The correct answer is $\boxed{\text{C. }0.3 \text{R}}$.
The center of gravity of a quadrant of a circle is located at a distance of $\frac{1}{3}$ of the radius from the center of the circle. This is because the quadrant of a circle can be thought of as two right triangles with a common hypotenuse. The center of gravity of a right triangle is located at a distance of $\frac{1}{2}$ of the hypotenuse from each of the two legs. Therefore, the center of gravity of the quadrant of a circle is located at a distance of $\frac{1}{2} \cdot \frac{2}{3} R = \frac{1}{3} R$ from the center of the circle.
Option A is incorrect because it is the distance from the center of the circle to the midpoint of the arc of the quadrant. Option B is incorrect because it is the distance from the center of the circle to the vertex of the quadrant. Option D is incorrect because it is the distance from the center of the circle to the point of tangency of the arc of the quadrant with the radius.