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For pure rolling motion, at the point of contact of the wheel at the ground the net velocity is 0, so there is no relative motion. But if at that instant no relative motion, then how can static friction act? And if static friction does not act the net torque will also be zero, so how can the body continue the rolling motion ?

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3 Answers 3

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But if at that instant no motion is there then how can static friction act ?

Static friction is defined as the friction present when the surfaces are not sliding past each other. For example, if you have a book at rest on a table that you are applying a force to, the static friction force is what is counteracting your force to keep the book at rest.

then how can the body continue the rolling motion ?

Static friction is not required to keep a body rolling. Static friction would only come into play if some other force/torque was attempting to change the angular velocity of the body. Again, the book example is a good one. Static friction is not required to keep the book at rest; the book will sit on the table if nothing else pushes on it. Only when you apply a force to the book will static friction come into play.

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And if static friction does not act the net torque will also be zero , then how can the body continue the rolling motion ?

Net torque is not required for pure rolling (rolling without slipping) to continue at constant angular velocity. If there is no net torque acting on the body, static friction is neither present nor needed to maintain pure rolling. Pure rolling will continue due to the rotational inertia of the body. Per Newton's 1st law for rotation (assuming no change in mass or its distribution):

An object at rest tends to remain at rest, and an object that is spinning tends to spin with a constant angular velocity, unless it is acted on by a nonzero net torque.

However, in order to initiate pure rolling, the body must be given angular acceleration. That requires a nonzero net torque plus static friction to prevent slipping.

Hope this helps.

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  • $\begingroup$ Thank you ! But what about the case when rolling motion occurs with angular acceleration and linear acceleration. For eg; if a an external force acts parallel to the horizontal surface ? in my book there is statement for the same ,as follow "Since there is no sliding therefore the component parallel to surfce msut be zero. However it has an acc. component towards the centre. The centre always moves parallel to the surface and does not change direction of its velocity, therefore its acceleration can only be parallel to surface" $\endgroup$ Jan 5, 2023 at 18:17
  • $\begingroup$ I don't understand the quoted material. Perhaps it needs more context. $\endgroup$
    – Bob D
    Jan 5, 2023 at 18:34
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Assume pure rolling (no slip) of a rigid body. For a rigid body there are no heating effects. For pure rolling the force of static friction does no work since there is zero velocity of the body at the point of contact where friction acts.

For pure rolling in a straight line on a level surface at constant velocity with no external forces applied, the force of static friction is zero, since the center of mass (CM) moves at constant velocity hence there can be no net external force. The body has constant linear velocity of the CM and constant rotational velocity with respect to the CM.

For pure rolling down an incline, the component of force of gravity down the plane increases the kinetic energy of the CM. The force of static friction decreases the kinetic energy of the CM, but it also provides a torque to increase the kinetic energy of rotation of the body about the CM and the net work from both of these is zero with no slip.

See Is work done by torque due to friction in pure rolling? and related questions on this exchange.

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  • $\begingroup$ Thank you ! But what about the case when pure rolling motion occurs with angular acceleration and linear acceleration. How is that even possible ? in my book there is statement for the same ,as follow "Since there is no sliding therefore the component parallel to surface be zero. However it has an acc. component towards the centre. The centre always moves parallel to the surface and does not change direction of its velocity, therefore its acceleration can only be parallel to surface" Can you please throw some light ? $\endgroup$ Jan 5, 2023 at 18:26
  • $\begingroup$ Please clarify the conditions: rolling in the horizontal plane in a circle, general rolling motion not in a circle not in the horizontal plane, any other forces besides friction and gravity, ...? $\endgroup$
    – John Darby
    Jan 5, 2023 at 18:50
  • $\begingroup$ When a body is in pure rolling motion, on a horizontal plane surface in translational direction but it has acceleration (the centre of mass of wheel moves with acceleration and angular acceleration on the wheel) How can pure rolling be possible in such a case ? $\endgroup$ Jan 6, 2023 at 14:11
  • $\begingroup$ I really need a picture, but I will assume rolling in a circle on a horizontal surface with constant speed. With pure rolling, in the direction of the roll there is no friction. In the direction toward the center of the circle a force of static friction provides the centripetal force for the center of mass moving in a circle. Gravity provides a torque and turns the rolling object (like a rolling bicycle leaning and turning); the torque from gravity changes the existing angular momentum of the rolling body. If this is not the case, please send a picture. $\endgroup$
    – John Darby
    Jan 6, 2023 at 14:45

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