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Common Exam 3, tomorrow 8:30 – 9:45 A.M. at KUPF 205 (Arrive by 8:15) Chaps. 6, 7, 8 Bring scientific calculators Today…. Chapter 9 Review for Chaps. 6, 7, 8 Quiz #11 Last class, we learned… Chapter 8. Potential Energy and Conservation of Energy Work and energy for System of objects External forces: Forces from outside the system Internal forces: Forces between objects within the system Apply Newton’s 3rd law Action & reaction forces have equal magnitude & opposite directions Analysis of the system as a whole Consider external forces only Analysis of each object within the system Consider both external and internal forces m2 Force m1 Horizontal frictionless surface 1 Chapter 9. Center of Mass and Linear Momentum Section 9-1, Motovation Section 9-2, Center of Mass Section 9-3, Newton’s second law for a system of particles Data : Fatality of a driver in head-on collision (With a passenger) < (Without a passenger) 2 Section 9-2, Center of Mass How should we define the position of the moving body ? What is h for U = mgh Take the average Position of mass ! Call “Center of Mass” (com) Center of Mass for fun : Equilibrium COM Unstable Equil. Equil. COM Stable Equil. Equil. 3 Center of Mass for a system of particles 2 bodies, 1 dimension COM should be on this line Center of Mass for a system of particles 2 bodies, 1 dimension 0 COM 4 Center of Mass and Uniform Gravity The CM of an object or a system is the point, where the object or the system can be balanced in the uniform gravitational field How can we find Center of Mass for many particles in 3D? 5 Center of Mass for a System of Particles 2 bodies, 1 dimension General case: n bodies, 3 dimensions n bodies, 3 dimensions Example: Center of Mass for 2 bodies in 1 dimension 0 3 kg 1 kg 1m 5m Where is the center of mass? 6 Sample Problem 9-1 Three particles of masses m1 = 1.2 kg, m2 = 2.5 kg, and m3 = 3.4 kg form an equilateral triangle of edge length a = 140 cm. Where is the center of mass of this system? (Hint: m1 is at (0,0), m2 is at (140 cm,0), and m3 is at (70 cm, 120 cm), as shown in the figure below.) Section 99-3, Newton’ Newton’s second law for a system of particles System of particles Fnet : Net force of all A firework rocket explodes external forces that act on the system (Internal forces are not included) M : Total mass of the system acom : Acceleration of the center of mass 7 Proof of M r CM = m 1r + m 2 r + m 3 r + ...+ m n r 1 2 3 n M v CM = m 1v + m 2v + m 3v + ...+ m n v 1 2 3 n +. M a CM = m 1a + m 2 a + m 3 a + ...+ m n a 1 M a CM = 2 F +F +F 1 2 3 3 + ...+ F n n Review for Chaps. 6, 7, 8 Ch. 6, Force and Motion-II Friction and Uniform Circular Motion Ch.7, Kinetic Energy and Work Ch.8, Potential Energy and Conservation of Energy 8 Units SI Unit (MKS Unit) Length: m Mass : kg Time : s Force : N Work, Energy : J Power : W Other relations 2π (radian) = 360o 1 T = 1000 kg HW9 External force: Gravity, Normal force from pulley Internal force: Tensions between rope and objects Extra problem: Find the work done on m2 by Tension 9 HW#9 See Lecture Note 9 Work done by a constant force W = Fd cos θ The horizontal force and the friction force are doing work on the object. Wtotal = W1 + W2 = K f − K i HW#8 First, find the normal force by balancing the forces along the vertical direction. Then, find the friction force. You are ready to find the work done by friction force. 10 HW#7 See Sample Prob 6-8 Period Angular velocity Velocity v = ω= 2π T 2π r =ω r T 11