A question about a wheel rolling without slipping while tied to a cube

[glen21082000]

Homework Statement

In what direction can the wheel start rolling if it is rolling without slipping

Relevant Equations

Vcm = ωR

 

[glen21082000]

I forgot to write my question, oops. Honestly I simply don't understand the given answer to the question. There is a mistake in the question and the solution - the R in the moment of the tension should be "r" instead of "R" since the string is pulled from the inner ring (look at the figure). Other than that I don't really understand what is going on in the solution and I really want to understand it so I would appreciate any form of help! TIA

 

[Gordianus]

This YouTube video may help:

 

[haruspex]

My first problem with the question is relating the description to the diagram. “The other end of the string” - two ends are already mentioned. How many ends does it have? - “is attached frictionlessly to the table” - how do you attach something frictionlessly? To a table?

Ignoring that, and correcting TR to Tr, which way is positive in the given torque equation? If f acts to the left and T to the right then the torque expression is written with clockwise positive, so a positive result means the pulley moves to the right.

Simpler, here, is to take torques about the point of contact. That way we don’t have to worry about the direction or magnitude of the friction. There is only one torque to consider, T(R-r), and it acts clockwise, so the pulley moves to the right.

Where is this question from?

 

[glen21082000]

This YouTube video may help:

Thank you very much! I have a question about the video you sent, in the video why is the friction left? Shouldn't the friction be dependent on whether the yoyo rolls to the right or to the left? If the yoyo rolls to the right then shouldn't the yoyo try to slip left at the contact point with the ground which will cause the static friction to be to the right?

Another question is about the torque calculation, is it okay using the center of the yoyo when calculating the torque or do we need to fix another point outside of the yoyo since the yoyo is non inertial.

Thanks a lot of the help!

 

[haruspex]

Thank you very much! I have a question about the video you sent, in the video why is the friction left?

Drawing it to the left merely says that we are taking left as positive for that force. All the equations derived from the diagram work perfectly well if f turns out negative, i.e. if friction turns out to act to the right.

Note that the question in this thread is for the case $\beta=0$, and in this case Prof Miller's solution is that the yoyo moves to the right (and, of course, friction is to the right).

Another question is about the torque calculation, is it okay using the center of the yoyo when calculating the torque or do we need to fix another point outside of the yoyo since the yoyo is non inertial.

In general, it is safe to use any point if its line of acceleration passes through the mass centre of the body. That's a bit complicated, so stick to either of two special cases:

• the mass centre of the body
• any point fixed in an inertial frame

With rolling contact it can be a little confusing: taking the instantaneous point of contact is ok if by that you mean a point fixed on the ground, not the locus of the point of contact, which could be accelerating.

Did you read post #4? Using torques about point of contact shows why $r_1\cos(\beta)-r_2$ is critical. If positive, the torque from T is clockwise about the point of contact, so the pulley rolls to the right.

 

[kuruman]

To @glen21082000:

Here is a question to test your understanding of how to analyze what's going on. On the right is shown a tricycle on a flat surface where it can roll without sliding. The handlebar is locked and prevented from turning away from the forward position.

You stand in front of the tricycle and perform two tasks separately and independently of each other as indicated below.

1. Tie a string on the pedal above the axle of the front wheel and pull forward and parallel to the ground as indicated by the yellow arrow.
2. Tie a string on the pedal below the axle of the front wheel and pull forward and parallel to the ground as indicated by the red arrow.

In which direction, forward or backward, will the tricycle move in each case and why?

Hint
One might argue thus: I know that if I ride the tricycle and push forward with my foot against the top pedal, I will move forward and if I push forward against the bottom pedal, I will move in reverse. A force is a force. Therefore in (1) the tricycle will move forward and in (2) the tricycle will move in reverse.

This argument is specious.

 

[haruspex]

To @glen21082000:
View attachment 361984Here is a question to test your understanding of how to analyze what's going on. On the right is shown a tricycle on a flat surface where it can roll without sliding. The handlebar is locked and prevented from turning away from the forward position.

You stand in front of the tricycle and perform two tasks separately and independently of each other as indicated below.

1. Tie a string on the pedal above the axle of the front wheel and pull forward and parallel to the ground as indicated by the yellow arrow.
2. Tie a string on the pedal below the axle of the front wheel and pull forward and parallel to the ground as indicated by the red arrow.

In which direction, forward or backward, will the tricycle move in each case and why?

Hint
One might argue thus: I know that if I ride the tricycle and push forward with my foot against the top pedal, I will move forward and if I push forward against the bottom pedal, I will move in reverse. A force is a force. Therefore in (1) the tricycle will move forward and in (2) the tricycle will move in reverse.

This argument is specious.

I see no evidence in post #2 that @glen21082000 does not already understand. He correctly identifies several errors in the given solution, not least that it arrives at the wrong answer.

 

[kuruman]

I see no evidence in post #2 that @glen21082000 does not already understand. He correctly identifies several errors in the given solution, not least that it arrives at the wrong answer.

I agree. However, by OP's own admission

Other than that I don't really understand what is going on in the solution and I really want to understand it so I would appreciate any form of help! TIA

OP wants to understand how the solution was crafted. My explanation is that the solution's author seems to be under the impression that the way to figure out which way the pulley will roll is to first imagine which way it would rotate about a fixed axle (clockwise or counterclockwise) when the string is pulled and then conclude that the center will move to the right if clockwise and to the left if counterclockwise. The tricycle question is a more subtle variation of this and I wanted to ensure that @glen21082000 really "got it."

 

[haruspex]

I agree. However, by OP's own admission

OP wants to understand how the solution was crafted. My explanation is that the solution's author seems to be under the impression that the way to figure out which way the pulley will roll is to first imagine which way it would rotate about a fixed axle (clockwise or counterclockwise) when the string is pulled and then conclude that the center will move to the right if clockwise and to the left if counterclockwise. The tricycle question is a more subtle variation of this and I wanted to ensure that @glen21082000 really "got it."

Ok, but that’s not how I read the logic of the given solution. Rather, I read it (correcting the R/r typo) as:

1. Suppose it moves left.
2. [missing statement: so we'll define f as positive left and T as positive right]
3. It follows from F=ma that f>T.
4. Since R>r, f>T implies fR>Tr.
5. fR>Tr implies a rotation consistent with the pulley moving left
6. The consistency implies our assumption (1) is correct.

Trouble is, 5 is false. Had that step been performed correctly we would have got the right answer.
Also, step 6 is a bit hokey. Merely observing that no contradiction arises from the tests we happened to apply is hardly rigorous.
Besides, it is a poor approach because had we happened to start with supposing it moves right, so f<T, we would not have been able to draw any inference on the comparison between fR and Tr.

 

[kuruman]

Trouble is, 5 is false. Had that step been performed correctly we would have got the right answer.

I would say, "Had the correct conclusion been drawn from step 4, we would have found a direct contradiction to the assumption in step 1." In other words, provide a reductio ad absurdum proof.

 

[haruspex]

I would say, "Had the correct conclusion been drawn from step 4, we would have found a direct contradiction to the assumption in step 1." In other words, provide a reductio ad absurdum proof.

I believe that is the same as I wrote, except that it is a RAD disproof of what the author claims.