C

Step 1: Calculate the force of friction using the formula:
$F_{f} = \mu \cdot m \cdot g$, where
$\mu = 0.4$, $m = 50 \text{ kg}$, and $g = 9.81 \text{ m/s}^{2}$.
$F_{f} = 0.4 \cdot 50 \cdot 9.81 = 196.2 \text{ N}$.

Step 2: Identify the centripetal force needed to keep the child on the disk, given by:
$F_{c} = m \cdot r \cdot \omega^{2}$, where $\omega = 2 \text{ rad/s}$.
Set the centripetal force equal to the maximum frictional force:
$196.2 = 50 \cdot r \cdot (2)^{2}$.

Step 3: Solve for radius $r$:
$196.2 = 50 \cdot r \cdot 4$
$196.2 = 200r$
$r = \frac{196.2}{200} = 0.981 \text{ m}$.

Therefore, the maximum radius R where the child can sit and still remain on the disk is approximately 0.98 m.