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I just want a clarity about work done by friction in pure rolling when external force is acting. So suppose we have a ring and an external force is applied at the centre, the ground has friction. After some time pure rolling starts such that $a = r* alpha$. After this is work done by friction still zero. Because although the point of contact is at rest but rotational speed is increasing due to angular acceleration so there should be a rotational work done by friction? Please if anyone can clarify

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If you want to write work in two parts, $$W_{tot} = \int \vec{F}\cdot d\vec{s}_{trans} + \int \vec{\tau}\cdot d\theta$$ Then yes, the part due to torque will be non zero. However, first term will be negative to make sure that the total work is zero.

To evaluate the first term, recall that we only have to consider translational motion. Accounting for translational motion of the ball and assuming constant friction force, $$dW_{trans} = - f_{fric} ds$$ Where ds is small translational displacement of the ball. Now, for the rotational work, $$dW_{rot} = f_{fric} r d\theta$$ But remember that for pure rolling, $ds=rd\theta$. So, $$dW_{rot}= f_{fric} ds$$ Thus, $$dW_{tot} = dW_{trans} + dW_{rot} = 0$$

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  • $\begingroup$ Thank You! Just one more thing, So if the lower surface was also moving. Let's say this ring was pure rolling on a moving plank, then the net work done by friction would not be zero right? because there is displacement of the plank with respect to ground! $\endgroup$ Oct 18, 2023 at 2:49
  • $\begingroup$ No, it'd still be zero. If a ring is pure rolling on a moving plank, it actually means that the ring is in pure roll with respect to the plank. So, application point of friction on the plank still has no velocity and thus 0 work. $\endgroup$ Oct 18, 2023 at 4:03
  • $\begingroup$ But with respect to the ground, the point is moving and it is getting displaced as well. With respect to the plank the point is at rest but not with respect to ground and if we calculate work done in ground frame then friction is there and there is also displacement with respect to ground !! $\endgroup$ Oct 18, 2023 at 7:43
  • $\begingroup$ The application point of the force is moving with the body as well $\endgroup$ Oct 18, 2023 at 7:59
  • $\begingroup$ Exactly the application point is moving! So then work done by friction should not be zero because it has a displacement right? $\endgroup$ Oct 18, 2023 at 12:49

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