A calculus problem by Hobart Pao

Given the vector function $\vec{r}(t) = \left \langle 2t, t^2, \dfrac{t^3}{3} \right \rangle$ , if the rate of change of radius of osculating circle to $\vec{r}(t)$ with respect to $t$ , at $t=1$ , is $R$ , find $\left \lfloor 1000R \right \rfloor$ .

Clarification:

The osculating circle to $\vec{r}(t)$ at time $t$ is the circle tangent to $\vec{r}(t)$ at time $t$ with radius $\dfrac{1}{\kappa (t)}$ , where $\kappa(t)$ is the curvature of $\vec{r}(t)$ at time $t$ .