# How can a mass slide up the inside of a rotating cone?

So I solved this problem in the picture for the maximum value of the period and got $$T = 2\pi\sqrt{\frac{h\tan{\beta}}{g}\frac{\sin{ \beta+\mu_s\cos{\beta}}}{\cos{\beta}-\mu_s\sin{\beta}}}$$

with friction acting up the slope. The minimum value of $$T$$ occurs when the friction acts down the slope. What I don’t understand is how this is possible since friction opposes motion and if you resolve the forces in the direction of the slope the net force does not act upwards, so how could the mass move up the slope? My tutor said the inertia of the mass carries it up the slope. Could someone please give me a more quantitative explanation of this?

• I got this result for the period T ? \begin{align*} & T^2=4\,\pi^2\frac{ \tan(\beta)\,h\,\left(\mu_s-\tan(\beta)\right)}{g\left(\mu_s\,\tan(\beta)+1\right)} \end{align*}
– Eli
Nov 2, 2021 at 17:36

If the rotation is very fast $$N$$ has to be large to provide the necessary centripetal force, so the upward force from $$N$$ can be bigger than the weight - and make the mass move upwards (this also means outwards if the mass is still in contact with the slope).
If the rotation is fast, $$F$$ in the diagram above is pointing down the slope and there seems to be no force that could move the mass to the right.