The rotor of a turbojet engine of an aircraft has a mass 180 kg and polar moment of inertia 10 kg m² about the rotor axis. The rotor rotates at a constant speed of 1100 rad/s in the clockwise direction when viewed from the front of the aircraft. The aircraft while flying at a speed of 800 km per hour takes a turn with a radius of 1.5 km to the left. The gyroscopic moment exerted by the rotor on the aircraft structure and the direction of motion of the nose when the aircraft turns, are

The correct option is: 1629.6 N-m and the nose goes down.

To find the gyroscopic moment (couple) and the behavior of the aircraft's nose during the turn, we will break down the calculation and logical derivation step-by-step.

1. Identify the Given Parameters:
- Mass of the rotor, $m=180\text{kg}$
- Polar moment of inertia of the rotor, $I=10{\text{kg m}}^2$
- Angular velocity of the rotor, $\omega =1100\text{rad/s}$ (Clockwise when viewed from the front)
- Linear speed of the aircraft, $v=800\text{km/h}$
- Radius of the turn, $R=1.5\text{km}=1500\text{m}$

2. Convert Linear Speed to Standard Units (m/s):
$v=800\times \frac{1000}{3600}\text{m/s}=222.22\text{m/s}$

3. Calculate the Angular Velocity of Precession (${\omega }_p$):
The angular velocity of precession is the rate at which the aircraft turns, given by:
${\omega }_p=\frac{v}{R}$
Substituting the values:
${\omega }_p=\frac{222.22}{1500}\approx 0.14815\text{rad/s}$

4. Calculate the Gyroscopic Couple (Moment):
The magnitude of the gyroscopic couple $C$ is given by the formula:
$C=I\times \omega \times {\omega }_p$
Substituting the known values:
$C=10\times 1100\times 0.14815=1629.63\text{N-m}$
Rounding to one decimal place gives $1629.6\text{N-m}$.

5. Determine the Direction of Motion of the Nose:
Let us establish the direction of the angular momentum vector and the effect of the turn:
- The rotor rotates clockwise when viewed from the front of the aircraft. By the right-hand rule, curl the fingers of your right hand clockwise when looking at the front of the aircraft. Your thumb points towards the rear of the aircraft. Therefore, the spin angular momentum vector ($\overrightarrow{H}$) points from front to rear (backward direction).
- The aircraft is turning to the left. When looking from the top, a left turn is a counter-clockwise (CCW) rotation.
- As the aircraft turns left, the nose moves to the left and the tail moves to the right. Since the angular momentum vector ($\overrightarrow{H}$) points towards the rear (tail), this vector rotates towards the right side of the aircraft.
- This change in direction generates an active gyroscopic couple vector pointing to the right.
- The reactive gyroscopic couple exerted by the rotor on the aircraft structure is equal and opposite to the active couple, so its vector points to the left (port side).
- Applying the right-hand rule to the reactive couple vector pointing to the left: point your right thumb to the left, and your fingers will curl in a direction that forces the nose down and the tail up.

Thus, the gyroscopic moment is 1629.6 N-m and the effect is that the nose goes down.