Experiment 2 Energy Transformations; 8.01 Week 07D2 Today’s Reading Assignment: MIT 8.01 Course Notes: Chapter 14 Potential Energy and Conservation of Energy Sections 14.5-14.7,14.9Announcements Exam 2 Thursday Oct 24 7:30-9:30 See Announcements Page for Room Assignments Problem Set 6 due Week 8 Tuesday at 9 pm in box outside 26-152Potential Energy and Force In one dimension, the potential difference is Force is the derivative of the potential energy Examples: (1) Spring Potential Energy: (2) Gravitational Potential Energy: U (x) − U (x0) = − FxAB∫dx Fx= −dUdx2spring1( )2U x kx=2,12x springdU dF kx kxdx dx⎛ ⎞= − = − = −⎜ ⎟⎝ ⎠1 2grav( ) =Gm mU rr−1 2 1 2,2r gravityGm m Gm mdU dFdr dr r r⎛ ⎞= − = − − = −⎜ ⎟⎝ ⎠Table Problem: Potential Energy Diagram A body of mass m is moving along the x-axis. Its potential energy is given by the function U(x) = b(x2-a2) 2 where b = 2 J/m4 and a = 1 m . a) On the graph of U vs. x, sketch the force F vs. x. b) What is an analytic expression for F(x)?Energy Diagram Choose zero point for potential energy: Potential energy function: Mechanical energy is represented by a horizontal line since it is a constant Kinetic energy is difference between mechanical energy and potential energy (independent of choice of zero point) U (x) =12kx2, U (x = 0) = 0 U (x = 0) = 0Emech= K (x) + U(x) =12mvx2+12kx2K (x) = Emech− U(x)Graph of Potential energy function U(x) vs. xConcept Question: Energy Diagram 1 A particle with total mechanical energy E has position x > 0 at t = 0 1) escapes to infinity in the negative x-direction 2) oscillates around a 3) oscillates around b 4) periodically revisits a and bConcept Question: Energy Diagram 2 A particle with total mechanical energy E has position x1< x < x2 at t = 0 1) escapes to infinity in the negative x-direction 2) oscillates around a 3) oscillates around b 4) periodically revisits a and bTable Problem: Energy Diagram The figure above shows a graph of potential energy U(x) verses position for a particle executing one dimensional motion along the x-axis. The total mechanical energy of the system is indicated by the dashed line. At t =0 the particle is somewhere between points A and G. For later times, answer the following questions. a) At which point will the magnitude of the force be a maximum? b) At which point will the kinetic energy be a maximum? c) At how many of the labeled points will the velocity be zero? d) At how many of the labeled points will the force be zero?Table Problem: Experiment 2 Cart-Spring on an Inclined Plane A cart of mass m slides down a plane that is inclined at an angle θ from the horizontal. The cart starts out at rest. The center of mass of the cart is a distance d from an unstretched spring with spring constant k that lies at the bottom of the plane. Assume that the inclined plane has a coefficient of kinetic friction µ. Find an equation whose solution describes how far the spring will compress when the cart first comes to rest.Experiment 2 Energy
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