What is the minimum velocity needed?

[nineteen]

Homework Statement

The statement will be included in the photo I insert.

Relevant Equations

F = ma
v^2 = u^2 + 2as

I tried to solve this problem and this is what I could come through:
When the object is moving, the force acting on object is the frictional force, so, it got to be μmg.
So, F = ma and as F is μmg
μmg = ma
μg = a

So, to find out the magnitude of the initial velocity v given to the smaller object, we can substitute the values in the motion equation v2 = u2 + 2as ------> v is final velocity here and u is initial velocity
and this is what we get : v = √2μgL --------------- 1 > v here is the initial velocity of the object as cited in the problem.

Then I went on to find the moment acting on the strip at the toppling instance.
MgXo = mg(L/2 - Xo)
MXo = m(L/2 - Xo) ------------------------ 2

When I combined the equations 1 and 2 and subjected v, I got something like this :
√2μg(MXo/m + 2Xo) = v

But the answer I got is nowhere to be seen in the options list of the multiple choice question. Can you guys please help me? Pleeeeeaaaaaasssseeee guide me...

 

[PeroK]

This problem looks ambiguous to me. Does the strip move frictionlessly across the table? But, then it would never stop and always topple.

And, if the strip does not move, then $m$ may not be large enough to make it topple.

Perhaps $\mu$ applies to the friction between the strip and the table as well?

In all the answers, if you take $\mu = 0$, you get $v =0$, which makes no sense. There ought to be some minimum friction.

I'm not sure about this problem.

 

[nineteen]

This problem looks ambiguous to me. Does the strip move frictionlessly across the table?

In all the answers, if you take $\mu = 0$, you get $v =0$, which makes no sense. There ought to be some minimum friction.

I'm not sure about this problem.

Yeah, I too had some confusions at the beginning, but, sir, there is friction between the small object and the strip. In fact, it is stated in the question right?

 

[PeroK]

Yeah, I too had some confusions at the beginning, but, sir, there is friction between the small object and the strip. In fact, it is stated in the question right?

Yes, that's clear. I've added a few more thoughts in my first post now. Let me see what I can figure out here.

 

[kuruman]

This problem looks ambiguous to me. Does the strip move frictionlessly across the table?

I think the strip stays in place while the block slides on top of it. The velocity must be zero when $\mu_k$ is zero because on a frictionless strip any non-zero velocity will cause the assembly to topple over.

To OP: Suppose the block didn't slide, but were placed on top of the strip. At what maximum distance from the left edge of the strip can he block be placed before the whole thing topples over? Once you determine the distance, the rest is kinematics.

On Edit: The strip must move.

 

[PeroK]

I think the strip stays in place while the block slides on top of it.
To OP: Suppose the block didn't slide, but were placed on top of the strip. At what maximum distance from the left edge of the strip can he block be placed before the whole thing topples over? Once you determine the distance, the rest is kinematics.

What happens if $M \gg m$? The answer formulas given generally give a large $v$, but that just means that the block shoots off the end!?

 

[nineteen]

Yes, that's clear. I've added a few more thoughts in my first post now. Let me see what I can figure out here.

Yeah thank you in advance. Please try to help me over here.

 

[nineteen]

I think the strip stays in place while the block slides on top of it. The velocity must be zero when $\mu_k$ is zero because on a frictionless strip any non-zero velocity will cause the assembly to topple over.

To OP: Suppose the block didn't slide, but were placed on top of the strip. At what maximum distance from the left edge of the strip can he block be placed before the whole thing topples over? Once you determine the distance, the rest is kinematics.

What max distance in the sense, it is the total distance from the left edge to the right edge right?

 

[nineteen]

What happens if $M \gg m$? The answer formulas given generally give a large $v$, but that just means that the block shoots off the end!?

Aren't we concerning about the instance where it just starts to topple? Where the small block stays at the right edge? Isn't that instance the key to all this? I am just wondering you know.

 

[PeroK]

Aren't we concerning about the instance where it just starts to topple? Where the small block stays at the right edge? Isn't that instance the key to all this? I am just wondering you know.

