Hooke’s Law and Centripetal Force – Impact on Normal Force

Question

The current total length of the string is $L+x$, which is partially inside a tube of length $b < L$. A mass of $m$ kg is rotating at the end, with angle $\beta$ and angular speed $\omega$ (gravitational acceleration $g$).

What is the magnitude of the normal force $N$ inside the tube, where the string comes out of it?

The hard think for me is how to apply Newton's second law, to the point in which the string is comming out of the tube to get an expression for $N$:
$$N = F_{\text{centripetal}}-F_{\text{elastic}}\hat{r} \\ N = m\omega^2(L-b+x)\sin(\beta)-kx\sin(\beta)$$

But as I understand, both normal force and centripetal force are reaction forces so I do not know if this is properly stated. Moreover I am only considering the elastic force "as is" due to the lower part of the string, but it is probably different

1. Does Hooke's Law affect the normal force in this case?
2. If yes, should Hooke's Law be applied considering just the lower part of the string, say using a different k like $k_1=k(L-b)/L$? I have never seen this, but maybe it makes sense?

Answer 1

I think this problem is not to apply Hooke's Law, but simply the tension of the string (not considered as an elastic spring.)

First, at the position of the mass $m$, in its rest frame, there are three forces:

1. $T_1$ the tension of the string, point upward along the string;
2. gravitational force $m g$, pointing vertical downward. along the $-\hat y$;
3. the centrifugal force (inertial force in rotational frame) $m\omega^2 r$, pointing outside, along the horizontal direction.

The tension $T_1$ is found from the condition that these three forces add up to zero.

Then at the bottom of the tube, there are another three forces:

1. The tension $T_1$ pointing downward along the string;
2. another tension $T_2$ for the string inside the tube, pointing vertical upward.
3. the normal force $N$ along the horizontal direction.

Given $T_1$ from above, we can obtain $T_2$ and $N$ from the force balance in both vertical and horizontal directions.