Finding out the rotational speed of a mass

[nazmulhasanshipon]

Homework Statement

If a $1.5$ kg mass is rotating at uniform speed on a table and another $2$ kg mass is connected with a string which goes through the hole on the table at the center of rotation of the $1.5$ kg mass making the $2$ kg mass hanging then what should be the speed of rotation of the $1.5$ kg mass to keep the $2$ kg mass stationary?

Coefficient of sliding friction of the $1.5$ kg mass is $μ_k=0.2$ and the string is inextensible.

Relevant Equations

Newtonian Mechanics

**My Attempt:** Here, I considered for 2 kg mass the resultant motion is zero, which means it's acceleration (a) would be zero. So, if we consider the Tension force from the 1.5 kg mass to keep 2 kg mass from falling is L, then 2×9.8-L=ma=2× 0 ⇒ L=19.6 N. But where does the tension force from 1.5 kg comes from? I thought it's from the centrifugal (or centripetal) force. But should I take into account friction now? If the friction is f_k then can I write this $\frac{mv^2}{r}-f_k-L=0$ in the next step to find out speed of the 1.5 kg mass?

 

[Delta2]

From what I know sliding friction acts in direction opposite to velocity, and since the velocity has tangential direction, sliding friction has tangential direction too. So the correct equation for me is $$\frac{mv^2}{r}-L=0$$, sliding friction has only tangential component, not radial component.

 

[nazmulhasanshipon]

Thank you so much. Another question of mine is: If the speed of 1.5kg mass decrease due to that friction, then how can I draw the effective Velocity vs. Time graph of the 2kg mass?

Thank you so much. Another question of mine is: If the speed of 1.5kg mass decrease due to that friction, then how can I draw the effective Velocity vs. Time graph of the 2kg mass?

 

[Delta2]

Thank you so much. Another question of mine is: If the speed of 1.5kg mass decrease due to that friction, then how can I draw the effective Velocity vs. Time graph of the 2kg mass?

I am afraid then the problem becomes quite more complex, the 2kg mass will do linear motion with non constant acceleration, and I am not sure at all what sort of motion the 1.5kg mass will do (I think its velocity and hence friction will have a normal as well as tangential component). I 'll invite some other users to this thread maybe they will be able to offer us their lights, @ergospherical ,@haruspex ,@TSny .

 

[nazmulhasanshipon]

I am afraid then the problem becomes quite more complex, the 2kg mass will do linear motion with non constant acceleration, and I am not sure at all what sort of motion the 1.5kg mass will do (I think its velocity and hence friction will have a normal as well as tangential component). I 'll invite some other users to this thread maybe they will be able to offer us their lights, @ergospherical ,@haruspex ,@TSny .

Thanks, remember the Coefficient of sliding friction of the 1.5 kg mass is $μ_k=0.2$ and the string is inflexible.

 

[jbriggs444]

Thanks, remember the Coefficient of sliding friction of the 1.5 kg mass is $μ_k=0.2$ and the string is inflexible.

I think we should interpret the string as being "inextensible" rather than inflexible. If the string is inflexible then the mass down below is irrelevant and you have a coat hangar through the table, not a string.

 

[nazmulhasanshipon]

I think we should interpret the string as being "inextensible" rather than inflexible. If the string is inflexible then the mass down below is irrelevant and you have a coat hangar through the table, not a string.

Sorry, I am not native english speaker. I translated it, maybe it was a mistranslation.

 

[kuruman]

For the 2-kg mass to be stationary, the centripetal force $F_c$ on the 1.5 kg mass must match the hanging weight. Now $F_c=\dfrac{mv^2}{r}$. Because friction changes the kinetic energy and hence the centripetal force, the value of the speed at which the centripetal force matches the hanging weight is only instantaneous. Thus the mass cannot be stationary if friction is taken into account.
To @nazmulhasanshipon: Are you sure you have translated the statement of the problem faithfully?

 

[nazmulhasanshipon]

For the 2-kg mass to be stationary, the centripetal force $F_c$ on the 1.5 kg mass must match the hanging weight. Now $F_c=\dfrac{mv^2}{r}$. Because friction changes the kinetic energy and hence the centripetal force, the value of the speed at which the centripetal force matches the hanging weight is only instantaneous. Thus the mass cannot be stationary if friction is taken into account.
To @nazmulhasanshipon: Are you sure you have translated the statement of the problem faithfully?

