Plank with block of mass on top executing vertical SHM

[andyrk]

Homework Statement

A plank on which a mass of mass m is put, is executing vertical SHM according to the equation y = sinωt + √3cosωt. At what time does the mass break of the plank and what is the value for the mass to break off ?

Homework Equations

The modified equation of motion looks like: y = 2sin(ωt + π/3)
Maximum chance of break off is at extreme position. So mg - N = mω²A

The Attempt at a Solution

.[/B]
My doubt is that why does the maximum chance for the mass to break off occur at an extreme position of the plank and that too at the positive extreme (A) and not the negative extreme (-A)? Can it be explained quantitatively rather than qualitatively?

 

[DEvens]

What has to happen for the block to leave the plank? What forces are involved?

I did not check the modified equation of motion, so you should do that if you have not already.

What is the N in mg - N = mω²A ?

Who says the block leaves at the maximum position?

 

[andyrk]

What has to happen for the block to leave the plank? What forces are involved?

I did not check the modified equation of motion, so you should do that if you have not already.

What is the N in mg - N = mω²A ?

Who says the block leaves at the maximum position?

The question asks the time when the block breaks off and the minimum ω needed to do so.
N is the normal reaction acting on the block because of the contact between block and plank.

 

[haruspex]

why does the maximum chance for the mass to break off occur at an extreme position of the plank

Put that information aside for the moment. At what point in the cycle do you think there's the greatest chance of the mass leaving the plank?

 

[andyrk]

At the topmost point of course. That's what my logic and intuition says. But I am not able to find a suitable quantitative explanation for this.

 

[haruspex]

At the topmost point of course. That's what my logic and intuition says. But I am not able to find a suitable quantitative explanation for this.

Think about forces. What does "stays on the plank" mean in terms of forces?

 

[andyrk]

Well, the block stays on the plank but its not glued to it or anything like that. So, it can break off if such a situation occurs, can't it?

 

[haruspex]

Well, the block stays on the plank but its not glued to it or anything like that. So, it can break off if such a situation occurs, can't it?

Exactly, it's not glued on. What are the forces on it? What will be the relationship between those forces when it separates from the plank?

 

[andyrk]

When it breaks off then the normal reaction between the block and the plank is 0. The forces acting on the block are represented by the following equation
mg - N = mω²A.

But my question is that why is the net acceleration of the block mω²A? Isn't that supposed to be towards the mean position and if it is then how would the block ever break off if its acceleration is towards the mean position, i.e vertically down at the topmost point and vertically upwards at the bottom-most point?

 

[haruspex]

When it breaks off then the normal reaction between the block and the plank is 0. The forces acting on the block are represented by the following equation
mg - N = mω²A.

In general, it is a function of time.

But my question is that why is the net acceleration of the block mω²A? Isn't that supposed to be towards the mean position and if it is then how would the block ever break off if its acceleration is towards the mean position, i.e vertically down at the topmost point and vertically upwards at the bottom-most point?

Because to become detached, the block only needs to accelerate downwards more slowly than the plank.

 

[andyrk]

Because to become detached, the block only needs to accelerate downwards more slowly than the plank.

But both the plank and the block have an acceleration of ω²A downwards at the topmost position. Then how can one be slower than the other?

 

[haruspex]

But both the plank and the block have an acceleration of ω²A downwards at the topmost position. Then how can one be slower than the other?

As long as the block has that acceleration downwards it will stay with the plank. But how is it guaranteed to have that acceleration down? Its acceleration is a consequence of the forces on it, not of a wish to stay in SHM.

 

[andyrk]

So

As long as the block has that acceleration downwards it will stay with the plank. But how is it guaranteed to have that acceleration down? Its acceleration is a consequence of the forces on it, not of a wish to stay in SHM.

So the forces acting on the block would be its own weight and the normal reaction then. But how does that determine whether the acceleration of the block would be lesser than the plank or not?

