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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

    W=∫Fxds

    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

    Mark44

    Staff: Mentor

    ds is the incremental change along the path.

    What diagram?
     
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