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disk

I'm trying to find the friction coefficient that makes the body roll without slipping but I just can't reach a value. The force is applied on a small central disk of radius $r=0,03\, m$ and mass $m=0,05\, kg$. Each big disk has a radius $r=0,05\, m$ and mass $m=0,01\, kg$. That's all the information that is given. My first attempt was to write all the equations that would be useful:

$$F=MA_{cm},\quad T=Ia,\quad V_{cm}=RW.$$

I understand each of the equations but as I substitute $F= 0,1-Fa$ and $T=FaR-0,1r$, I just can't work out the friction force. My question is am I supposed to try the forces approach or the energy approach, since the total energy is conserved? I've tried both approaches but it seems like I am forgetting something. I've been trying to find solutions but most of the problems involve inclined planes, which is different. Thank you very much!

lucas
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George Sailor
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  • Is it possible for you to post the whole question, exactly as written in the book you are studying? – sammy gerbil Jun 26 '16 at 11:05
  • Of course!

    a. What is the direction of the motion;

    b. What is the acceleration of the COM;

    c.What is the minimum coefficient of friction to ensure that the disks don't slip.

     – George Sailor Jun 26 '16 at 19:50
  • Sorry, I meant the whole of the conditions of the problem - especially how F is applied. Although, I notice you say "that is all the information that is given." Nothing else which is relevant? – sammy gerbil Jun 27 '16 at 00:25
  • Nothing else is given, I am supposed to find everything based only on this information... – George Sailor Jun 28 '16 at 15:32

2 Answers2

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Transmit the force $F$ to the center of mass and add its torque. Consider to the relative motion between disc and ground. Then you can recognize the correct direction of friction force (friction force opposes relative motion). Free body diagram of disc is as below. (See this answer to understand better.)

enter image description here

Equations of motion are: $$F-F_f=ma$$ $$N=mg$$ $$F_f(0.05)-T=I\alpha$$ $$a=0.05\alpha$$

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lucas
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  • Alright, but if I transmit the force to the center of mass as you suggest, then the torque exerted by that force will be zero because it is applied at the axis of rotation, and then you will have to remove it from the net torque equation, isn't it? Well I understand that the friction force opposes the motion, because otherwise the Vcm due to the translational motion wouldn't be cancelled out at the contact point – George Sailor Jun 25 '16 at 20:22
  • $T$ is the torque due to force $F$ before transmitting. When you want to transmit a force, you must add its torque with respect to the point that is transmitted to. In other words, both of systems must be equal with respect to equation motions. – lucas Jun 25 '16 at 20:29
  • You can solve the problem without transmitting the force. Just consider that we have: $$\Sigma F_x=ma$$ $$\Sigma F_y=0$$ $$\Sigma T_{\textrm{cm}}=I_{\textrm{cm}}\alpha$$ $$a=r\alpha$$ – lucas Jun 25 '16 at 20:34
  • Oh alright, so what is it used for? We already know that the point where the external force is applied doesn't matter for the Acm... Oh and I got a negative value, Ff=-0,69, so it means the friction force is pointing in the same direction as the applied force! – George Sailor Jun 26 '16 at 08:18
  • For determining direction of friction force, you must answer this question: "Which relative motion between two surfaces impeding to occur and what forces or torques cause that?" In your case, the cylinder tends to move in direction of force $F$ and rotate clockwise. Hence, friction force will act in a direction that opposes the motions. I.e. Friction force opposes both translation and rotation of the cylinder due to $F$ – lucas Jun 26 '16 at 08:28
  • @GeorgeSailor : If the disk is rolling without slipping, I think $F_f$ must be +ve, ie point to the right. It would help if you post your calculation of Ff=-0,69 at the end of your question. – sammy gerbil Jun 28 '16 at 17:01
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You say "rolling without slipping." But what do you think will happen in this situation? Do you think the large disk will "roll without slipping"? Can you do an experiment at home to find out?

You can use whatever method gets you to the correct answer. You are given forces (F, weight) so try using forces. Try solving the equations lucas gives you.

UPDATE

Sorry, I have made a mistake here. I was suggesting that it is impossible for the disk to roll without slipping. This is incorrect : it can do so.

I was assuming that F is applied via a string wrapped around the inner disk, and that this string does not slip. In that case it would be impossible for the large disk to move to the left without slipping against the ground, unless the string were wound in the opposite direction so that F is applied above the CM of the disks.

However, there is nothing in the question to say that the string does not slip against the inner disk. Alternatively F could be applied in some other way, not using a string. In this case the outer disk can roll to the left without slipping against the ground.

sammy gerbil
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