Maximum height a block reaches after separating from a curved moving ramp

AI Thread Summary
The discussion revolves around the mechanics of a block separating from a moving curved ramp and the implications of friction and normal forces on its motion. The initial approach using conservation of energy led to an incorrect height calculation, prompting a reevaluation of the role of friction and momentum. It was established that if friction is present, it affects the velocity of the blocks, but the problem states there is no friction, allowing for momentum conservation to be applied. The participants debated whether the blocks would maintain the same horizontal velocity while on the curved section, with the consensus leaning towards them sticking together due to the geometry of the ramp. Ultimately, the conversation highlights the complexity of the problem and the importance of correctly interpreting the forces at play.
ItsukaKitto
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Homework Statement
A body of mass M with a small block of mass m placed on it, rests on a smooth horizontal plane. The surface of the body M is horizontal near the smaller mass and gradually curves to become vertical. The block is set in motion in the horizontal direction with a velocity v. To what height, relative to the initial level, will the block rise after breaking off the body M? Friction is assumed to be absent.
Relevant Equations
F = ma, momentum conservation
Conservation of energy
Total work = ∆K.E.
Diagram attached at the endI personally think there's something wrong with this question, and I'd like if someone can tell me whether it's the question that is wrong or my approach.

If I attempt the solution thinking that M should be stationary, the solution is simple. 0 - 1/2 mv^2 = -mgh, gives h = v^2/(2g). However, this is not the right answer, so I assumed the lower block moves (because this problem was given in a section titled "momentum" ), but for mass M to move, there should be a force on it...which can only arise if there is friction.

If I assume friction is present, I get the velocity
w = mv/(M+m) as the common velocity for m and M.

And to obtain the height, I can use the work energy theorem:
>>> W(gravity) + W(friction) = 1/2 m(v1)^2 + 1/2Mw^2 - 1/2mv^2
note: here v1 is the velocity of the block as it breaks off with M.

I found that W(friction) = -v^2(Mm)/2(M+m)
So substituting and setting v1 = w, (as I think, the horizontal component when the block is breaking off, wouldn't change) I get:
>>>2*W(gravity) = w^2(m + M) - mv^2 + v^2(Mm)/(M+m)

>>>-2mgh = v^2(m^2)/(m+M) - mv^2 + v^2 (Mm)/(M+m)

>>> Meaning that h is precisely zero...(is it?)

When I checked the answer to this problem, it was actually Mv^2/2g(m+M) and I found out a way they could have arrived at this:
Using conservation of energy, and ignoring the work done by friction:
1/2 mv^2 = 1/2 m(v1)^2 + 1/2 Mw^2 + mgh
Now setting v1 = w, we get:
mv^2 - (m+M) w^2 = 2mgh

>> h = Mv^2 /2g(M+m)

I don't think this approach is right, because if they are considering friction, why are they ignoring the work it does?
 

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ItsukaKitto said:
but for mass M to move, there should be a force on it...which can only arise if there is friction.
There is also the normal force. If you assume that the small block has enough initial velocity to get past the curved section then you have enough information to calculate what you need to know.
 
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jbriggs444 said:
There is also the normal force. If you assume that the small block has enough initial velocity to get past the curved section then you have enough information to calculate what you need to know.
Thanks for replying, I've a few questions...
It's not possible for the normal force to cause the lower block to have a horizontal velocity, right?
Do you think the "most accurate" answer should actually be v^2/2g then, assuming no friction?
In saying that the block should have enough initial velocity to get past the curved section, are you referring to the case where friction is present?
 
ItsukaKitto said:
It's not possible for the normal force to cause the lower block to have a horizontal velocity, right?
Why not? In what directions does it act?
 
jbriggs444 said:
Why not? In what directions does it act?
In the curved part, the normal force would have a horizontal component, and the lower block will have a constant velocity in the horizontal direction...then I suppose I can use momentum conservation to find it. I'm not sure however, will the horizontal velocity of both the blocks be the same then? I see if they're stuck together till the end that must be true, should I just assume this?
 
ItsukaKitto said:
In the curved part, the normal force would have a horizontal component, and the lower block will have a constant velocity in the horizontal direction...then I suppose I can use momentum conservation to find it. I'm not sure however, will the horizontal velocity of both the blocks be the same then? I see if they're stuck together till the end that must be true, should I just assume this?
You are told that there is no friction. This is a useful piece of information. It let's you know that you can use momentum conservation. The total momentum of lower plus upper block (in the horizontal direction at least) must be conserved because no external force has a horizontal component. As you suspected.

If the lower block manages to reach the place where the curved surface becomes purely vertical then yes, the horizontal velocity of both blocks will be the same.
 
jbriggs444 said:
You are told that there is no friction. This is a useful piece of information. It let's you know that you can use momentum conservation. The total momentum of lower plus upper block (in the horizontal direction at least) must be conserved because no external force has a horizontal component. As you suspected.

If the lower block manages to reach the place where the curved surface becomes purely vertical then yes, the horizontal velocity of both blocks will be the same.
I see, so would I have to assume the blocks stick together through the entire time the smaller one is on the curved part? Most likely they would have different velocities in that part, and I think whether they will stick together here, depends on the geometry of the big block too. Is this true?
 
ItsukaKitto said:
I see, so would I have to assume the blocks stick together through the entire time the smaller one is on the curved part? Most likely they would have different velocities in that part, and I think whether they will stick together here, depends on the geometry of the big block too. Is this true?
It is difficult to imagine them flying apart since the curve is concave throughout.
 
jbriggs444 said:
It is difficult to imagine them flying apart since the curve is concave throughout.
For the sake of conserving momentum here, I think that's a reasonable assumption. Thank you for your help.
 
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