Solution:

The acceleration due to gravity \( g' \) on the new planet can be calculated using the formula:

\[
g' = \frac{GM}{R^2}.
\]

For the new planet:

- Its radius \( R' = 3R \) (3 times the radius of Earth).
- Its mass \( M' \) is given by \( M' = \text{density} \times \text{volume} = \rho \times \frac{4}{3} \pi (R')^3 = \rho \times \frac{4}{3} \pi (3R)^3 = 27 \times \rho \times \frac{4}{3} \pi R^3 = 27M \) (mass is 27 times that of Earth).

Substituting into the formula:

\[
g' = \frac{G(27M)}{(3R)^2} = \frac{27GM}{9R^2} = 3g.
\]

Thus, \( g' = 3g \).