Problem Solving 5 Energy Principle 8.01 Week 07D3 Today’s Reading Assignment: Chapter 14 Potential Energy and Conservation of Energy, Sections 14.9Announcements Exam 2 Week 8 Thursday Oct 24 7:30-9:30 See Announcements Page for Room Assignments Sunday Tutoring in 26-152 from 1-5 pm Problem Set 6 due Week 8 Tuesday at 9 pm in box outside 26-152Table Problem: Ramping Up An object of mass m is released from an initial state of rest from a spring of constant k that has been compressed a distance x0. After leaving the spring (at the position x = 0 when the spring is unstretched) the object travels a distance d along a horizontal track that has a coefficient of friction that varies with position as . Following the horizontal track, the object enters a quarter turn of a frictionless loop whose radius is R. Finally, after exiting the quarter turn of the loop the object travels vertically upward to a maximum height, h, (as measured from the horizontal surface). The magnitude of the gravitational acceleration is g. Find the maximum height, h, that the object attains in terms of the given quantities. µ=µ0+µ1(x / d)Table Problem: Block Sliding off Hemisphere A small point like object of mass m rests on top of a sphere of radius R. The object is released from the top of the sphere with a negligible speed and it slowly starts to slide and g is the magnitude of the gravitational acceleration. (a) Determine the angle with respect to the vertical at which the object will lose contact with the surface of the sphere. (b) What is the speed of the object at the instant it loses contact with the surface of the sphere. θ1 v1Table Problem: Energy Diagram A particle of mass m moves in one dimension. Its potential energy is given by where U0 and a are positive constants. The mechanical energy E of the particle is constant such that . a) Draw an energy diagram showing the potential energy U(x), the kinetic energy K(x), and the mechanical energy E. b) Find the force on the particle, Fx(x), as a function of position x. c) Find the speed of the particle at the origin x = 0 such that the when the particle reaches the positions x = ± a, it will reverse its motion. U (x) = −U0e− x2/a2 −U0< E <
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