Yes, that was an assignment problem. My english wasn't very good but I mentioned all the information presented there. By the way, 2 kg mass must have Tension force on the 1.5 kg mass, which provides the centripetal force. Right? But does those two force act in opposite direction?

 

[kuruman]

There is one tension in the string connecting the two masses. The tension force acting on the 2-kg mass is up in the opposite direction from gravity. The tension force acting on the 1.5-kg mass is directed towards the center.

 

[haruspex]

a 1.5 kg mass is rotating at uniform speed on a table
...
what should be the speed of rotation of the $1.5$ kg mass to keep the $2$ kg mass stationary?

Coefficient of sliding friction of the $1.5$ kg mass is $μ_k=0.2$ and the string is inextensible.

The question strikes me as ambiguous. Is the table stationary or rotating?
If stationary then, as @kuruman notes, the hanging mass cannot remain stationary. It will only be instantaneously so.

But if the table is rotating and the mass stationary wrt it then the relevant coefficient is static friction, not kinetic, and the answer will be a range, not a single value.

@nazmulhasanshipon , please confirm that the original question a) does not make it clear whether the table is rotating and b) does specify only kinetic friction.

From what I know sliding friction acts in direction opposite to velocity, and since the velocity has tangential direction, sliding friction has tangential direction too. So the correct equation for me is $$\frac{mv^2}{r}-L=0$$, sliding friction has only tangential component, not radial component.

Again, it depends how we interpret the question.
Friction acts to oppose relative motion of the surfaces in contact. If the platform is rotating at steady speed then the static friction is purely radial.

But if the table is static then, as you say, at the instant that the hanging block is stationary, friction will be kinetic and entirely tangential. That makes its value irrelevant, suggesting the other interpretation.

 

[nazmulhasanshipon]

@haruspex , the table is static. If the table is static and we take the friction into account, then it is apparent that the rotating body will slow down, as it slows down the radius will decrease. Will there be any radial friction then?

The question strikes me as ambiguous. Is the table stationary or rotating?
If stationary then, as @kuruman notes, the hanging mass cannot remain stationary. It will only be instantaneously so.

But if the table is rotating and the mass stationary wrt it then the relevant coefficient is static friction, not kinetic, and the answer will be a range, not a single value.

@nazmulhasanshipon , please confirm that the original question a) does not make it clear whether the table is rotating and b) does specify only kinetic friction.

Again, it depends how we interpret the question.
Friction acts to oppose relative motion of the surfaces in contact. If the platform is rotating at steady speed then the static friction is purely radial.

But if the table is static then, as you say, at the instant that the hanging block is stationary, friction will be kinetic and entirely tangential. That makes its value irrelevant, suggesting the other interpretation.

 

[kuruman]

@haruspex , the table is static. If the table is static and we take the friction into account, then it is apparent that the rotating body will slow down, as it slows down the radius will decrease. Will there be any radial friction then?

If there is a radial component to the velocity, there will be a radial component to the kinetic friction. This problem is fine as long as the table contact with the 1.5-kg mass is frictionless. Once you introduce friction, the 1.5-kg mass does not execute uniform circular motion and there is no answer to the question "What should be the speed of rotation of the 1.5-kg mass to keep the 2-kg mass stationary?" because the 2-kg mass cannot be stationary. So what's your pleasure?

 

[nazmulhasanshipon]

If there is a radial component to the velocity, there will be a radial component to the kinetic friction. This problem is fine as long as the table contact with the 1.5-kg mass is frictionless. Once you introduce friction, the 1.5-kg mass does not execute uniform circular motion and there is no answer to the question "What should be the speed of rotation of the 1.5-kg mass to keep the 2-kg mass stationary?" because the 2-kg mass cannot be stationary. So what's your pleasure?

If we take into account friction and forget about keeping it stationary then, due to friction the radius decrease and the mass also comes near the center gradually, does this mean that it will experience radial friction when at the same time it is rotating and it's radius is decreasing?
I know because of circular motion it will feel tangential friction. But if the radius is decreasing, will it experience radial friction along the radius?

 

[kuruman]

At any point on the trajectory of the mass, draw an arrow in the direction of the velocity. The force of kinetic friction will be directed at 180^o from that. If the velocity has a radial component, the kinetic friction will have a radial component. It's as simple as that.

