1. ## Cylindrical polar coordinates

In terms of cylindrical polar coordinates the whirlpool-like surface
z*p^1/2 = −1, z < 0 is formed by rotating a branch of a function z = −1/x^1/2 about a vertical asymptote (the z-axis). A particle of mass m slides on this smooth surface and experiences the action of the gravitational force.

Part 1 of the question was to show that the Kinetic energy, T + the potential energy, V was equal to E (i.e. that energy is conserved). This was done ok.

Can anyone try to explain what the following part of the question means?

For what value of z0 the motion at constant height z0 is possible, i.e. z(t) = z0 for any t?

2. Originally Posted by Unoticed
In terms of cylindrical polar coordinates the whirlpool-like surface
z*p^1/2 = −1, z < 0 is formed by rotating a branch of a function z = −1/x^1/2 about a vertical asymptote (the z-axis). A particle of mass m slides on this smooth surface and experiences the action of the gravitational force.

Part 1 of the question was to show that the Kinetic energy, T + the potential energy, V was equal to E (i.e. that energy is conserved). This was done ok.

Can anyone try to explain what the following part of the question means?

For what value of z0 the motion at constant height z0 is possible, i.e. z(t) = z0 for any t?
I will try to do so. The particle will slide down the whirlpool unless the vertical component of the normal force it experiences can counterbalance the weight of the particle. (The particle will then move in a circle, not down into the whirlpool.)

This can't be done for any y value of the whirlpool, it has to be where the normal component matches the weight. Hint: take your coordinate directions such that +y is straight up and +x is toward the center of the whirlpool and take components of the normal force, just like you would if this were a problem about a car going in a circle on a graded surface.

-Dan

3. Ok, thanks for that. I understand what you mean but I don't really get how to start solving it. What is z0 supposed to be? and z(t)?