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Pure rolling is governed by Newton's laws as applicable to pure translation and pure rotation under the condition of rolling.

Essentials of pure rolling motion are no different than that of pure translational and rotational motions, except that these two basic forms of motions occur simultaneously. A clear understanding of the two basic motion forms, therefore, is a perquisite for a clear understanding of pure rolling motion (referred simply as rolling also).

There are two distinct framework associated with the study of rolling motion :

  • Uniform rolling
  • Accelerated rolling

Independence of analysis

Rolling, being combination of translation and rotation, involves two “causes”, which might change its velocity. Two causes act to produce “effects” independently, but in tandem to satisfy the condition of rolling (we shall subsequently derive this condition in the module).

A net force causes acceleration of the center of mass of the rigid body. A rolling motion involves rigid body of finite size and, therefore, its translation should always be referred to the center of mass. Further, when we consider the effect of force, we treat translation as if the rigid body were not rotating at all.

Independence of analysis

Forces are analyzed as if rigid body were not rotating at all.
Torques are analyzed as if rigid body were not translating at all.

Similarly, a net torque causes rotational acceleration of the rigid body about its central axis passing through center of mass. When we consider the effect of torque, we treat rotation as if the axis of rotation were not translating at all.

In simple words, the analysis of rolling can be done independently for two motions types as if other motion did not exist. This independence of analysis of motion allows us to apply the familiar laws of motion for analyzing each motion types. We are required only to combine the results to describe rolling motion.

Force and torque

Treatment of force with respect to a rigid body capable of both translation and rotation is different than the case when only one type of motion is involved (i.e. not the combination). In pure translation along a straight line, the rigid body is constrained (or otherwise) not to rotate; similarly in pure rotation about a fixed axis, the rigid body is constrained not to translate.

A force, whose line of action passes through center of mass, is capable to produce only translational acceleration ( a C ). A force, whose line of action does not pass through center of mass, works as “force” to produce translational acceleration ( a C ) and simultaneously as “torque” to produce angular acceleration (α).

Force and torque

A force through the COM only produces linear acceleration.
A force not through the COM produces both linear and angular accelerations.

Since there may be multiple effects (more than one) of a single force, it is always desirable to clearly understand the roles of the forces operating on the rolling body to accurately analyze its motion.

Rolling and newton’s first law

A pure rolling is equivalent to pure translation and pure rotation. It, therefore, follows that a uniform rolling (i.e. rolling with constant velocity) is equivalent to uniform translation (constant linear velocity) and uniform rotation (constant angular velocity).

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Source:  OpenStax, Physics for k-12. OpenStax CNX. Sep 07, 2009 Download for free at http://cnx.org/content/col10322/1.175
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