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Question #42397

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The Question

A gyroscope consists of a thin uniform ring of initial mass $M_0$ and radius $R$ , mounted on a massless shaft of length $l$ . The shaft is pivoted at a fixed origin $O$ . The ring spins about the shaft with an initial angular velocity $\omega_0$ . A constant downward force $F$ is applied at the center of the ring (ignore gravity). The ring loses mass at a constant rate $\mu$ such that its mass at time $t$ is $M(t) = M_0 - \mu t$ . The detached mass is ejected radially outwards relative to the ring's center. Assuming the rapid spin approximation ( $\omega_0 \gg \Omega_p$ ), find the total angle of precession $\Delta \phi$ of the shaft from $t = 0$ to $t = \frac{M_0}{2\mu}$ .

Options

A.
$\frac{F l \ln 2}{\mu R^2 \omega_0}$
B.
$\frac{F l}{2 \mu R^2 \omega_0}$
C.
$\frac{3 F l}{4 \mu R^2 \omega_0}$
D.
$\frac{F l}{\mu R^2 \omega_0}$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Variable_Constraint

Develop pure symbolic logic skills by mastering algebraic constraints without relying on numerical data.

Concepts Behind This Question

Gyroscopic Stability|Variable Mass Systems|Vector Cross-Products

Why This Question Is Tricky

Common Mistakes

This question tests the application of the angular momentum theorem to a variable mass system undergoing gyroscopic precession.

A common trap is assuming that because there is no external torque along the spin axis, the spin angular momentum $L$ is conserved, leading to a constant precession rate (Option B).

However, students must recognize that the radially ejected mass carries away angular momentum.

Applying the generalized Newton's Second Law for rotation reveals that the spin angular velocity $\omega$ remains constant, causing $L(t)$ to decrease proportionally with mass.

Consequently, the precession frequency $\Omega_p = \tau / L(t)$ increases over time, requiring integration to find the true total precession angle.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	System of Particles and Rotational Motion - Class 11 Part 1
Concept	Gyroscopic Stability
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free