 

[haruspex]

So

So the forces acting on the block would be its own weight and the normal reaction then. But how does that determine whether the acceleration of the block would be lesser than the plank or not?

The plank will follow SHM, so you know the acceleration of that. When will the forces acting on the block fall short of what is required? Consider what the maximum downward acceleration of the block can be.

 

[andyrk]

The plank follows SHM, and so at all instants when the plank is moving up, it has acceleration towards the mean position and since the block is sitting on top of it, it too has an acceleration towards the mean position even though it is going up. This case is a bit confusing. Because I compare it with the case of an elevator which is accelerating upwards and a man is standing inside it on a weighing scale. As the elevator accelerates upwards, the weighing scale shows the weight of the man more than what it actually is. This is because the normal reaction faced by the man is more than his weight downward (and hence contributing to an acceleration equal to that of the elevator). But this happens when the elevator (in this case a plank) is accelerating upwards. But the plank is accelerating downwards even though it is going up and so the normal reaction experienced by the block is less than its weight downwards and so it has no reason to break off even at the top most point.

What I am trying to come at is that even though the whole system (plank + block) are going up, their net acceleration is down. But I am not able to figure out the case when the accelerations of the two are both downwards but unequal.

 

[haruspex]

The plank follows SHM, and so at all instants when the plank is moving up, it has acceleration towards the mean position and since the block is sitting on top of it, it too has an acceleration towards the mean position even though it is going up. This case is a bit confusing. Because I compare it with the case of an elevator which is accelerating upwards and a man is standing inside it on a weighing scale. As the elevator accelerates upwards, the weighing scale shows the weight of the man more than what it actually is. This is because the normal reaction faced by the man is more than his weight downward (and hence contributing to an acceleration equal to that of the elevator). But this happens when the elevator (in this case a plank) is accelerating upwards. But the plank is accelerating downwards even though it is going up and so the normal reaction experienced by the block is less than its weight downwards and so it has no reason to break off even at the top most point.

What I am trying to come at is that even though the whole system (plank + block) are going up, their net acceleration is down. But I am not able to figure out the case when the accelerations of the two are both downwards but unequal.

Answer this: what is the maximum downward acceleration the block can ever have (given that it is not glued on).

 

[andyrk]

Answer this: what is the maximum downward acceleration the block can ever have (given that it is not glued on).

mω²A + mg - N?

 

[haruspex]

mω²A + mg - N?

No. You're overthinking it. There are only two forces acting on the block. The downward force is constant. The upward force has a minimum.

 

[andyrk]

mg - N?

 

[haruspex]

mg - N?

Ok, but the maximum value of that is?

 

[andyrk]

When the mass breaks off, i.e N = 0. So mg - N = mg. That's the maximum value.

 

[haruspex]

When the mass breaks off, i.e N = 0. So mg - N = mg. That's the maximum value.

Right. So the maximum downward acceleration is ?

 

[andyrk]

Right. So the maximum downward acceleration is ?

I just told you, mg?

 

[haruspex]

I just told you, mg?

That's a force.

 

[andyrk]

Oh yeah. So it would be g.

 

[haruspex]

Oh yeah. So it would be g.

Right. And the maximum downward acceleration that is required (at a certain point in the cycle) to stay in contact with the plank is?

 

[andyrk]

Right. And the maximum downward acceleration that is required (at a certain point in the cycle) to stay in contact with the plank is?

You mean the minimum, right? It is ω²A.

 

[haruspex]

You mean the minimum, right? It is ω²A.

That's what I was after, but is the maximum that is ever required. So if the required downward acceleration to stay in contact exceeds the maximum downward acceleration available, what equation do you have?

 

[andyrk]

That's what I was after, but is the maximum that is ever required. So if the required downward acceleration to stay in contact exceeds the maximum downward acceleration available, what equation do you have?

You mean the downward acceleration of the block exceed the maximum downward acceleration of the plank? Does that mean the block presses even harder to the plank so any possibilities of breaking off the plank are ruled out?

