Cylinder lying on conveyor belt

[JD_PM]

Right, but you need another equation. Express the fact of rolling contact in terms of the radius, the rotation rate and the two velocities.

You mean:

$$v_{CM} + V(t) = R \omega$$

 

[haruspex]

You mean:

$$v_{CM} + V(t) = R \omega$$

Since both those linear velocities are in the lab frame, and I assume are positive in the same direction, adding them looks wrong.

 

[JD_PM]

Since both those linear velocities are in the lab frame, and I assume are positive in the same direction, adding them looks wrong.

My bad, regarding left as positive, we would have:

$$v_{CM} - V(t) = R \omega$$

Together with the equation obtained by conservation of AM:

$$v_{CM} = \frac{I_{CM} \omega}{RM}$$

Getting as the final answer:

$$v_{CM} = \frac{1}{2}[(\frac{I_{CM}}{R^2M} + 1) R\omega + V(t)]$$

 

[haruspex]

My bad, regarding left as positive, we would have:

$$v_{CM} - V(t) = R \omega$$

Together with the equation obtained by conservation of AM:

$$v_{CM} = \frac{I_{CM} \omega}{RM}$$

Getting as the final answer:

$$v_{CM} = \frac{1}{2}[(\frac{I_{CM}}{R^2M} + 1) R\omega + V(t)]$$

Two problems there.
Your first two equations are not handling signs consistently. If velocities are positive to the left then for your first equation to be right anticlockwise must be positive for the rotation. The second equation was obtained using a different pair of definitions.

Secondly, you are asked to express v_CM in terms of k (where I=Mk²), R, M and V(t). ω should not feature.

 

[JD_PM]

Two problems there.
Your first two equations are not handling signs consistently. If velocities are positive to the left then for your first equation to be right anticlockwise must be positive for the rotation. The second equation was obtained using a different pair of definitions.

Secondly, you are asked to express v_CM in terms of k (where I=Mk²), R, M and V(t). ω should not feature.

OK I redid the problem from scratch:

From second (translation) Newton's law:

$$a_o= \frac{f}{M}$$

From second (rotation) Newton's law:

$$\tau = I \alpha = fa$$

$$k^2 \alpha= a_o a$$

$$\alpha = \frac{a_o a}{k^2}$$

Where $a_o$ is the acceleration of the cylinder measured from the ground.

Assuming that the acceleration of the belt is constant ($a_b$):

$$ \frac{V}{t} = a_b$$

The acceleration of the cylinder with respect to the ground accounts for the acceleration of the belt and the angular acceleration of the cylinder (which has a negative sign because I considered the cylinder spinning clockwise i.e. the belt accelerating to the left and regarding that direction as the positive one). Therefore:

$$a_o = a_b - a\alpha$$

$$a_o = \frac{V}{t} -a\frac{a_o a}{k^2}$$

$$a_o = \frac{Vk^2}{(k^2 + a^2)t}$$

We know by kinematics that:

$$v_{cm} = \frac{Vk^2}{k^2 + a^2}$$

The issue here is that I had to assume the acceleration of the belt being constant so as to solve the problem. The answer makes sense to me but I would like to know how to solve it without assuming that $a_b = const$. (Recall that @kuruman suggested on #2 comment that we should not assume it).

 

[haruspex]

OK I redid the problem from scratch:

From second (translation) Newton's law:

$$a_o= \frac{f}{M}$$

From second (rotation) Newton's law:

$$\tau = I \alpha = fa$$

$$k^2 \alpha= a_o a$$

$$\alpha = \frac{a_o a}{k^2}$$

Where $a_o$ is the acceleration of the cylinder measured from the ground.

Assuming that the acceleration of the belt is constant ($a_b$):

$$ \frac{V}{t} = a_b$$

The acceleration of the cylinder with respect to the ground accounts for the acceleration of the belt and the angular acceleration of the cylinder (which has a negative sign because I considered the cylinder spinning clockwise i.e. the belt accelerating to the left and regarding that direction as the positive one). Therefore:

$$a_o = a_b - a\alpha$$

$$a_o = \frac{V}{t} -a\frac{a_o a}{k^2}$$

$$a_o = \frac{Vk^2}{(k^2 + a^2)t}$$

We know by kinematics that:

$$v_{cm} = \frac{Vk^2}{k^2 + a^2}$$

The issue here is that I had to assume the acceleration of the belt being constant so as to solve the problem. The answer makes sense to me but I would like to know how to solve it without assuming that $a_b = const$. (Recall that @kuruman suggested on #2 comment that we should not assume it).

