The figure shows an arrangement of a heavy propeller shaft in a ship. The combined polar mass moment of inertia of the propeller and the shaft is 100 kg.m². The propeller rotates at ω = 12 rad/s. The waves acting on the ship hull induces a rolling motion as shown in the figure with an angular velocity of 5 rad/s. The gyroscopic moment generated on the shaft due to the motion described is ______ N.m. (round of to the nearest integer).

The correct answer is 6000.

Analysis of the Figure and Motion:
The provided image shows two views of the ship's propeller shaft assembly:
1. The Side view shows the propeller attached to a horizontal shaft aligned along the z-axis (axis of spin). The propeller and shaft rotate with an angular velocity $\omega$ about this axis.
2. The End view shows the "Ship hull" facing front-on, with the vertical y-axis and horizontal x-axis. It depicts a "Rolling" motion of the ship's hull about the longitudinal z-axis (the axis of the propeller shaft) represented by a curved arrow, with a rolling angular velocity ${\omega }_p$.

Given Data:
- Combined polar mass moment of inertia of the propeller and shaft, $I=100{\text{kg·m}}^2$
- Propeller spin velocity, $\omega =12\text{rad/s}$
- Rolling angular velocity (precession velocity in this context), ${\omega }_p=5\text{rad/s}$

Calculation of the Gyroscopic Moment:
The magnitude of the gyroscopic moment (or gyroscopic couple) $T$ generated on the shaft is given by the formula:

$$T=I\cdot \omega \cdot {\omega }_p$$
Substituting the given values into the formula:

$$T=100\cdot 12\cdot 5$$
$$T=6000\text{N·m}$$
Therefore, the gyroscopic moment generated on the shaft due to the described motion is 6000 N·m.