Rolling motion  (Page 2/4)

Rolling motion

$$\begin{array}{l}x=\theta R\end{array}$$

where "θ" is the angle subtended by the arc and "R" is the radius of the disk. Now differentiating the above relation with respect to time, we have :

$$\begin{array}{l}\frac{đx}{đt}=\left(\frac{đ\theta }{đt}\right)R\\ \Rightarrow v=\omega R\end{array}$$

where "v" is the velocity of moving frame of reference (velocity of the observer) and "ω" is the angular velocity of the rotating disk. However, we have assumed that frame of reference is moving with a velocity equal to that of center of mass, " $v_C$ ". Hence,

$$\begin{array}{l}\Rightarrow v_C=\omega R\end{array}$$

We note from the figure that the linear velocity is along x-axis and is positive. On the other hand, angular velocity is clock-wise and is negative. In order to account for this, we introduce a negative sign in the relation as :

This equation relates the linear velocity of center of mass," $v_C$ ", and angular velocity of the disk,"ω". We can drop negative sign if speeds (not velocities) are considered. Importantly, we must note here that this relation connects velocities which are measured in different frames of reference. The linear velocity of center of mass is measured in the ground reference, whereas angular velocity is measured in the moving reference.

We must also understand here that this relation does not relate linear velocities of the particles constituting the body to that of angular velocity. The linear velocities are, as a matter of fact, different for different particles. The velocities of the particles constituting the body depends on their relative position with respect to the center of mass or the point of contact. Thus, we should bear in mind that above relation specifically connects the linear velocity of center of mass ( $v_C$ ) to the angular velocity of the rotating disk ("ω" ), which is same for all particles, constituting the disk.

The relation “ $v_C=\omega R$ ” is the defining relation for pure rolling. The moment we refer a motion as pure rolling motion, then two velocities are related precisely as given by this equation. The caution is that it is not same as the similar relation available for pure rotational motion :

$$\begin{array}{l}v=\omega r\end{array}$$

This relation is a relation for any particle constituting the body. It is not so for pure rolling. The linear velocity in the equation ( $v_C$ ) of pure rolling motion refers to the linear velocity of a specific point i.e. center of mass (not the velocity of any point,”v”) and equation involves radius of the disk, “R”, (not any “r” denoting the distance of any particle from the axis of rotation). These distinctions should always be kept in mind. In order to emphasize the distinction, we may write :

$$\begin{array}{l}\Rightarrow \omega \ne \frac{v_C}{r}\mathrm{;\; if}r\ne R\end{array}$$

Description of rolling in ground frame

The description of rolling is different for different observers (also read frames of reference). We have observed that the motion of disk is pure rotation as seen in the moving reference attached to the rolling disk. The disk rotates about an axis, passing through the center of mass and perpendicular to its surface. When seen from the ground, however, the axis of rotation is not fixed as in pure rotation; rather it translates along x-direction with a velocity " $v_C$ ". It means that the rolling motion is not a pure rotation.