 

[jbriggs444]

At any point on the trajectory of the mass, draw an arrow in the direction of the velocity. The force of kinetic friction will be directed at 180^o from that. If the velocity has a radial component, the kinetic friction will have a radial component. It's as simple as that.

There is a somewhat interesting result of this fact.

Consider a moving object. Its velocity has an x component. Its velocity has a y component. If the object is subject to kinetic friction, the magnitude of the kinetic friction is fixed. It does not depend on speed.

If you make the object move faster in the x direction then the force component in the -x direction increases.
If you make the object move faster in the y direction then the force component in the -y direction increases.

But if magnitude of the vector sum is constant, the interesting thing is that if you increase the velocity in the x direction, the force in the y direction must decrease. This is why you can do doughnuts in the parking lot. Spinning your wheels fast in the x direction let's them slip quite easily when they move slowly in the y direction.

 

[kuruman]

There is a somewhat interesting result of this fact.

Consider a moving object. Its velocity has an x component. Its velocity has a y component. If the object is subject to kinetic friction, the magnitude of the kinetic friction is fixed. It does not depend on speed.

If you make the object move faster in the x direction then the force component in the -x direction increases.
If you make the object move faster in the y direction then the force component in the -y direction increases.

But if magnitude of the vector sum is constant, the interesting thing is that if you increase the velocity in the x direction, the force in the y direction must decrease. This is why you can do doughnuts in the parking lot. Spinning your wheels fast in the x direction let's them slip quite easily when they move slowly in the y direction.

I never thought of the doughnut as an application. I was always annoyed by that fact because it stands in the way of getting closed-form solutions to projectile motion with air resistance.

 

[nazmulhasanshipon]

There is a somewhat interesting result of this fact.

Consider a moving object. Its velocity has an x component. Its velocity has a y component. If the object is subject to kinetic friction, the magnitude of the kinetic friction is fixed. It does not depend on speed.

If you make the object move faster in the x direction then the force component in the -x direction increases.
If you make the object move faster in the y direction then the force component in the -y direction increases.

But if magnitude of the vector sum is constant, the interesting thing is that if you increase the velocity in the x direction, the force in the y direction must decrease. This is why you can do doughnuts in the parking lot. Spinning your wheels fast in the x direction let's them slip quite easily when they move slowly in the y direction.

Okay, but if the mass is subject to kinetic friction, the radius of it's rotation will decrease and the object will be spinning simultaneously. I understand, due to spin there would be tangential friction. But the object is moving to the center simultaneously too, does that mean it will experience a radial friction i.e friction along the radius because as you can see the hanging mass will make it's radius decrease if the spinning body experience friction?

 

[kuruman]

Okay, but if the mass is subject to kinetic friction, the radius of it's rotation will decrease and the object will be spinning simultaneously. I understand, due to spin there would be tangential friction. But the object is moving to the center simultaneously too, does that mean it will experience a radial friction i.e friction along the radius because as you can see the hanging mass will make it's radius decrease if the spinning body experience friction?

You can answer that question yourself. See #15.

 

[nazmulhasanshipon]

You can answer that question yourself. See #15.

I understood that, but for the problem I mentioned in the last comment, does my velocity has radial component? (I have given all the information I was given. This question was set by Bangladesh ministry of education as an assignment. Everyone knows the problem has no numerical solution but I have to point out the concept behind it and bring some new formulas out of it. Some of the teacher said a simplified version of this problem is available on [this book](https://salmanisaleh.files.wordpress.com/2019/02/fundamentals-of-physics-textbook.pdf) )

 

[haruspex]

Okay, but if the mass is subject to kinetic friction, the radius of it's rotation will decrease and the object will be spinning simultaneously. I understand, due to spin there would be tangential friction. But the object is moving to the center simultaneously too, does that mean it will experience a radial friction i.e friction along the radius because as you can see the hanging mass will make it's radius decrease if the spinning body experience friction?

The radius is not necessarily decreasing. Maybe the initial condition includes a velocity component away from the hole, so the radius increases at first and decreases later. While the radius increases the friction will have a radial component towards the hole, at maximum radius only tangential friction, then radially outward friction as the radius decreases.

 

[kuruman]

Here is a screenshot from the reference you gave. This version is fine because there is no friction and has an answer to the question being asked. Whoever added the complication of friction, did not modify the question being asked to take friction into account.

