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Warmup problem for line integrals of conservative force

  1. Oct 6, 2011 #1
    1. The problem statement, all variables and given/known data

    A sleeve of mass m is constrained to move without
    friction along the x-axis. The sleeve is connected to the point (0, 2) on the y-axis by a spring as shown in
    the diagram below. Assume that Hooke’s “Law” is a good approximation for the restoring force exerted by
    the spring, i.e. that F = −k4l, where k > 0 and 4l denotes the extension/compression of the spring from
    its equilibrium (unextended) length, directed along the axis l of the spring. In this problem, assume that the
    equilibrium (unextended) length of the spring is 1 unit.
    Using an appropriate integration, compute the work W(x) necessary to move the mass from x = 0 to a
    point x 6= 0. (Because of symmetry, you need only to consider the case x > 0.) Hint: Diagrams of forces,
    projections, etc., could be very helpful here. What is the significance of the quantity W(x) in terms of energy?
    (A simple answer will do.)
    Note: You have computed the line integral of a nonconstant force that is not directed along the direction of
    motion of an object. Later in this course, we shall extend the process to motion along curves.

    2. Relevant equations


    3. The attempt at a solution

    I know how to solve most of this question, just that I do not know what ds is. Can somebody help me?
  2. jcsd
  3. Oct 6, 2011 #2


    Staff: Mentor

    ds is the incremental change along the path.

    What diagram?
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