# Rotation and varying friction coefficient

(see attachment)

## The Attempt at a Solution

I am not able to understand the question and build a scenario in my mind. The question asks the distance travelled when the cylinder starts slipping. I can't think of the situation when the cylinder "slips". I am not sure which equations to start with and i suppose this question involves some integration too.

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Doc Al
Mentor
How much friction force is required to prevent slipping?

How much friction force is required to prevent slipping?
I am not sure but do i have to make the equations for torque and the forces to find the frictional force?

Doc Al
Mentor
I am not sure but do i have to make the equations for torque and the forces to find the frictional force?
Yes. Apply Newton's 2nd law for rotation and translation.

Yes. Apply Newton's 2nd law for rotation and translation.
Let the mass of cylinder as M, radius R, friction force f, α as angular acceleration, θ as angle of inclination, and a as the linear acceleration, we have
$$fR=\frac{MR^2}{2}α$$
Since, the cylinder does not slip, the equation α=a/R is applicable, so
$$f=Ma$$
Now,
$$Mg\sin(\theta)-f=Ma$$
$$Mg\sin(\theta)=2Ma$$
$$a=\frac{g\sin(\theta)}{2}$$
Therefore, the friction force required to prevent slipping is
$$f=\frac{Mg\sin(\theta)}{2}$$
Is this correct?

Doc Al
Mentor
Let the mass of cylinder as M, radius R, friction force f, α as angular acceleration, θ as angle of inclination, and a as the linear acceleration, we have
$$fR=\frac{MR^2}{2}α$$
Good.
Since, the cylinder does not slip, the equation α=a/R is applicable, so
$$f=Ma$$
Redo that step and all that follows.

But you're on the right track. Once you have the correct expression for the friction force, ask yourself what minimum value of μ is required to provide that force.

Redo that step and all that follows.
Oops, made a small mistake there. The f would be
$$f=\frac{Ma}{2}$$
Solving using the same method as before, i get
$$f=\frac{Mg\sin(\theta)}{3}$$

Doc Al said:
...ask yourself what minimum value of μ is required to provide that force.
I still can't get the meaning of the problem. Doesn't the cylinder starts slipping the instant it is released or does "slipping" have a different meaning here? Doc Al
Mentor
Oops, made a small mistake there. The f would be
$$f=\frac{Ma}{2}$$
Solving using the same method as before, i get
$$f=\frac{Mg\sin(\theta)}{3}$$
Good.

I still can't get the meaning of the problem. Doesn't the cylinder starts slipping the instant it is released or does "slipping" have a different meaning here?
As long as the surfaces are capable of providing the needed friction force (which you have just calculated), there will be no slipping. But on this surface, the value of μ (and thus the maximum available static friction) decreases with distance down the incline. At some point the surfaces will not be able to provide the needed friction and the cylinder will begin slipping. Find that point.

ehild
Homework Helper
Oops, made a small mistake there. The f would be
$$f=\frac{Ma}{2}$$
Solving using the same method as before, i get
$$f=\frac{Mg\sin(\theta)}{3}$$

I still can't get the meaning of the problem. Doesn't the cylinder starts slipping the instant it is released or does "slipping" have a different meaning here? The cylinder starts rolling when released if the static friction is enough. Remember the force of static friction ≤ μ FN (FN is the normal force).

ehild

As long as the surfaces are capable of providing the needed friction force (which you have just calculated), there will be no slipping. But on this surface, the value of μ (and thus the maximum available static friction) decreases with distance down the incline. At some point the surfaces will not be able to provide the needed friction and the cylinder will begin slipping. Find that point.
The cylinder starts rolling when released if the static friction is enough. Remember the force of static friction ≤ μ FN (FN is the normal force).

ehild
Thanks you both for the explanation, i have got the right answer. To find that point, i equate the friction force provided by the surface to the friction force required to prevent slipping.
$$μN=\frac{Mg\sin(\theta)}{3}$$
Here N is the normal reaction due to the inclined plane and is equal to Mgcos(θ).
$$\frac{2-3x}{\sqrt{3}}Mg\cos(\theta)=\frac{Mg\sin(\theta)}{3}$$
Solving, i get
$$x=\frac{1}{3}$$

Thanks once again!