Edit: I remember puzzling over this problem as an undergraduate more than 50 years ago. It was in the 1964 edition of "Physics" by Halliday and Resnick.

 

[nazmulhasanshipon]

Here is a screenshot from the reference you gave. This version is fine because there is no friction and has an answer to the question being asked. Whoever added the complication of friction, did not modify the question being asked to take friction into account.

Edit: I remember puzzling over this problem as an undergraduate more than 50 years ago. It was in the 1964 edition of "Physics" by Halliday and Resnick.

View attachment 286780

@kuruman where can I find it's solution, sir?

 

[haruspex]

@kuruman where can I find it's solution, sir?

The solution to the frictionless version? Consider the tension and the balance of forces on each mass.

 

[kuruman]

@kuruman where can I find it's solution, sir?

I didn't keep my solution from more than 50 years ago. I think you should follow @haruspex's advice.

 

[nazmulhasanshipon]

The solution to the frictionless version? Consider the tension and the balance of forces on each mass.

@haruspex, for the rotating mass, should I consider the pseudo centrifugal force acting in outward direction and tension force acting towards the center?

 

[kuruman]

@haruspex, for the rotating mass, should I consider the pseudo centrifugal force acting in outward direction and tension force acting towards the center?

Not if you write the forces for each mass in the inertial frame of the table which (I think) is what @haruspex had in mind.

 

[nazmulhasanshipon]

The solution to the frictionless version? Consider the tension and the balance of forces on each mass.

Would the tension on the string and the central force exerted on the rotating mass be same when the cylinder is at rest?

 

[nazmulhasanshipon]

Not if you write the forces for each mass in the inertial frame of the table which (I think) is what @haruspex had in mind.

@kuruman sir, why should I consider the inertial frame of the table?

 

[kuruman]

Would the tension on the string and the central force exerted on the rotating mass be same when the cylinder is at rest?

Yes.

@kuruman sir, why should I consider the inertial frame of the table?

Because the rotating mass moves relative to it with the speed you are looking for?
Because then you don't have to worry about fictitious forces?
Because the cylinder is at rest relative to it?

You can do it in the non-inertial frame but why bother?

 

[nazmulhasanshipon]

Yes.

Because the rotating mass moves relative to it with the speed you are looking for?
Because then you don't have to worry about fictitious forces?
Because the cylinder is at rest relative to it?

You can do it in the non-inertial frame but why bother?

@kuruman sir, if I consider forces in the inertial reference frame of the table, then wouldn't centrifugal force be a fictitious force?

 

[kuruman]

@kuruman sir, if I consider forces in the inertial reference frame of the table, then wouldn't centrifugal force be a fictitious force?

It would and what does that mean if you were to draw a free body force diagram in the inertial frame?

 

[nazmulhasanshipon]

It would and what does that mean if you were to draw a free body force diagram in the inertial frame?

I don't know, @kuruman . Would you clarify sir?

 

[kuruman]

Please draw two force diagrams, one for the hanging mass and one for the mass on the table. In each of these diagrams draw the forces acting on each mass and label them with unique symbols that make it clear what they stand for. Post your force diagrams here. We'll take it from there.

 

[nazmulhasanshipon]

Please draw two force diagrams, one for the hanging mass and one for the mass on the table. In each of these diagrams draw the forces acting on each mass and label them with unique symbols that make it clear what they stand for. Post your force diagrams here. We'll take it from there.

Is it okay sir? @kuruman

 

[kuruman]

View attachment 286980

Is it okay sir? @kuruman

On the left diagram, you show two vertical forces. These forces arise from physical entities, one entity for one force. Specifically, you have the Earth that exerts the "down" force and the string that exerts the "up" force.

Now that you understand this, please explain to me what two physical entities are responsible for the two horizontal forces that you drew in the diagram on the right. They must be items that you can see and touch.

 

[nazmulhasanshipon]

On the left diagram, you show two vertical forces. These forces arise from physical entities, one entity for one force. Specifically, you have the Earth that exerts the "down" force and the string that exerts the "up" force.

Now that you understand this, please explain to me what two physical entities are responsible for the two horizontal forces that you drew in the diagram on the right. They must be items that you can see and touch.

@kuruman sir Maybe the physical entities are: the string that provides the centripetal force and the rotation which provides centrifugal but should we consider centrifugal force too although it's a fictions force?

