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Curvilinear One-Dimensional Systems

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?


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.