Doc Al
Mentor
Good! Here is my attempt at the problem and i am getting an incorrect result ...Kindly check my work and let me know where am i making mistake

Since at the point of slipping maximum static friction acts

$f = μN$

where $N = Mgcosθ$

Thus $f =μMgcosθ$

but at the same time ,

$f = \frac{Ma}{2}$

So, $\frac{Ma}{2}=μMgcosθ$
$a=2μgcosθ$

now θ =60°

$a=μg$
$μ =\frac{2-3x}{\sqrt3}$

$a=(\frac{2-3x}{\sqrt3})g$

Writing $a=v\frac{dv}{dx}$

$v\frac{dv}{dx}=(\frac{2-3x}{\sqrt3})g$

$vdv={(\frac{2-3x}{\sqrt3})g}dx$

$\int_{0}^{v}vdv=\int_{0}^{x}{(\frac{2-3x}{\sqrt3})g}dx$

$\frac{v^2}{2}=\frac{2}{\sqrt3}gx - \frac{3}{2\sqrt3}g{x^2}$

Hence ,$v^2=\frac{4}{\sqrt3}gx - \frac{3}{\sqrt3}g{x^2} (1)$

Now, we will apply Conservation of energy

When the cylinder has moved by a distance x along the incline ,then

Loss in potential energy =Gain in Kinetic energy

Since rolling without slipping occurs,we can put v=ωR and $I = \frac{MR^2}{2}$

$mgxsinθ = \frac{1}{2}Mv^2 + \frac{1}{2}Iω^2$

Solving ,we get $v^2=\frac{2}{\sqrt3}gx (2)$

Equating (1) and (2) we get x=2/3 which is incorrect.

Where am i getting it wrong???

ehild
Homework Helper
You can not apply conservation of energy.

ehild

You can not apply conservation of energy.

ehild
why cant we apply conservation of energy?? friction is not doing any work since the cylinder rolls without slipping ...energy is definitely conserved till the point cylinder starts to slip

ehild
Homework Helper
Well, yes, you are right... The energy is conserved.

You used the maximum static friction to calculate the speed. a=2Fs/m, but Fs is not μgsosθ during rolling. It reaches that value only at the end when the cylinder starts to slip.
You can get the acceleration a from the kinematic equations for translation of the CM and rotation about the CM.

ehild

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Well, yes, you are right... The energy is conserved.

You used the maximum static friction to calculate the speed. a=2Fs/m, but Fs is not μgsosθ during rolling. It reaches that value only at the end when the cylinder starts to slip.
You can get the acceleration a from the kinematic equations for translation of the CM and rotation about the CM.

ehild
Okay...Then what should I equate 'a' (accelerataion) with ? Kindly let me know how then will i be able to calculate the speed 'v' at distance 'x' .I have showed you the complete working ...

ehild
Homework Helper
You do not need v really. But if you want it, find the acceleration first and integrate. What equations do you have for the motion of the cylinder?

ehild

For translation motion
$N=Mgcosθ$
$Mgsinθ - f = Ma$

For rotational motion

$fR = Iα$

For rolling without slipping $α = \frac{a}{R}$

Putting these values , we get

$a=\frac{2}{3}gsinθ$

Integrating , we get $v^2=\frac{2}{\sqrt3}gx$ which is the same result which we get from applying conservation of energy

What then is the condition mathematically , which we can apply in finding when the cylinder starts to slip ?

ehild
Homework Helper
Well, a can be determined from conservation of energy. You know a, then you can find the force of static friction. What is it?

ehild

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ehild said:
Well, a can be determined from conservation of energy.
How can we find a from conservation of energy ???

ehild said:
You know a, then you can find the force of static friction. What is it?
Okay...then we are on the same lines as what has been done earlier by Pranav...i.e equating $\frac{Mgsinθ}{3} =μN$ .Isnt it??

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ehild
Homework Helper
yes, it is. At the brink of slipping, Fs=μN.

As for acceleration, you can find v from conservation of energy, and then you can differentiate... A bit complicated, and has no sense to do it now. The velocity was not needed at all.

ehild