 

[kuruman]

@kuruman sir Maybe the physical entities are: the string that provides the centripetal force and the rotation which provides centrifugal but should we consider centrifugal force too although it's a fictions force?

The string is something that you and I can see and touch. I cannot see or touch "the rotation which provides centrifugal", can you? If not, then you should not consider it in the free body diagram drawn in an inertial frame. Fictitious forces get their name because they don't exist in inertial free body diagrams but appear in non-inertial frames. They are not forces generated by things that you can see and touch; they are mass times acceleration which is moved over to the "other side" of Newton's second law and just looks like a force.

The time has come for you to write Newton's second law for the hanging mass and the rotating mass. Two equations, please.

 

[nazmulhasanshipon]

The string is something that you and I can see and touch. I cannot see or touch "the rotation which provides centrifugal", can you? If not, then you should not consider it in the free body diagram drawn in an inertial frame. Fictitious forces get their name because they don't exist in inertial free body diagrams but appear in non-inertial frames. They are not forces generated by things that you can see and touch; they are mass times acceleration which is moved over to the "other side" of Newton's second law and just looks like a force.

The time has come for you to write Newton's second law for the hanging mass and the rotating mass. Two equations, please.

I think for the hanging mass there is gravitational force which is $F_g=m_1g$ and the tension the string provides is $T$ but I don't know how to write $T$ in equation just yet. For the rotating mass, I think the centripetal force is provided by the tension of the string which means $$T=\frac{m_2v^{2}}{r}$$ but I don't know what balances this force. My brain tells me centrifugal force balances it.

 

[kuruman]

Please tell your brain to start thinking. Where is it written that the force on the rotating mass needs to be balanced? What does Newton's second law have to say on the matter? Remember, you are doing this in the inertial frame.

 

[nazmulhasanshipon]

Please tell your brain to start thinking. Where is it written that the force on the rotating mass needs to be balanced? What does Newton's second law have to say on the matter? Remember, you are doing this in the inertial frame.

@kuruman sir, I don't know what Newton's second law says about this. Would you please tell me?

 

[kuruman]

@kuruman sir, I don't know what Newton's second law says about this. Would you please tell me?

Do you know what Newton's second law says in general, not necessarily about this?

 

[nazmulhasanshipon]

@kuruman sir, The second law states that the rate of change of momentum of an object is directly proportional to the force applied.

 

[kuruman]

@kuruman sir, The second law states that the rate of change of momentum of an object is directly proportional to the force applied.

Have you ever solved any problems requiring the use of the second law? If so, is that what you used?

 

[nazmulhasanshipon]

Do you know what Newton's second law says in general, not necessarily about this?

And @kuruman sir one of my seniors was saying that if the table isn't frictionless then at first the rotating mass would try to get away and feel frictional force in the inward direction then $$T+f_k-\frac{m_2v^2}{r}=0$$ would be true for the rotating body. He also added after sometime it will try to go the inward direction towards center so $$T-f_k-\frac{m_2v^2}{r}=0$$ would be true. So, if you were given $m_2, T, f_k$ and $r$ then you will find the speed of the rotating body $v$ ranges between some value. But I disagree, if something like this happen then the radius vector $r$ will not be a constant. I mean how can there be friction in the radial direction if $r$ is constant. What will you say about it sir?

 

[nazmulhasanshipon]

Have you ever solved any problems requiring the use of the second law? If so, is that what you used?

I used $$\sum \vec{F}=m \vec{a}$$

 

[kuruman]

I used $$\sum \vec{F}=m \vec{a}$$

Very good. That's what I was looking for. Now answer this. What is $\Sigma \vec F$ for the rotating mass?

 

[nazmulhasanshipon]

Very good. That's what I was looking for. Now answer this. What is $\Sigma \vec F$ for the rotating mass?

I can imagine if the table is frictionless then at first someone implied some toque and made it to rotate and then I can see the string provides the centripetal force.

 

[nazmulhasanshipon]

Very good. That's what I was looking for. Now answer this. What is $\Sigma \vec F$ for the rotating mass?

And also sir, can you please recommend me some books where I can learn extensively about motion like Fundamentals of Physics by David Halliday?

 

[kuruman]

And also sir, can you please recommend me some books where I can learn extensively about motion like Fundamentals of Physics by David Halliday?

That's as good as any and you already have the link. I have used it both as a student and as an instructor. So what is $\Sigma \vec F$ for the rotating mass?