Two discs are mounted on thin, lightweight rods oriented through their centers and normal to the discs. These axles are constrained to be vertical at all times, and the discs can pivot frictionlessly on the rods. The discs have identical thickness and are made of the same material, but have differing radii $r_\text{1}$ and $r_\text{2}$. The discs are given angular velocities of magnitudes $\omega_\text{1}$ and $\omega_\text{2}$, respectively, and brought into contact at their edges. After the discs interact via friction it is found that both discs come exactly to a halt. Which of the following must hold? Ignore effects associated with the vertical rods.
[asy]
//Code by riben, Improved by CalTech_2023
// Solids
import solids;

//bigger cylinder
draw(shift(0,0,-1)*scale(0.1,0.1,0.59)*unitcylinder,surfacepen=white,black);
draw(shift(0,0,-0.1)*unitdisk, surfacepen=black);
draw(unitdisk, surfacepen=white,black);
draw(scale(0.1,0.1,1)*unitcylinder,surfacepen=white,black);


//smaller cylinder
draw(rotate(5,X)*shift(-2,3.2,-1)*scale(0.1,0.1,0.6)*unitcylinder,surfacepen=white,black);
draw(rotate(4,X)*scale(0.5,0.5,1)*shift(1,8,0.55)*unitdisk, surfacepen=black);
draw(rotate(4,X)*scale(0.5,0.5,1)*shift(1,8,0.6)*unitdisk, surfacepen=white,black);
draw(rotate(5,X)*shift(-2,3.2,-0.2)*scale(0.1,0.1,1)*unitcylinder,surfacepen=white,black);



// Lines
draw((0,-2)--(1,-2),Arrows(size=5));
draw((4,-2)--(4.7,-2),Arrows(size=5));
// Labels
label("r1",(0.5,-2),S);
label("r2",(4.35,-2),S);
// Curved Lines
path A=(-0.694, 0.897)--
(-0.711, 0.890)--
(-0.742, 0.886)--
(-0.764, 0.882)--
(-0.790, 0.873)--
(-0.815, 0.869)--
(-0.849, 0.867)--
(-0.852, 0.851)--
(-0.884, 0.844)--
(-0.895, 0.837)--
(-0.904, 0.824)--
(-0.879, 0.800)--
(-0.841, 0.784)--
(-0.805, 0.772)--
(-0.762, 0.762)--
(-0.720, 0.747)--
(-0.671, 0.737)--
(-0.626, 0.728)--
(-0.591, 0.720)--
(-0.556, 0.715)--
(-0.504, 0.705)--
(-0.464, 0.700)--
(-0.433, 0.688)--
(-0.407, 0.683)--
(-0.371, 0.685)--
(-0.316, 0.673)--
(-0.271, 0.672)--
(-0.234, 0.667)--
(-0.192, 0.664)--
(-0.156, 0.663)--
(-0.114, 0.663)--
(-0.070, 0.660)--
(-0.033, 0.662)--
(0.000, 0.663)--
(0.036, 0.663)--
(0.067, 0.665)--
(0.095, 0.667)--
(0.125, 0.666)--
(0.150, 0.673)--
(0.187, 0.675)--
(0.223, 0.676)--
(0.245, 0.681)--
(0.274, 0.687)--
(0.300, 0.696)--
(0.327, 0.707)--
(0.357, 0.709)--
(0.381, 0.718)--
(0.408, 0.731)--
(0.443, 0.740)--
(0.455, 0.754)--
(0.458, 0.765)--
(0.453, 0.781)--
(0.438, 0.795)--
(0.411, 0.809)--
(0.383, 0.817)--
(0.344, 0.829)--
(0.292, 0.839)--
(0.254, 0.846)--
(0.216, 0.851)--
(0.182, 0.857)--
(0.153, 0.862)--
(0.124, 0.867);
draw(shift(0.2,0)*A,EndArrow(size=5));
path B=(2.804, 0.844)--
(2.790, 0.838)--
(2.775, 0.838)--
(2.758, 0.831)--
(2.740, 0.831)--
(2.709, 0.827)--
(2.688, 0.825)--
(2.680, 0.818)--
(2.660, 0.810)--
(2.639, 0.810)--
(2.628, 0.803)--
(2.618, 0.799)--
(2.604, 0.790)--
(2.598, 0.778)--
(2.596, 0.769)--
(2.606, 0.757)--
(2.630, 0.748)--
(2.666, 0.733)--
(2.696, 0.721)--
(2.744, 0.707)--
(2.773, 0.702)--
(2.808, 0.697)--
(2.841, 0.683)--
(2.867, 0.680)--
(2.912, 0.668)--
(2.945, 0.665)--
(2.973, 0.655)--
(3.010, 0.648)--
(3.040, 0.647)--
(3.069, 0.642)--
(3.102, 0.640)--
(3.136, 0.632)--
(3.168, 0.629)--
(3.189, 0.627)--
(3.232, 0.619)--
(3.254, 0.624)--
(3.281, 0.621)--
(3.328, 0.618)--
(3.355, 0.618)--
(3.397, 0.617)--
(3.442, 0.616)--
(3.468, 0.611)--
(3.528, 0.611)--
(3.575, 0.617)--
(3.611, 0.619)--
(3.634, 0.625)--
(3.666, 0.622)--
(3.706, 0.626)--
(3.742, 0.635)--
(3.772, 0.635)--
(3.794, 0.641)--
(3.813, 0.646)--
(3.837, 0.654)--
(3.868, 0.659)--
(3.886, 0.672)--
(3.903, 0.681)--
(3.917, 0.688)--
(3.931, 0.697)--
(3.943, 0.711)--
(3.951, 0.720)--
(3.948, 0.731)--
(3.924, 0.745)--
(3.900, 0.757)--
(3.874, 0.774)--
(3.851, 0.779)--
(3.821, 0.779)--
(3.786, 0.786)--
(3.754, 0.792)--
(3.726, 0.797)--
(3.677, 0.806)--
(3.642, 0.812);
draw(shift(0.7,0)*B,EndArrow(size=5));
[/asy]