 

[haruspex]

You mean the downward acceleration of the block exceed the maximum downward acceleration of the plank? Does that mean the block presses even harder to the plank so any possibilities of breaking off the plank are ruled out?

No, the other way around. When the maximum downward acceleration available to the block (g) is less than the downward acceleration required to stay in contact (which can be up to $\omega^2A$) the block will lose contact with the plank, no?

 

[andyrk]

Right! So if g < ω²A, then the block leaves contact. But this is a bit confusing. We came to the conclusion that the downward acceleration available to the block is g after we put N = 0 in the equation: mg - N = mω²A. That means, we have already assumed that the block leaves contact before even proving it. And even if we do prove it before hand (I don't know how), then the equation simply reduces down to mg = mω²A and so: g = ω²A. So how do we know if the block leaves contact or not?

And btw, mg - N = mω²A equation holds for the block at the topmost extreme point because the whole system is having an acceleration of ω²A towards the mean position.

 

[haruspex]

Right! So if g < ω²A, then the block leaves contact. But this is a bit confusing. We came to the conclusion that the downward acceleration available to the block is g after we put N = 0 in the equation: mg - N = mω²A. That means, we have already assumed that the block leaves contact before even proving it. And even if we do prove it before hand (I don't know how), then the equation simply reduces down to mg = mω²A and so: g = ω²A. So how do we know if the block leaves contact or not?

And btw, mg - N = mω²A equation holds for the block at the topmost extreme point because the whole system is having an acceleration of ω²A towards the mean position.

We showed that the maximum downward acceleration available to the block is g. We showed that at a certain point in the cycle the downward acceleration needed to be ω²A if the block is to remain in contact. Therefore, for the block to remain in contact through the cycle g >= ω²A. Where's the confusion?

 

[andyrk]

We showed that g is the maximum acceleration of the block by putting N = 0 in mg - N right? But then how do we know that N = 0 in the first place?

 

[haruspex]

We showed that g is the maximum acceleration of the block by putting N = 0 in mg - N right? But then how do we know that N = 0 in the first place?

No, we showed it was the maximum downward acceleration by recognising that N >= 0.

 

[andyrk]

Finally. I get it! Thanks a lot for the help :D

 

[Jediknight]

so this posts a month old, I am probably good to propose some answers, a(max)=ω^2(A)=g at the minimum frequency to make block leave plank, so just algebra from there right

 

[haruspex]

so this posts a month old, I am probably good to propose some answers, a(max)=ω^2(A)=g at the minimum frequency to make block leave plank, so just algebra from there right

Yes.

 

[andyrk]

While pondering on this problem once again, I got some more doubts with it. Why is mg - N = mω²x and not: mg - N = -mω²x, since acceleration is
-ω²x and not ω²x?

 

[haruspex]

While pondering on this problem once again, I got some more doubts with it. Why is mg - N = mω²x and not: mg - N = -mω²x, since acceleration is
-ω²x and not ω²x?

It might depend on the conventions adopted for the positive direction and what 'g' means. If we take up as positive and g as a signed value (i.e. its value will be negative) then the equation is mg+N=-mω²x. Other conventions can lead to other forms. Since N will be set to zero, it may be that the equation was simply wrong in a way that didn't matter.

 

[andyrk]

It might depend on the conventions adopted for the positive direction and what 'g' means. If we take up as positive and g as a signed value (i.e. its value will be negative) then the equation is mg+N=-mω²x. Other conventions can lead to other forms. Since N will be set to zero, it may be that the equation was simply wrong in a way that didn't matter.

The convention is that up is positive and down is negative.

 

[andyrk]

No, the other way around. When the maximum downward acceleration available to the block (g) is less than the downward acceleration required to stay in contact (which can be up to $\omega^2A$) the block will lose contact with the plank, no?

