Question

Working from first principles show that the gyroscopic torque Ta generated by a body of polar moment of inertia I spinning about one axis at an angular velocity an whilst precessing at a about a perpendicular axis is lo Clearly indicate the direction of [10 marks] A submerged submarine is travelling at 36 kmh in a circular path of unknown radius R as shown in Fig. Q2a. A transversely mounted single-rotor gyroscope is carried onboard. The submarine's motion causes reaction forces of magnitude F-1.2N at the rotors supporting bearings which are spaced a distance d-150 mm along the rotor shaft, as illustrated in Fig. Q2b. (a) the gyroscope has a mass of 0.2 kg, a radius of gyration of 50 mm and spins at 30,000 revlmin, determine the radius R of the submarine's circular path [10 marks Determine the direction of the gyroscope's spin when viewed from the starboard (b) side of the submarine. [5 marks) Stem (rear) Bow (front) (a) Top view on submarine Inertia, I Port (Left) (Right) Spin axis (b) Stem (rear) view on gyroscope Fig. Q2

step by step explanation please

Answer 1

Solution.

Ineta, I tA) Legt) Al in auis

Mass of gyroscope, M= 0.2 Kg

Radius of gyration of gyroscope, R_{​​​​​​g}= 50 mm

Moment of inertia, I of the gyroscope = MR_{​​​​​g}^​​​2 = 0.2* (0.05)² =0.00005 Kgm^{​​​​2}

Radius of circular path=R

speed of submarine, v= 36 km/hr

Angular velocity of single rotor gyroscope, w_{​​​​​​1}= 500 rad/s

Angular velocity of submarine, w_{​​​​​​2}= 10/R rad/s

Gyroscopic torque, T_{​​​​​​g}= Iw_{​​​​1}​w_​​​2 = F.d

0.0005*50* 10/R = 1.2 * 0.15

R= 13.88 m ........ (a)

b) To find out the direction of the spin look at the following figure.

View Acve Reaetive uw2

When rotor is viewed from the starboard side, it rotates in anticlockwise direction. While the submarine takes a right turn. Due to this we get an active gyroscopic couple as shown and in turn, a reactive gyroscopic couple which actually move the submarine.

Reactive gyroscopic couple is upwards and when we use right hand thumb rule, we get that starboard moves downwards and port moves upwards which means a rolling movement of the submarine.