# Gyroscope

 Question 1
A massive uniform rigid circular disc is mounted on a frictionless bearing at the end E of a massive uniform rigid shaft AE which is suspended horizontally in a uniform gravitational field by two identical light inextensible strings AB and CD as shown, where G is the center of mass of the shaft-disc assembly and $g$ is the acceleration due to gravity. The disc is then given a rapid spin $\omega$ about its axis in the positive x-axis direction as shown, while the shaft remains at rest. The direction of rotation is defined by using the right-hand thumb rule. If the string AB is suddenly cut, assuming negligible energy dissipation, the shaft AE will A rotate slowly (compared to $\omega$) about the negative z-axis direction B rotate slowly (compared to $\omega$) about the positive z-axis direction C rotate slowly (compared to $\omega$) about the negative y-axis direction D rotate slowly (compared to $\omega$) about the positive y-axis direction
GATE ME 2022 SET-2   Theory of Machine
Question 1 Explanation: The spin vector will chase the couple on torque vector and produce precision in system.
Hence precision will be $-y$ direction. Rotate slowly (compared to $\omega$) about negative $z-$axis direction.
 Question 2
The figure shows a schematic of a simple Watt governor mechanism with the spindle $O_1O_2$ rotating at an angular velocity $\omega$ about a vertical axis. The balls at P and S have equal mass. Assume that there is no friction anywhere and all other components are massless and rigid. The vertical distance between the horizontal plane of rotation of the balls and the pivot $O_1$ is denoted by $h$. The value of $h=400$ mm at a certain $\omega$. If $\omega$ is doubled, the value of $h$ will be _________ mm. A 50 B 100 C 150 D 200
GATE ME 2022 SET-1   Theory of Machine
Question 2 Explanation:
$h_1 = 400 mm, h_2 = ?$
$\omega _1=\omega \;\;\;\omega _2=2\omega$
For Watt governor,
\begin{aligned} h&=\frac{g}{\omega ^2}\\ h\propto \frac{1}{\omega ^2}\\ \Rightarrow h_1\omega _1^2&=h_2\omega _2^2\\ 400 \times \omega ^2&=h_2 \times (2\omega )^2\\ h_2&=100mm \end{aligned}
 Question 3
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 $kg.m^2$. The propeller rotates at $\omega =12$ rad/s. The waves acting on the ship hull induces 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 _______N.m (round off to the nearest integer). A 0 B 1 C 2 D 4
GATE ME 2021 SET-1   Theory of Machine
Question 3 Explanation:
As the axes of the rolling of the ship and the axes of the rotor are parallel, there is no precession of the axis of spin.

Gyroscopic couple $=I\cdot (\vec{\omega _s} \times \vec{\omega _p})=0$
 Question 4
A uniform disc with radius r and a mass of m kg is mounted centrally on a horizontal axle of negligible mass and length of 1.5r. The disc spins counter-clockwise about the axle with angular speed $\omega$, when viewed from the right-hand side bearing, Q. The axle precesses about a vertical axis at $\omega_p=\omega /10$ in the clockwise direction when viewed from above. Let $R_P \; and \; R_Q$ (positive upwards) be the resultant reaction forces due to the mass and the gyroscopic effect, at bearings P and Q, respectively. Assuming $\omega^2 r=300m/s^2$ and $g=10m/s^2$, the ratio of the larger to the smaller bearing reaction force (considering appropriate signs) is_______ A -2 B -3 C -6 D -9
GATE ME 2019 SET-2   Theory of Machine
Question 4 Explanation:
Given:
\begin{aligned} &\omega_{\mathrm{p}}=\frac{\omega}{10} \\ &\omega^{2}=300 \mathrm{m} / \mathrm{s}^{2} \\ &\mathrm{g}=10 \mathrm{m} / \mathrm{s}^{2} \\ &\Sigma \mathrm{F}_{\mathrm{y}}=0 \Rightarrow \mathrm{R}_{\mathrm{p}}+\mathrm{R}_{\mathrm{Q}}=\mathrm{mg} \ldots(i)\\ &\Rightarrow \mathrm{R}_{\mathrm{Q}}(1.5 \mathrm{r})=(\mathrm{mg} \times 1.5) \frac{\mathrm{r}}{2}+\mathrm{I} \omega . \omega_{\mathrm{p}} \\ &\Rightarrow(1.5 \mathrm{r}) \mathrm{R}_{\mathrm{Q}}=\mathrm{mg} \times \frac{1.5 \mathrm{r}}{2}+\frac{\mathrm{mr}^{2}}{2} \frac{\omega^{2}}{10} \\ &\Rightarrow(1.5 \mathrm{R}) \mathrm{R}_{\mathrm{Q}}=\mathrm{mg} \times \frac{1.5 \mathrm{r}}{2}+\frac{\mathrm{m}}{2} \frac{\left(\mathrm{r} \omega^{2}\right) \mathrm{r}}{10} \\ &\Rightarrow 1.5 \mathrm{R}_{\mathrm{Q}}=\frac{1.5 \mathrm{mg}}{2}+\frac{\mathrm{m}}{2} \times \frac{300}{10} \\ &\Rightarrow R_{Q}=15 \mathrm{m}\ldots(ii) \end{aligned}
From equation (i) and (ii)
$\mathrm{R}_{\mathrm{P}}+15 \mathrm{m}=10 \mathrm{m}$
$R_{p}=-5 \mathrm{m}$
$\Rightarrow \frac{R_{\text {larger }}}{R_{\text {smaller }}}=\frac{R_{Q}}{R_{p}}=\frac{15 \mathrm{m}}{-5 \mathrm{m}}=-3$
 Question 5
The rotor of a turbojet engine of an aircraft has a mass 180 kg and polar moment of inertia 10 $kg\cdot m^2$ 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
 A 1629.6 N-m and the nose goes up B 1629.6 N-m and the nose goes down C 162.9 N-m and the nose goes up D 162.9 N-m and the nose goes down
GATE ME 2019 SET-1   Theory of Machine
Question 5 Explanation:
\begin{aligned} \text{mass of rotor }&=180 \mathrm{kg} \\ \mathrm{I}&=10 \mathrm{kg}-\mathrm{m}^{2} \\ \omega_{\mathrm{s}}&=1100 \mathrm{rad} / \mathrm{s} \\ \mathrm{v}&=800 \mathrm{kmph} \\ \mathrm{R}&=1.5 \mathrm{km} \\ \end{aligned} $\mathrm{I}=\mathrm{I} \omega_{\mathrm{S}} \omega_{\mathrm{p}}=1629.62 \mathrm{N}-\mathrm{m}$
Nose goes down
 Question 6
A car is moving on a curved horizontal road of radius 100 m with a speed of 20 m/s. The rotating masses of the engine have an angular speed of 100 rad/s in clockwise direction when viewed from the front of the car. The combined moment of inertia of the rotating masses is 10 kg-$m^{2}$. The magnitude of the gyroscopic moment (in N-m) is __________
 A 100 B 200 C 300 D 400
GATE ME 2016 SET-1   Theory of Machine
Question 6 Explanation:
\begin{aligned} R&=100\;m \\ v&=20\;m/sec \\ \omega _p &=\frac{v}{R} \\ &= 0.2\; rad/sec\\ \omega _s &=100\; rad/sec \\ I&= 10\; Kg-m^2\\ &\text{Gyroscopic moment,}\\ I\omega _s\omega _p &=10 \times 0.2 \times 100\; N-m\\ &=200\; N-m \end{aligned}
There are 6 questions to complete.