To avoid assuming constant acceleration you will need to avoid involving acceleration at all.
You were nearly there with post #63. You just need to get your use of signs consistent and express the answer in terms of the variables given in the problem statement.

 

[JD_PM]

To avoid assuming constant acceleration you will need to avoid involving acceleration at all.
You were nearly there with post #63. You just need to get your use of signs consistent and express the answer in terms of the variables given in the problem statement.

OK, Fixing signs and $k$ issues one gets:

$$v_{CM} = V(t) - R \omega$$

$$v_{CM} = - \frac{k^2 \omega}{R}$$

$$v_{CM} = \frac{1}{2}[-(\frac{k^2}{R^2} + 1) R\omega + V(t)]$$

The only thing still disturbs me is that $\omega$ is still there. I am thinking now

OK It is just about using $v_{CM} = \omega R$ again. Actually doing so I get a pretty nice solution for $v_{CM}$:

$$v_{CM} = \frac{V(t)}{\frac{k^2}{R^2} + 3}$$

In terms of dimensions makes sense because k has dimensions of length, then the right hand side of the equation has dimensions of velocity.

WoW it has been a long journey but I enjoyed it a lot! :)

 

[kuruman]

What's counter-intuitive about this problem is that if the belt brakes suddenly, the cylinder will stop moving just as suddenly.

 

[JD_PM]

What's counter-intuitive about this problem is that if the belt brakes suddenly, the cylinder will stop moving just as suddenly.

Mmm true but I think that is due to not considering an initial velocity on the cylinder. If we were to consider one different from zero, we would get an extra term that would make $v_{CM}$ different from zero even when $V(t) = 0$

 

[haruspex]

it has been a long journey

And not quite over. The answer you got in post #65 was correct, but it assumed constant acceleration. You need to get the same answer without that assumption.

In post #63, the signs on ω in the first two equations were inconsistent. In post #67 you have changed the sign on both occurrences of ω, so they are still inconsistent.

Once you have those two equations right, you don't need the third equation in those two posts. In fact, I am not sure how you arrived at it. Just eliminate ω between the first two.

 

[kuruman]

Mmm true but I think that is due to not considering an initial velocity on the cylinder. If we were to consider one different from zero, we would get an extra term that would make $v_{CM}$ different from zero even when $V(t) = 0$

The problem specifies that the bottle (cylinder) and the belt are initially at rest.

 

[JD_PM]

And not quite over. The answer you got in post #65 was correct, but it assumed constant acceleration. You need to get the same answer without that assumption.

In post #63, the signs on ω in the first two equations were inconsistent. In post #67 you have changed the sign on both occurrences of ω, so they are still inconsistent.

Once you have those two equations right, you don't need the third equation in those two posts. In fact, I am not sure how you arrived at it. Just eliminate ω between the first two.

Yeah the following equation is wrong (algebraic mistake):

$$v_{CM} = \frac{1}{2}[-(\frac{k^2}{R^2} + 1) R\omega + V(t)]$$

Then, combining the following two equations so as to solve for $v_{CM}$ one gets the same result than #65:

$$v_{CM} = V(t) - R \omega$$

$$v_{CM} = - \frac{k^2 \omega}{R}$$

$$v_{cm} = \frac{Vk^2}{k^2 + R^2}$$

 

[haruspex]

Yeah the following equation is wrong (algebraic mistake):

$$v_{CM} = \frac{1}{2}[-(\frac{k^2}{R^2} + 1) R\omega + V(t)]$$

Then, combining the following two equations so as to solve for $v_{CM}$ one gets the same result than #65:

$$v_{CM} = V(t) - R \omega$$

$$v_{CM} = - \frac{k^2 \omega}{R}$$

$$v_{cm} = \frac{Vk^2}{k^2 + R^2}$$

Now you have arrived.

 

[JD_PM]

Now you have arrived.