How do we know that the downward acceleration of the plank would be greater than the block? We just saw that g = $\omega^2A$. And also, the downward acceleration of the block at the topmost point is $\omega^2A$. So shouldn't the two be same?

 

[andyrk]

Anybody there?

 

[haruspex]

How do we know that the downward acceleration of the plank would be greater than the block? We just saw that g = $\omega^2A$. And also, the downward acceleration of the block at the topmost point is $\omega^2A$. So shouldn't the two be same?

We don't know the plank's downward acceleration is greater than the block's. I said if it is ever greater.
g = $\omega^2A$ Is the limiting condition, i.e. when it is in imminent danger of leaving the plank at some point in the oscillation. It is not a general fact.
I don't know what you mean about "the two" being the same. What two?

Seems to me you have a basic problem with handling conditionals, the difference between a general statement and a statement that only applies when certain conditions are met. I don't think I can make it any clearer. If you still don't get it, I can only recommend that you read and reread my posts.

 

[andyrk]

I don't know what you mean about "the two" being the same. What two?

By two I mean the downward acceleration of the block and the plank at the topmost point. If they are the same, then why would the block leave contact? Now you are saying if they are same. But what I am trying to say that they have to be same, there's no if associated with it, it just has to be the same.

 

[haruspex]

By two I mean the downward acceleration of the block and the plank at the topmost point. If they are the same, then why would the block leave contact? Now you are saying if they are same. But what I am trying to say that they have to be same, there's no if associated with it, it just has to be the same.

The value of g is fixed. Give the plank enough amplitude and its acceleration at the top of the cycle will exceed g. The block will have less downward acceleration than the plank and lose contact with it. The normal force cannot be negative.

 

[andyrk]

The value of g is fixed. Give the plank enough amplitude and its acceleration at the top of the cycle will exceed g. The block will have less downward acceleration than the plank and lose contact with it. The normal force cannot be negative.

The acceleration of the plank at the top is ω²A, right? And the acceleration of the block is g, right? And we also proved that g = ω²A (by putting N = 0 in the force equation of the block). So, we just showed that the accelerations are same at the topmost point and not different. But wait. Had they been the same, then the block wouldn't have left contact and hence N wouldn't have been 0. So what am I doing wrong here?

 

[haruspex]

The acceleration of the plank at the top is ω²A, right? And the acceleration of the block is g, right? And we also proved that g = ω²A (by putting N = 0 in the force equation of the block). So, we just showed that the accelerations are same at the topmost point and not different. But wait. Had they been the same, then the block wouldn't have left contact and hence N wouldn't have been 0. So what am I doing wrong here?

We could only write N=0 in that equation on the basis that the block was almost leaving the plank but not quite. If the block leaves the plank then N will still be zero, but the equation we plugged it into is no longe valid. The equation assumes they remain in contact.

 

[andyrk]

I am unable to make any sense of what you are saying. Are you then saying that the downward acceleration of the block is less than g and that of the plank is g? If block leaves contact at the top and its acceleration is g, then does that mean that the acceleration of the plank is greater than g, i.e. greater than ω²A? But how is this possible??

 

[haruspex]

I am unable to make any sense of what you are saying. Are you then saying that the downward acceleration of the block is less than g and that of the plank is g? If block leaves contact at the top and its acceleration is g, then does that mean that the acceleration of the plank is greater than g, i.e. greater than ω²A? But how is this possible??

At the top position, the downward acceleration of the plank will always be ω²A.
The downward acceleration of the block at that position will be min{ω²A, g}. If ω²A <= g they stay in contact. If ω²A > g they will lose contact.

 

[andyrk]

The downward acceleration of the block at that position will be min{ω2A, g}

But we just found out that ω²A = g by putting N = 0 in the equation mg - N = mω²A. So isn't ω²A = g at the topmost position? Which means that min{ω2A, g} = g or ω²A since they are one and the same thing? So that means that they should not leave contact but that is not the case as it is clearly mentioned that the block leaves contact. This is what is confusing me.