Growth of pepperweed in a rotating reference frame I currently have an exercise in my first physics semester that I don't think I can solve yet with which we have seen.
Question is as follow: Pepperweed seed was planted near the rim of a horizontal platform (radius R) rotating at a constant angular velocity ω around a vertical axis Oz passing through the center O of the disk. Determine the shape z(r) of the pepperweed stem. Help: A pepperweed stem always grows in such a way that the net normal force on each of its mass elements is zero. 
Now, I'm confused about some things. First: If we are in the reference frame each mass element would be under influence of the centripetal and the centrifugal force, but don't these cancel each other out so it would just grow normally? Also, how does a normal force of a mass element work since it's just a infinitesimally small point? I'm confused how to make a graph out of it and how to calculate it. Is mass point in this sense here meant as a mass of certain minimal length and you combine infinitely many of these? In that case again, I don't understand how to form a graph out of it.
 A: Your 1st question is answered by Do centripetal and reactive centrifugal forces cancel each other out?
Your 2nd question asks how to find the equation $z(r)$ of the stem of the growing plant :
If the platform is stationary the weed grows vertically upward against gravity. If the platform rotates there is a centrifugal acceleration acting horizontally outwards, so the weed will grow inwards against this also. At each radius the plant stem points opposite to the vector sum of gravitational and centrifugal accelerations. 
Gravity is constant but the centrifugal acceleration increases with distance from the axis. The plant stem grows more vertical as it grows inwards. Your graph plots the height $z$ of the plant against its distance $r$ from the axis.
If the weed grows at a finite rate, then its tip is moving with a finite velocity. There is also a Coriolis force acting perpendicular both to this velocity $v$ and also to the angular velocity $\omega$ of the platform. As a result the tip spirals in a helix as it grows upwards and inwards. However, you are probably expected to ignore this effect because the speed of growth $v$ is (presumably) very much smaller than $r\omega$, so the radius of the spiral will be negligible.   
A: The centripetal force is the force needed to make an object go around a circular orbit. The centrifugal force is the fictitious force observed in a rotating frame of reference - it appears you need to apply a force on an object just to stop it from moving.
While these two forces have the same magnitude, for the weed in the rotating frame of reference  (which doesn't know it is going around in circles) only the centrifugal component matters. And so as it tries to grow "vertically", it will actually grow at an angle to the vertical plane that depends on both the rate of rotation, and the radius.
You should be able to write an expression for the angle as a function of radius; this leads to a differential equation; and this equation is what you need to solve.
