The rotor of an aeroplane engine has mass moment of inertia 1.0 kg-m². The engine rotates at speed of 500 RPM in clockwise direction if viewed from front of aeroplane. If aeroplane flying at 1200 km/hr turns with a radius of 2 km at same elevation. Then magnitude of gyroscopic moment by rotor on aeroplane structure is

The correct option is 8.73 N-m.

To find the magnitude of the gyroscopic couple (or gyroscopic moment) exerted by the rotor on the aeroplane structure, we use the standard formula for the gyroscopic couple:
$$C=I\cdot \omega \cdot {\omega }_p$$
where:
• $I$ is the mass moment of inertia of the rotor.
• $\omega$ is the angular velocity of the rotor.
• ${\omega }_p$ is the angular velocity of precession (associated with the turning of the aeroplane).

Let's calculate each of these values step-by-step from the given data:
1. Mass moment of inertia ($I$):
Given directly as:
$$I=1.0\text{kg-m}{2}^{}$$

2. Angular velocity of the rotor ($\omega$):
The rotational speed of the engine is given as $N=500\text{RPM}$.
The relation between angular velocity and RPM is:
$$\omega =\frac{2\pi N}{60}$$
Substituting $N=500$:
$$\omega =\frac{2\cdot \pi \cdot 500}{60}=\frac{50\pi }{3}\approx 52.36\text{rad/s}$$

3. Angular velocity of precession (${\omega }_p$):
The linear speed of the aeroplane is $v=1200\text{km/hr}$. First, convert this speed into meters per second ($\text{m/s}$):
$$v=1200\cdot \frac{1000}{3600}=1200\cdot \frac{5}{18}=\frac{1000}{3}\approx 333.33\text{m/s}$$
The turn radius is given as $R=2\text{km}=2000\text{m}$.
The angular velocity of precession is:
$${\omega }_p=\frac{v}{R}=\frac{1000/3}{2000}=\frac{1}{6}\approx 0.1667\text{rad/s}$$

4. Magnitude of the gyroscopic couple ($C$):
Now substitute these values into the couple formula:
$$C=1.0\cdot 52.36\cdot 0.1667$$
$$C=1.0\cdot \left(\frac{50\pi }{3}\right)\cdot \left(\frac{1}{6}\right)=\frac{50\pi }{18}=\frac{25\pi }{9}$$
Using $\pi \approx 3.14159$:
$$C\approx \frac{78.54}{9}\approx 8.73\text{N-m}$$
Thus, the magnitude of the gyroscopic moment acting on the aeroplane structure is indeed 8.73 N-m.