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When finding the velocity of a rolling disc using newton's second law method we need to take into account the friction. We use $$ mg \sin(\theta) - F = ma $$ and $$ Fr = I \alpha$$ $$I = \dfrac{mr^2}{2}$$

However, if we are finding the velocity using the energy method. We use $ T = \dfrac{1}{2} mv^2 + \dfrac{1}{2} Iw^2 $ and $ V = mgh$ .

After some manipulation of either set of equations we can achieve $v = \sqrt{\dfrac{4sfg \sin\theta}{3}}$

So why is it that there is no account for friction when using the energy method?

Wasserwaage
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user154844
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We don't explicitly account for the work done by friction because friction actually does no net work. In terms of energy dissipation by friction, if there is rolling with no slipping, then there is no energy loss. However, it is the case that friction is the force that is responsible for allowing some of our potential energy to be converted into kinetic energy due to rotation. Since we are converting between types of energy with no loss in energy, then we can just consider the kinetic and potential energies. See below to see how we do account for this in the "rotational kinetic energy" term.

If you want to, you can consider the work done by friction in terms of the rotation (note, this is not the total work done by friction here, as that is $0$). The work done by a torque is given by $$W=\int\tau\ \text d\theta$$ Assuming a constant friction force, we get a constant torque, and so $$W_{f}=Fr\Delta\theta=I\alpha\Delta\theta$$

Now, under constant angular acceleration, and assuming the disk was released from rest, $\Delta\theta=\frac{\omega^2}{2\alpha}$, therefore $$W_f=\frac12I\omega^2$$

So you see, we actually do account for this work by using this "rotational kinetic energy" term.

BioPhysicist
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  • @AaronStevens hey i had just a query why does rolling friction cause the disc to gain kinetic energy without producing heat? Does only dynamic friction produce heat? – Nobody recognizeable Jan 06 '19 at 13:13
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    @Nobodyrecognzeable If you have a new question, then post a new question. The comments on an answer is not the place to ask and answer a separate question. Also, search this site before posting. I'm sure that question has been asked here before. – BioPhysicist Jan 06 '19 at 13:47
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    @Nobodyrecognizeable Some people who have answered the question before have said there is no rubbing between the surface and the disc when rolling so there is no heat produced. Not sure if this is correct though. – user154844 Jan 06 '19 at 15:30
  • @aaronstevens so by friction you are meaning staric friction right? https://physics.stackexchange.com/questions/452458/why-does-not-the-rotational-friction-produce-heat-and-waste-energy – Nobody recognizeable Jan 07 '19 at 01:47
  • @user154844 well if there is no slipping of disk then nor dynamic nor static friction should come into account. I guess you are not referring rolling friction at all. See:https://physics.stackexchange.com/questions/452458/why-does-not-the-rotational-friction-produce-heat-and-waste-energy – Nobody recognizeable Jan 07 '19 at 01:50
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    @Nobodyrecognizeable Yes I am considering static friction – BioPhysicist Jan 07 '19 at 01:58
  • @AaronStevens another query if there was dynamic friction then there would also be no loss at all as the disk isn't slipping or sliding . Is this right? – Nobody recognizeable Jan 07 '19 at 02:09
  • @Nobodyrecognizeable with dynamic friction the surfaces are sliding past each other... So yeah there would be energy loss – BioPhysicist Jan 07 '19 at 02:42
  • @AaronStevens so if the dynamic friction there then even while just rolling would cause energy loss? – Nobody recognizeable Jan 07 '19 at 03:08
  • @Nobodyrecognizeable Yes, because it would be rolling with slipping. If you have further questions please post a new question – BioPhysicist Jan 07 '19 at 03:33