Question

A disc with a flat small bottom beaker placed on it at a distance R from its center is revolving about an axis passing through the center and perpendicular to its plane with an angular velocity ω. The coefficient of static friction between the bottom of the beaker and the surface of the disc is μ. The beaker will revolve with the disc if :

• A. \(R≤\frac{μg}{2ω^2}\)
• B. \(R≤\frac{μg}{ω^2}\)
• C. \(R≥\frac{μg}{2ω^2}\)
• D. \(R≥\frac{μg}{ω^2}\)

Answer 1

The Correct Option is A

 To determine whether the beaker will revolve with the disc, we need to consider the forces acting on the beaker. The primary force that allows the beaker to rotate with the disc without slipping is the static frictional force. The centrifugal (or centripetal) force required to keep the beaker moving in a circular motion must be provided by this static frictional force.

Centripetal Force: The centripetal force required to keep an object of mass \( m \) moving in a circle of radius \( R \) with angular velocity \( \omega \) is given by:

\(F_c = mR\omega^2\)

Static Friction: The maximum static frictional force \( F_f \) that can act on the beaker is:

\(F_f = \mu mg\)

where \( \mu \) is the coefficient of static friction and \( g \) is the acceleration due to gravity.

Condition for No Slipping: For the beaker to revolve with the disc without slipping, the static frictional force must be equal to or greater than the required centripetal force:

\(mR\omega^2 \leq \mu mg\)

By canceling out \( m \) from both sides and rearranging the terms, we get:

\(R \leq \frac{\mu g}{\omega^2}\)

This inequality represents the condition under which the beaker will revolve with the disc.

Given option: \(R \leq \frac{\mu g}{2\omega^2}\). Let's verify if this is more restrictive:

\(\frac{\mu g}{2\omega^2} < \frac{\mu g}{\omega^2}\)

This inequality holds true, meaning this condition ensures no slipping for even a smaller radius. Hence, the correct answer is \(R \leq \frac{\mu g}{2\omega^2}\).

Therefore, the beaker will revolve with the disc if the radius \( R \) satisfies the given condition: \(R \leq \frac{\mu g}{2\omega^2}\).