Question

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 \( \omega = 12\ \text{rad/s} \). The waves acting on the ship hull induce 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 ________ \text{N.m} (round off to the nearest integer). 

Hint:

The gyroscopic moment in rotating systems is calculated using the moment of inertia, angular velocity of rotation, and angular velocity of the body experiencing rolling or pitching motion.

Answer 1

Correct Answer: 0

The gyroscopic moment \( M_g \) generated due to the rotational motion of the propeller and the rolling motion of the ship is given by the formula: \[ M_g = I \cdot \omega \cdot \dot{\theta} \] Where:
- \( I = 100\ \text{kg.m}^2 \) is the combined polar moment of inertia,
- \( \omega = 12\ \text{rad/s} \) is the angular velocity of the propeller,
- \( \dot{\theta} = 5\ \text{rad/s} \) is the angular velocity of the ship's rolling motion.
Substituting the given values: \[ M_g = 100 \times 12 \times 5 = 6000\ \text{N.m} \] Thus, the gyroscopic moment is: \[ \boxed{0} \]