# Question on Moment of inertia about center of mass of a smooth plane curve.

This question is in continuation to this and motivated to answer this question. If I have an answer to it, then I can prove a special case of this question.

Consider two smooth plane curves $C \equiv (X_C(s),Y_C(s))$ and $S \equiv (X_S(s),Y_S(s))$ represented in arc length parametrization. Curve $C$ is asymptotic to a straight line $(a,s)$ in arc length parametrization, where $a$ is a constant. As $s\to\infty$ the curve $S$ approaches (converges) to a point in the plane, as $s\to\infty$.

Now we define moment of intertia about its center of mass, of a segment of curve $C$ between $s=s_1$ and $s = s_2$ as $$I_C(s_1,s_2) = \int_{s1}^{s_2} ((X_C(s)-X_{C_{cm}})^2 + (Y_C(s)-Y_{C_{cm}})^2) ds$$ where $(X_{C_{cm}},Y_{C_{cm}})$ is the center of mass of the segment under consideration given by $X_{C_{cm}} = \frac{1}{s_2-s_1}\int_{s_1}^{s_2}X_C(s)ds$ and $Y_{C_{cm}} = \frac{1}{s_2-s_1}\int_{s_1}^{s_2}Y_C(s)ds$. Similarly $$I_S(s_1,s_2) = \int_{s1}^{s_2} ((X_S(s)-X_{S_{cm}})^2 + (Y_S(s)-Y_{S_{cm}})^2) ds$$ where $X_{S_{cm}} = \frac{1}{s_2-s_1}\int_{s_1}^{s_2}X_S(s)ds$ and $Y_{S_{cm}} = \frac{1}{s_2-s_1}\int_{s_1}^{s_2}Y_S(s)ds$.

I'd like to prove/disprove the following statement.

Statement : Given any $s = s_1$ we can always find a sufficiently large $L$ such that for all $s_2 > L$, we have $I_C(s_1,s_2) > I_S(s_1,s_2)$.

• mathoverflow.net/a/175862/14414 Jul 12 '14 at 4:30
• In case it has already occurred to you to do so: It might be helpful to first simplify the problem through the parallel axis theorem, which states that the moment of inertia $I^{(\text{p})}$ about a point $\text{p}$ is equal to the moment of inertia $I^{(\text{cm})}$ about the center of mass $\text{cm}$ (specially, a parallel axis through the center of mass) plus the squared perpendicular distance between the points $\text{p}$ and $\text{cm}$: $I^{(\text{p})}=I^{(\text{cm})}+\|\text{p}-\text{cm}\|^2$. This means it suffices to solve the special case in which the C.o.M. is at the origin. Jul 12 '14 at 5:24
• ^In case it hasn't occurred to you, I meant to say. Oops Jul 12 '14 at 5:37
• @DavidH : I know. The CoM can safely assumed to be origin without any loss to the problem. Infact thats what RobJohn did in his answer : math.stackexchange.com/a/618993/2987 Jul 12 '14 at 5:41
• @DavidH : In case you were wondering, Altogether elimination of CoM is not possible$(X_{cm},Y_{cm})$ is the center of mass of the segment of curve we are considering, and not a fixed constant for all segments of the curve. It changes when we change when we move from one segment of the curve to another segment of the curve. Ofcourse it also changes when we move to another curve. Jul 12 '14 at 5:50