A disc rotating about its axis with angular speed ω0 is placed lightly (without any translationa

step 1 In this question it is told that a disk rotating about its axis with an angular speed of omega -

A disc rotating about its axis with angular speed ω0 is...

A disc rotating about its axis with angular speed $\omega_{0}$ is placed lightly (without any translational push) on a perfectly frictionless table. The radius of the disc is $R$. Let $\mathrm{k}, v_{8}$ and $\mathrm{\psi}$ be the magnitudes of linear velocities of the points $A, B$ and $C$ on the disc as shown. Then (a) $v_{A}>v_{B}>v_{C}$ (b) $v_{A}<v_{B}<v_{C}$ (c) $v_{A}=v_{B}<v_{C}$ (d) $v_{A}=v_{B}>v_{C}$

A disc rotating about its axis with angular speed $\omega_{0}$ is placed lightly (without any translational push) on a perfectly frictionless table. The radius of the disc is $R$. Let $\mathrm{k}, v_{8}$ and $\mathrm{\psi}$ be the magnitudes of linear velocities of the points $A, B$ and $C$ on the disc as shown. Then
(a) $v_{A}>v_{B}>v_{C}$
(b) $v_{A}<v_{B}<v_{C}$
(c) $v_{A}=v_{B}<v_{C}$
(d) $v_{A}=v_{B}>v_{C}$

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Key Concepts

Superposition of Translational and Rotational Motion

For a rigid body that may have both rotational and translational motions, the actual velocity of any point is the vector sum of the translational velocity of the center of mass and the rotational velocity about the center of mass. In scenarios where there is no translational motion, as when a spinning disc is placed on a frictionless surface, the observed speeds at each point solely come from the rotational motion.

Angular to Linear Velocity Relationship

The relationship between angular velocity (?) and linear velocity (v) in a rotating object is given by v = ?r, where r is the distance from the rotation axis. This fundamental equation is crucial for comparing the speeds of different points on a rotating rigid body, as it directly relates the rotational motion to the linear speeds at various radial positions.

Rigid Body Rotation

In rigid body rotation, every point of the object rotates about a fixed axis with the same angular velocity. Although the entire body has a common rate of rotation, individual points move with linear speeds that depend on their distance from the axis. This concept explains why points further from the axis have higher linear speeds, while points closer to the axis (including the center of rotation) have lower speeds.

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