Question

Free ends of two inextensible cords wrapped on a massive disc of radius $r=10\sqrt{3}$ cm are affixed to the ceiling at two different points. Both the cords and the disc are in the same vertical plane. At an instant after the disc is released, angles made by the cord with the ceiling are $\theta=67^{\circ}$ and $\phi=53^{\circ}$ and angular velocity of the disc is $\omega=5.0$ rad/s. Find magnitude of velocity of the centre of the disc at this instant.

Answer 1

No consistent solution based on standard physics principles

Answer 2

This problem is ill-posed under standard interpretations of inextensible strings and unwinding motion. The standard constraints lead to a contradiction, implying a zero velocity for the center of mass. If a non-zero velocity is expected, the problem relies on an unstated assumption or simplification not evident from the text or diagram.

Detailed analysis:

• Assumptions: The typical assumptions for such problems include inextensible cords, a rigid disc, and the cords remaining taut.

• Constraints: The key constraints are derived from the inextensibility of the cords. The component of the velocity of the point of contact on the disc perpendicular to the cord must be zero. This arises because the cord's end is fixed, and any relative motion perpendicular to the cord would imply the cord is stretching or compressing.

• Equations: Applying these constraints leads to two equations:

  1. $v_x \sin\theta - v_y \cos\theta = 0$
  2. $v_x \sin\phi + v_y \cos\phi = 0$

  where $v_x$ and $v_y$ are the horizontal and vertical components of the velocity of the disc's center, and $\theta$ and $\phi$ are the angles the cords make with the ceiling.

• Contradiction: These two equations imply that both $v_x$ and $v_y$ must be zero, resulting in a zero velocity for the center of the disc. This contradicts the physical scenario of the disc unwinding due to gravity and its initial angular velocity.

• Possible Resolutions (but insufficient):

  • Assuming the velocity of the center of mass is purely vertical doesn't resolve the issue.
  • Considering the instantaneous center of rotation (ICR) also leads to a zero velocity for the center of the disc.
  • The problem might be simplified by assuming that the velocity of the center of mass is purely vertical, and then the horizontal components of the forces from the strings must balance. But that's dynamics.

• Conclusion: The provided problem statement is either flawed, requires a non-standard interpretation, or relies on an implicit assumption not specified, making it impossible to derive a consistent, physically meaningful solution using standard mechanics principles.