19.4 Understanding rolling motion  (Page 4/4)

It must, however, be kept in mind that the equation of rolling was developed for the case of rolling. As such, any derivation based on this relation will be valid only for pure rolling that does not involve sliding. Now, differentiating the equation with respect to time, we have :

$$\begin{array}{l}\frac{d{v}_C}{dt}=\left(\frac{d\omega }{dt}\right)R\end{array}$$

This is the conditional relation between linear and angular accelerations that should be maintained for the accelerated body to be in pure rolling. We call this relation as "equation of accelerated rolling" to distinguish the same with the "equation of rolling" derived earlier.

Like in the case of equation of rolling motion, this relation connects quantities, which are measured in two different references. Linear acceleration of center of mass is measured with respect to ground, whereas the angular acceleration is measured with respect to moving axis of rotation. This relation, therefore, also runs from sign syndrome. However, this should not cause concern as we shall use this relation mostly for magnitude purpose. If a particular condition (general derivation) requires to adjust the sign, then we will put a negative sign on the right hand of the equation to account for the direction.

The fact that axis of rotation is accelerated poses a serious problem with respect to application of Newton’s second law for the analysis of rotation in the accelerated frame. Note here that rotation takes place in accelerated frame of reference – not the translation, which takes place in the inertial frame of ground. Thus, application of Newton's second law for translation is valid.

We know that Newton’s second law for rotation is valid only in inertial frame of reference. However, an accelerated frame of reference can be rendered to an equivalent inertial frame of reference by applying a force (called pseudo force) at the center of mass. This pseudo force is equal to product of the mass of the rigid body and its linear acceleration. Whatever be its magnitude, the important point is that this pseudo force acts through center of mass. Since force through center of mass does not constitute torque, the angular velocity of the rotating body is not affected.

We, therefore, conclude that application of Newton's second law of rotation even in accelerated frame of reference is valid for rolling.

Laws governing rolling

Corresponding to two motion types involved with pure rolling, there are two Newton’s second laws governing the motion. One (Newton’s law second law of translation) governs the linear motion of the center of mass, whereas the other (Newton’s law second law of rotation) governs the rotational motion of the rolling disk.

For pure translation of the center of mass, the second law of Newton's law is :

Here, "∑F" denotes the resultant force and " $a_C$ " denotes the linear acceleration of the center of mass of the disk. For the sake of simplicity, we only consider rolling in one direction only and as such may avoid using vector notation.

Similarly, for pure rotation of the disk about an axis passing through the center of mass and perpendicular to its surface, the second law of Newton's law for rotation is :

where "I" denotes the moment of inertia and "α" denotes angular acceleration about the axis of rotation through the center of mass.

Analysis of rolling, therefore, is carried out in terms of Newton’s second law of motion for translation and rotation. The application of the laws, however, is conditioned by a third relation between linear and angular accelerations,

$$\begin{array}{l}{a}_C=\alpha R\end{array}$$

What it means that we should apply Newton’s second laws for rotation and translation in conjunction with the equation of accelerated rolling.

Summary

1: Uniform rolling means pure rolling with constant velocity.

2: Rolling with constant velocity means constant linear velocity of COM and constant angular velocity of rigid body about the axis of rotation.

3: There is no friction involved in uniform rolling.

4: An accelerated rolling is described by “equation of accelerated rolling” as given by :

$$\begin{array}{l}{a}_C=\alpha R\end{array}$$

5: The equation of accelerated rolling underlines the fact that there can not be selective acceleration. In other words, if there is linear acceleration, there is corresponding angular acceleration as well.

6: Governing laws of rolling motion are Newton’s second law of motion for translation and rotation. These two laws are subject to the condition imposed by equation of accelerated rolling as given above.

(i) Newton’s second law for translation

$$\begin{array}{l}a=\frac{\sum F}{M}\end{array}$$

(ii) Newton’s second law for rotation

$$\begin{array}{l}\mathrm{\alpha }=\frac{\sum \tau }{I}\end{array}$$