A point object of mass $m$ is connected to a cylinder of radius $R$ via a massless rope. At time $t = 0$ the object is moving with an initial velocity $v_0$ perpendicular to the rope, the rope has a length $L_0$, and the rope has a non-zero tension. All motion occurs on a horizontal frictionless surface. The cylinder remains stationary on the surface and does not rotate. The object moves in such a way that the rope slowly winds up around the cylinder. The rope will break when the tension exceeds $T_{max}$. Express your answers in terms of $T_{max}$, $m$, $L_0$, $R$, and $v_0$. [asy]
size(200);
real L=6;
filldraw(CR((0,0),1),gray(0.7),black);
path P=nullpath;
for(int t=0;t<370;++t)
{ 
pair X=dir(180-t)+(L-t/180)*dir(90-t);
if(X.y>L) X=(X.x,L);
P=P--X;
}
draw(P,dashed,EndArrow(size=7));
draw((-1,0)--(-1,L)--(2,L),EndArrow(size=7));
filldraw(CR((-1,L),0.25),gray(0.7),black);[/asy]What is the kinetic energy of the object at the instant that the rope breaks?

$ \textbf{(A)}\ \frac{mv_0^2}{2} $

$ \textbf{(B)}\ \frac{mv_0^2R}{2L_0} $

$ \textbf{(C)}\ \frac{mv_0^2R^2}{2L_0^2} $

$ \textbf{(D)}\ \frac{mv_0^2L_0^2}{2R^2} $

$ \textbf{(E)}\ \text{none of the above} $