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    Curvilinear One-Dimensional Systems

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    These problems are on curvilinear one dimensional systems and are giving me a lot of difficulty, if you could provide help along with visuals to help explain that would be very helpful.

    1) Which of the following forces is conservative? (a) F = k (x, 2y, 3z) where k is constant. (b) F = k (y, x,
    0). (c) F = k (-y, x, 0). For those which are conservative, find the corresponding potential energy U, and
    verify by direct differentiation that F = - ∫U.

    2) The figure shows a child's toy, which has the shape of a cylinder mounted on top of a hemisphere. The
    radius of the hemisphere is R and the CM of the whole toy is at height h above the floor. (a) Write down
    the gravitational potential energy when the toy is tipped to an angle  from the vertical.(b) For what
    values of R and h is the equilibrium  = 0 stable?

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    Solution Preview

    Please see the attached file.

    If F is conservative, then we can write where U is a scalar function.

    Therefore the curl of a conservative force must obey:

    For a general vector field we have:


    Therefore is conservative.

    Therefore is conservative.

    is not a conservative vector ...

    Solution Summary

    This solution provides step by step equations for questions regarding curvilinear one dimensional systems.