How do you know $m$ is massive enough? If $m$ is small (as it suggests in the question), you could balance it on the end and the strip won't topple.

I tried assuming the block could keep going (on a massless extension of the strip), and that gives one of the answers!

I think this problem is nonsense!

 

[PeroK]

Okay. Here's the thing. Assume that the small block, if placed on the end of the strip, is massive enough to cause it to topple.

I.e. calculate $d$, the distance that the block has to travel to topple the block. Then assume that $d \le L$.

But, looking at the diagram and the relative sizes of the block and strip, that is far from a realistic assumption.

 

[kuruman]

I took back my earlier post saying that the strip doesn't move. Both the block and strip must slide forward. We are looking for a minimum initial speed. This means minimum distance of the block from the left end of the strip. Note that if the strip's midpoint and the block are simultaneously above the edge, the assembly is just about ready to tip. So the key is to find how much time is required for the block to travel distance L/2 on the strip and the strip to travel distance x0/2 on the table. Because nothing is said about the strip-table contact, I would assume it is frictionless and conserve momentum of the block+strip system.

 

[PeroK]

Because nothing is said about the strip-table contact, I would assume it is frictionless and conserve momentum of the block+strip system.

But, once it's moving at all it's not going to stop! I've got an answer based on the assumption in post #11.

 

[PeroK]

Then I went on to find the moment acting on the strip at the toppling instance.
MgXo = mg(L/2 - Xo)
MXo = m(L/2 - Xo) ------------------------ 2

This is all okay, except you have asssumed that the block ends up at the end of the strip. It may not have to go that far.

 

[nineteen]

This is all okay, except you have asssumed that the block ends up at the end of the strip. It may not have to go that far.

I see, thanks for showing me the path on how to solve this, sir. After you implied, I could just have a glimpse about how the small block wouldn't need to travel so far to make the strip topple.

By the way, what is your opinion on assuming that the strip would also move?

 

[nineteen]

I took back my earlier post saying that the strip doesn't move. Both the block and strip must slide forward. We are looking for a minimum initial speed. This means minimum distance of the block from the left end of the strip. Note that if the strip's midpoint and the block are simultaneously above the edge, the assembly is just about ready to tip. So the key is to find how much time is required for the block to travel distance L/2 on the strip and the strip to travel distance x0/2 on the table. Because nothing is said about the strip-table contact, I would assume it is frictionless and conserve momentum of the block+strip system.

Is it okay to assume something that isn't mentioned in the question? I mean the frictionless contact between the strip and the table?

 

[haruspex]

After you implied, I could just have a glimpse about how the small block wouldn't need to travel so far to make the strip topple.

Not sure what you are saying. Are you asking for an explanation of why the block does not necessarily have to reach the end of the strip?

By the way, what is your opinion on assuming that the strip would also move?

@PeroK and I agree that the strip should be assumed stationary.
We also need to assume the strip has no height.

 

[PeroK]

After you implied, I could just have a glimpse about how the small block wouldn't need to travel so far to make the strip topple.

I would draw a new diagram:

First, I'd draw a long, thin strip, sticking a long way over the edge of the table. I.e. $x_0$ is small compared to $L$.

Second, I'd assume $m$ is relatively large compared to $M$. Certainly not "small" in the usual sense that it is much less than $M$.

 

[kuruman]

Is it okay to assume something that isn't mentioned in the question? I mean the frictionless contact between the strip and the table?

It is as OK as assuming that the strip remains at rest while the block slides over it. Nothing is mentioned about the contact between strip and table. One interpretation assumes that the coefficient of static friction between strip and table is zero and the other that it is very very big. There is no language in the statement of the problem that distinguishes between the two. When there is ambiguity in a multiple choice question, it is good practice to assume each possible interpretation and see if one gets multiple matches.

 

[haruspex]

One interpretation assumes that the coefficient of static friction between strip and table is zero

I think we can rule that out.
As @PeroK notes, that would mean there is no nonzero minimum velocity.

 

[kuruman]

I think we can rule that out.
As @PeroK notes, that would mean there is no nonzero minimum velocity.

Yes, of course. I must have had a mental lapse.