GATE ME 2022 Set 2 — Question 17
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Theory of Machines & Vibrations → Balancing, Flywheels & Gyroscopes → Gyroscope & Gyroscopic Couple
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Question
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 is the acceleration due to gravity. The disc is then given a rapid spin 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
Correct answer
(A) rotate slowly (compared to ω) about the negative z-axis direction
Solution
1.Angular Momentum (): The disc spins in the positive -direction. Thus, .
2.Torque (): When string AB is cut, the shaft is supported only at C. The pivot is at C. The center of mass G is to the left of C. The position vector from C to G is (where is the distance CG). The force of gravity is (assuming is up).
3.Calculating Torque: . The torque is in the negative -direction (into the page).
4.Precession: The rate of change of angular momentum is equal to the torque: . This means the angular momentum vector (initially along ) will move towards the direction of the torque ().
5.Direction of Precession: A rotation from the axis towards the axis in the plane corresponds to a rotation about the negative -axis (downward) according to the right-hand rule. Thus, the shaft precesses about the negative -axis.
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