(A) $\omega_\text{1}^2r_\text{1}=\omega_\text{2}^2r_\text{2}$
(B) $\omega_\text{1}r_\text{1}=\omega_\text{2}r_\text{2}$
(C) $\omega_\text{1}r_\text{1}^2=\omega_\text{2}r_\text{2}^2$
(D) $\omega_\text{1}r_\text{1}^3=\omega_\text{2}r_\text{2}^3$
(E) $\omega_\text{1}r_\text{1}^4=\omega_\text{2}r_\text{2}^4$
1. **Identify the forces and torques involved:**
   The friction force \( F \) is the same for both discs. The torque \( \tau \) due to friction is given by:
   \[
   \tau = Fr
   \]
   Therefore, for the two discs, we have:
   \[
   \tau_1 = F r_1 \quad \text{and} \quad \tau_2 = F r_2
   \]
   This implies:
   \[
   \frac{\tau_1}{r_1} = \frac{\tau_2}{r_2} \implies \frac{\tau_1}{\tau_2} = \frac{r_1}{r_2}
   \]

2. **Relate the torques to angular momentum:**
   The friction force brings both discs to a halt at the same time \( t \). The change in angular momentum \( L \) is given by:
   \[
   L_1 = \tau_1 t \quad \text{and} \quad L_2 = \tau_2 t
   \]
   Therefore:
   \[
   \frac{L_1}{L_2} = \frac{\tau_1 t}{\tau_2 t} = \frac{\tau_1}{\tau_2}
   \]
   From the previous step, we have:
   \[
   \frac{\tau_1}{\tau_2} = \frac{r_1}{r_2} \implies \frac{L_1}{L_2} = \frac{r_1}{r_2}
   \]

3. **Express angular momentum in terms of moment of inertia and angular velocity:**
   Angular momentum \( L \) is also given by:
   \[
   L = I \omega
   \]
   where \( I \) is the moment of inertia and \( \omega \) is the angular velocity. For the two discs:
   \[
   \frac{L_1}{L_2} = \frac{I_1 \omega_1}{I_2 \omega_2}
   \]
   Therefore:
   \[
   \frac{r_1}{r_2} = \frac{I_1 \omega_1}{I_2 \omega_2}
   \]

4. **Determine the moments of inertia:**
   The moment of inertia \( I \) for a disc is given by:
   \[
   I = \frac{1}{2} m r^2
   \]
   Therefore, for the two discs:
   \[
   I_1 = \frac{1}{2} m_1 r_1^2 \quad \text{and} \quad I_2 = \frac{1}{2} m_2 r_2^2
   \]
   Substituting these into the previous equation, we get:
   \[
   \frac{r_1}{r_2} = \frac{\frac{1}{2} m_1 r_1^2 \omega_1}{\frac{1}{2} m_2 r_2^2 \omega_2} = \frac{m_1 r_1^2 \omega_1}{m_2 r_2^2 \omega_2}
   \]

5. **Relate the masses to the radii:**
   Since the discs have identical thickness and are made of the same material, their masses are proportional to the square of their radii:
   \[
   m \propto r^2 \implies \frac{m_1}{m_2} = \frac{r_1^2}{r_2^2}
   \]
   Substituting this into the previous equation, we get:
   \[
   \frac{r_1}{r_2} = \frac{r_1^2 r_1^2 \omega_1}{r_2^2 r_2^2 \omega_2} = \frac{r_1^4 \omega_1}{r_2^4 \omega_2}
   \]
   Simplifying, we find:
   \[
   \omega_1 r_1^3 = \omega_2 r_2^3
   \]

Conclusion:
\[
\boxed{\omega_1 r_1^3 = \omega_2 r_2^3}
\]