Gyroscopic Effects Questions and Answers

Answer: C

Precession is the slow change in the direction of the axis of rotation of a spinning object. In a gyroscope a torque applied perpendicular to the spin axis makes the axis turn about a third axis, which is the precession. Wobbling about the axis is nutation, the spinning motion is the spin, and the applied force is the torque and not the precession, so the other options describe other things.

The rate of precession equals the applied torque divided by the angular momentum of the spinning rotor, so a faster spin gives a slower precession.

Answer: A

The shaft will rotate slowly about the negative z-axis direction: once AB is cut, the spinning disc makes the assembly precess about the vertical through C instead of falling.

With AB gone, string CD alone holds the assembly. G lies between A and C, so the weight \(W\) acts a distance \(a\) on the negative-x side of C. Taking x towards E, z up and y completing a right-handed set, the gravity torque about C is

\[ \begin{aligned} \vec{\tau} &= (-a\hat{i}) \times (-W\hat{k}) \\ &= aW(\hat{i} \times \hat{k}) = -aW\hat{j} \end{aligned} \]

The spin angular momentum is \(\vec{H} = I\omega\hat{i}\). For steady precession at rate \(\Omega\hat{k}\), \(\vec{\Omega} \times \vec{H} = \vec{\tau}\):

\[ \begin{aligned} \Omega I\omega\,(\hat{k} \times \hat{i}) &= \Omega I\omega\,\hat{j} \\ &= -aW\hat{j} \end{aligned} \]

so \(\Omega = -aW/(I\omega)\): rotation about the negative z axis, and slow because the rate falls as \(\omega\) rises.

Rotation about the y axis would be the shaft simply swinging down under gravity, which is what happens without the spin. The positive z direction would need the spin, or G's side of C, to be reversed.

Answer: A

The gyroscopic moment is about 8.73 N m.

The spin speed is \(\omega = 2\pi(500)/60 = 52.36\) rad/s. The precession rate is the aeroplane's angular velocity in the turn, \(\omega_{p} = v/R\), with \(v = 1200/3.6 = 333.3\) m/s:

\[ \begin{aligned} \omega_{p} &= 333.3/2000 = 0.1667\ \text{rad/s} \\ C &= I\omega\omega_{p} = 1.0 \times 52.36 \times 0.1667 = 8.73\ \text{N m} \end{aligned} \]

17.46 and 26.19 are two and three times the value, and 4.37 half of it. The clockwise sense of rotation decides the direction of the moment (nose up or down), not its size.