A non-uniform sphere has its centre of mass at a distance r...

A non-uniform sphere has its centre of mass at a distance $r / 2$ from the centre, where $r$ is the radius of the sphere. This sphere rolls without slipping on a horizontal surfacc such that the angular velocity is $\omega$ which is constant. Consider the instant shown in figure. Sclect the correct statement. (a) The speed of the centre $C$ is constant. (b) The specd of the centre of mass is constant. (c) The speed of the centre of mass is $\frac{r}{2} \omega$ at the instant shown. (d) The speed of the centre $C$ is $\frac{r}{2} \omega$ at the instant shown.

A non-uniform sphere has its centre of mass at a distance $r / 2$ from the centre, where $r$ is the radius of the sphere. This sphere rolls without slipping on a horizontal surfacc such that the angular velocity is $\omega$ which is constant. Consider the instant shown in figure. Sclect the correct statement.
(a) The speed of the centre $C$ is constant.
(b) The specd of the centre of mass is constant.
(c) The speed of the centre of mass is $\frac{r}{2} \omega$ at the instant shown.
(d) The speed of the centre $C$ is $\frac{r}{2} \omega$ at the instant shown.

Key Concepts

Angular Velocity and Linear Speed Relationship

There exists a key relationship in rotational kinematics between angular velocity and linear speed. For rolling motion without slipping, the linear speed of a point at a distance R from the axis of rotation is given by v = ?R. This concept is crucial for understanding how different points on a rolling object move.

Kinematics of Rolling with an Offset Centre of Mass

When an object has a non-uniform mass distribution, its centre of mass is not at the geometric centre. In such cases, the motion of the centre of mass relative to the object’s geometric centre must be analyzed separately. The overall kinematics of the object involves both the rotation about the geometric centre and the translation of the centre of mass, which can lead to differing instantaneous speeds at different points on the body.

Rolling without Slipping

Rolling without slipping occurs when an object rotates in such a way that the point of contact with the surface has zero relative velocity. This condition creates a fixed link between the object's angular velocity and its translational speed, typically described by the relation v = ?R, where R is the effective radius.

Centre of Mass

The centre of mass is the weighted average location of all the mass in an object. It is the point at which the object’s mass can be considered to be concentrated for translational motion analysis. In non-uniform objects, the centre of mass does not coincide with the geometric centre, impacting the overall dynamics.

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