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 Taylor series. 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. Thus:
$$\vec{x_c} = \frac{1}{6}\vec{j_{c_0}}t^3 + \frac{1}{24}\vec{s_c}t^4$$
$$\frac{d}{dt}\vec{v_t} = \frac{d}{dt}(\vec{ω}r) = \vec{a_t} = \vec{α}r$$
$$\frac{d}{dt}\vec{a_t} = \frac{d}{dt}(\vec{α}r) = \vec{j_t} = \vec{ζ}r$$
$$\vec{ω} = \vec{ω}_0 + \vec{α}t + \frac{1}{2}\vec{ζ}t^2$$
$$\frac{d}{dt}\vec{a_c} = \frac{d}{dt}(\frac{{\vec{v_t}}^2}{r}) = \frac{d}{dt}(\vec{ω}^2r) = \vec{j_c} = 2\vec{ω}\vec{α}r$$
$$\vec{s_c} = 2r(\vec{ω}\vec{ζ}+\vec{α}^2)$$
$$\vec{F_c} = M\vec{a_c} = M(\vec{j_{c_0}}t+\frac{1}{2}\vec{s_c}t^2) = M(\vec{j_{c_0}}t+r(\vec{ω}\vec{ζ}+\vec{α}^2)t^2)$$
So far, I believe that my interpretation of the fourth-degree polynomial of position. However, relating the three tension vectors in a way that makes any sort of sense becomes my issue.
How I have approached this tension vector so far is by considering the unit vectors relative to the end of the rod, such that $\hat{i}$ is always along the rod, $\hat{j}$ is pointing towards the tangential movement along the end of the rod, and $\hat{k}$ points towards the ground.
I cannot figure out how to relate each vector direction. I know that, as my pointmass is pulled upward and inward by the tension, the radius from the axis of rotation increases for the pointmass, thus increasing the angular acceleration and therefore the tension force along the y-axis is increasing. I know that the tension force along the x-axis also increases, as the x-component distance is becoming larger and therefore increasing the centripetal force. I do not know how to relate the z-axis with other two axes. Here in lies my main issue: relating my tension force components to the variables listed above.
My question is two-fold: are my assumptions about the fourth-degree polynomial of centripetal position and the third-degree polynomial of rotational position equations correct, and how may I relate my tension forces in a way that is theoretically correct?
(full disclosure: I am in AP Physics C with prior AP Calc BC knowledge as well as the beginnings of 3D vector calculus, so if a higher-level calculus is used (I know that tensor calculus might be involved?), please link me to a source that can give me fundamentals of the mathematics necessary)