# 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 = - âˆ«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?

https://brainmass.com/physics/equilibrium/curvilinear-one-dimensional-systems-104848

#### 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:

Hence:

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.