving at 0.30 m/s relative to the Hubble Telescope. What is her subsequent velocity relative to the telescope? (During this operation the astronaut is tethered to the ship; for our calculations assume that the tether remains slack.) April 28, 2010 Physics 121B - Lecture 12 25/28 Example: A Runaway Railroad Car A runaway 14,000-kg railroad car is rolling horizontally at 4.00 m/s toward a switchyard. As it passes by a grain elevator, 2000 kg of grain suddenly drops into the car. How long does it take the car to cover the 500-m distance from the elevator to the switchyard? Assume that the grain falls straight down and that slowing due to rolling friction or air drag is negligible. April 28, 2010 Physics 121B - Lecture 12 26/28 Example: Radioactive Decay A thorium-227 nucleus (mass 227 u) at rest decays into a radium-223 nucleus (mass 223 u) by emitting an alpha particle (mass 4.00 u) . The kinetic energy of the a particle is measured to be 6.00 MeV. What is the kinetic energy of the recoiling radium nucleus? April 28, 2010 Physics 121B - Lecture 12 27/28 End of Lecture 12 For the next lecture, read T&M Chapter 8.2-3. Homework Assignment #5 should be submitted on the Tycho system by 11:59 PM on Friday, April 30. As of 9:10 AM today, 152/176 Physics 121B students have registered their clickers. If you have not already done so, register your clicker at: http://courses.washington.edu/p121bs10/clicker.htm Isolation Emech Eth Esys Wext April 28, 2010 Physics 121B - Lecture 12 29/28 Problem Solving Strategy Picture: Determe that the net external force Fext (or Fext x) on the system is negligible for some time interval. (If the net force is NOT determined to be negligible, do not proceed.) Solve: 1. Draw a sketch showing the system before and after the time interval. Include coordinate axes and label the initial and final velocity vectors. 2. Equate the initial momentum to the final momentum and express this as a vector equation (or one or more scalar equations involving x, y, and z components.) 3. Substitute the given information into the equation(s) and solve for the quantity or quantities of interest. Check: Make sure you include any minus signs that accompany velocity components, because momentum can have either sign. April 28, 2010 Physics 121B - Lecture 12 30/28 Example: Moving a Sled A sled is coasting on a horizontal snow-covered surface with an initial speed of 4.0 m/s. If the coefficient of friction between the sled and the snow is 0.14, how far will the sled travel before coming to rest? Wext Emech Etherm (U K ) f k srel ; f k k Fn k mg 0 0 K k mgsrel srel 1 2 mv 2 v2 4.0 m/s 5.8 m k mg 2k g 2 0.14 9.81 m/s 2 April 28, 2010 Physics 121B - Lecture 12 31/28 Example: A Skateboard Workout A 40.0-kg skateboarder on a 3.00-kg board is training with two 5.00-kg weights. Beginning from rest, she throws the weights horizontally, one at a time, from her board The speed of each weight is 7.00 m/s relative to her after it is thrown. Assume the board rolls without friction. (a) How fast is she moving in the opposite direction after throwing the first weight? (b) After throwing the second weight? April 28, 2010 Physics 121B - Lecture 12 32/28