The problem has been solved.

However I am still interested in the approach @kuruman suggested on 36#; solving the problem from a non-inertial reference frame. I guess this approach is based on the second Newton's Law for rotation:

$$f - F = I \alpha$$

Where $f$ is the friction and $F$ is a fictitious force.

 

[kuruman]

It's not much different from what you did. The acceleration of the belt is $\frac{dV}{dt}$. In the non-inertial frame it manifests itself as a fictitious force $ma$ acting on the CM in the opposite direction as the acceleration of the belt. The torque equation about the point of contact on the belt gives $MaR=(Mk^2+MR^2)\alpha$ (note the use of he parallel axis theorem). Let $a'_{cm}$ be the acceleration of the CM in the non-inertial frame. Then $a'_{cm}=\alpha/R$. Put this back in the torque equation, cancel the masses and get $$a'_{cm}=\frac{R^2}{k^2+R^2}a=\frac{R^2}{k^2+R^2}\left(-\frac{dV}{dt}\right).$$ The negative sign is introduced because $a'_{cm}$ and $\frac{dV}{dt}$ are in opposite directions. In the inertial frame, the acceleration of the CM is $$a_{cm}=a+a'_{cm}=\frac{dV}{dt}-\frac{R^2}{k^2+R^2}\frac{dV}{dt} =\frac{k^2}{k^2+R^2}\frac{dV}{dt} $$The velocity of the CM is$$v_{cm}(t)=\int a_{cm}dt=\frac{k^2}{k^2+R^2}\int \frac{dV}{dt}dt=\frac{k^2}{k^2+R^2}V(t).$$I am not convinced that this is a better way to approach the problem however.

 

[haruspex]

It's not much different from what you did. The acceleration of the belt is $\frac{dV}{dt}$. In the non-inertial frame it manifests itself as a fictitious force $ma$ acting on the CM in the opposite direction as the acceleration of the belt. The torque equation about the point of contact on the belt gives $MaR=(Mk^2+MR^2)\alpha$ (note the use of he parallel axis theorem). Let $a'_{cm}$ be the acceleration of the CM in the non-inertial frame. Then $a'_{cm}=\alpha/R$. Put this back in the torque equation, cancel the masses and get $$a'_{cm}=\frac{R^2}{k^2+R^2}a=\frac{R^2}{k^2+R^2}\left(-\frac{dV}{dt}\right).$$ The negative sign is introduced because $a'_{cm}$ and $\frac{dV}{dt}$ are in opposite directions. In the inertial frame, the acceleration of the CM is $$a_{cm}=a+a'_{cm}=\frac{dV}{dt}-\frac{R^2}{k^2+R^2}\frac{dV}{dt} =\frac{k^2}{k^2+R^2}\frac{dV}{dt} $$The velocity of the CM is$$v_{cm}(t)=\int a_{cm}dt=\frac{k^2}{k^2+R^2}\int \frac{dV}{dt}dt=\frac{k^2}{k^2+R^2}V(t).$$I am not convinced that this is a better way to approach the problem however.

Reading that made me realize that in non-inertial frames it is valid to deal with fictitious changes in momentum. If the frame has acceleration a=a(t) over an interval then conservation of momentum works by adding the "fictitious" term mΔv=m∫a.dt.
Something similar may be possible for angular momentum, and if such a rule were taken as standard you could avoid the integration step above.

 

[JD_PM]

The acceleration of the belt is $\frac{dV}{dt}$.

Aren't you assuming constant acceleration with this statement?

 

[haruspex]

Aren't you assuming constant acceleration with this statement?

No, that is a pefectly general equation.

 

[kuruman]

Reading that made me realize that in non-inertial frames it is valid to deal with fictitious changes in momentum. If the frame has acceleration a=a(t) over an interval then conservation of momentum works by adding the "fictitious" term mΔv=m∫a.dt.
Something similar may be possible for angular momentum, and if such a rule were taken as standard you could avoid the integration step above.

Yes, having to do the integral is exactly why I am not convinced that my method is a better way to approach the problem. It has the advantage that it is based on a "snapshot" FBD and does not require justifying conservation of momentum over an interval Δt but, overall, I think it is lengthier.