Understanding a rotational system with centripetal snap, angular jerk, and a third-dimensional tension vector
In a recent experiment in my physics class, we were given the task of finding the experimental rotational moment of inertia in a T-stand with two masses attached to the ends of a certain length. I have drawn a diagram below with the hopes of aiding in the understanding of the problem:
As you can see, I have two point masses hanging off the ends of the T-stand of identical mass M with identical string length S. Mass is distributed uniformly within the T-stand. I am applying a force F tangentially to the edge of the bottom of the T-stand, for which the radius is R.
We were directed as a class to consider that centripetal acceleration as a constant and the tension force as though it only moved in the upward and inward directions. It was then that I made the mistake of pointing out that making these assumptions would not result in accurate data, as we were varying the torque applied to the T-stand. My physics teacher knew that I already had experience with three-dimensional vectors (though very limited) and understood that position (as we had seen it so far) was only the second-degree polynomial of an infinite <PERSON> series. 766
It was then I was instructed to "go and see if you can figure it out".
The issue of centripetal jerk is apparent when considering the movement of the masses as the F is constant, implying a tangential acceleration. The equation for centripetal acceleration assumes a constant rotational velocity, which this does not have. Thus, we cannot use $a_c = ωr$ (where ω is a constant), but its derivative, $j_c = αr$. Furthermore, the rotational moment of inertia is also increasing with no change in torque applied; thus, angular acceleration is also approaches 0 as time approaches infinity. We cannot even consider centripetal jerk; instead, we must consider centripetal snap: $s_c = ζr$.
This issue would not be too difficult were it not for the three-dimensional tension vector. Since there is a tangential acceleration, there must be a tangential force at the point masses, thus making not only an upward and inward tension force, but also an outward tension force.
Here is what I know so far about the tension vector and the string associated:
$$S^2 = {x_S}^2 + {y_S}^2 + {z_S}^2$$
Where xS, yS, and zS are proportionally related to the force vectors:
$$||\vec{F_T}||^2 = ||\vec{F_{T_x}}||^2 + ||\vec{F_{T_y}}||^2 + ||\vec{F_{T_z}}||^2$$
Other misc. knowns (I hope):
$$\vec{τ} = \vec{F}R = I\vec{α}$$
$$Δ\vec{τ} = 0$$
$$\vec{x_c} = \vec{x_{c_0}} + \vec{v_{c_0}}t + \frac{1}{2}\vec{a_{c_0}}t^2 + \frac{1}{6}\vec{j_{c_0}}t^3 + \frac{1}{24}\vec{s_c}t^4$$
Note that $\vec{x_{c_0}}$, $\vec{v_{c_0}}$, and $\vec{a_{c_0}}$ all have magnitude of zero. 766