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 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 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 positive y-axis direction
• D. rotate slowly (compared to $\omega$) about the negative y-axis direction

Hint:

For rapidly spinning rotors, any external torque causes slow gyroscopic precession about the torque direction—not the spin axis.

Answer 1

The Correct Option is A

When the disc spins rapidly with angular velocity $\omega$ along the positive x-axis, it possesses angular momentum $\mathbf{L}$ directed along the +x direction by the right-hand thumb rule.
Step 1: Effect of cutting string AB.
Initially, both strings AB and CD support the shaft horizontally. After AB is cut, only string CD supports the assembly, producing a net gravitational torque about point C. The weight acts downward at the center of mass G, creating a torque vector $\boldsymbol{\tau}$ pointing along the negative z-axis (using $\boldsymbol{\tau} = \mathbf{r} \times \mathbf{F}$).
Step 2: Gyroscopic precession.
A spinning body under a torque perpendicular to its angular momentum undergoes gyroscopic precession. The precession angular velocity is \[ \boldsymbol{\Omega} = \frac{\boldsymbol{\tau}}{L}. \] Since $\mathbf{L}$ is along +x and $\boldsymbol{\tau}$ is along –z, the precession occurs about the –z direction.
Step 3: Nature of rotation.
Because $|\mathbf{L}|$ is very large compared to the torque (disc spins rapidly), the precession is slow compared to $\omega$. Therefore, the shaft rotates slowly about the negative z-axis direction.
Final Answer: (A) rotate slowly about the negative z-axis direction