Height block reaches on accelerating quarter pipe

[Scronin267]

Homework Statement

A block in the shape of a quarter circle moves at constant acceleration across the floor. At some moment a small block of mass m, which is at rest with respect to the floor, is placed atop the horizontal part of the large block. There is no friction in this problem. Assuming the small block does not fly off the large block, what is the greatest altitude increase it will achieve? What will happen after this height is obtained?

Homework Equations

potential energy U = mgh
kinetic energy T = 1/2mv^2
langrangian L = T-U, ∂L/∂q=d/dt ∂L/∂q'

The Attempt at a Solution

I do not know how to handle the kinetic energy of the large block... because we don't know the initial speed. I am also not sure if the lagrange approach is correct, but finding the forces along the path seems overly tedious. Can someone point me in the right direction? Should I be looking at momentum instead.

 

[Scronin267]

Homework Statement

A block in the shape of a quarter circle moves at constant acceleration across the floor. At some moment a small block of mass m, which is at rest with respect to the floor, is placed atop the horizontal part of the large block. There is no friction in this problem. Assuming the small block does not fly off the large block, what is the greatest altitude increase it will achieve? What will happen after this height is obtained?[/B]

Homework Equations

potential energy U = mgh
kinetic energy T = 1/2mv^2
langrangian L = T-U, ∂L/∂q=d/dt ∂L/∂q'[/B]

The Attempt at a Solution

I do not know how to handle the kinetic energy of the large block... because we don't know the initial speed. I am also not sure if the lagrange approach is correct, but finding the forces along the path seems overly tedious. Can someone point me in the right direction? Should I be looking at momentum instead.
[/B]

 

[Nidum]

I haven't so far been able to think of any plausible interpretation of this problem .

Can you explain more ? Was there a diagram provided ?

Edit : I see that you have now provided a diagram . Block is somewhat different to what was originally described .

 

[Scronin267]

I haven't so far been able to think of any plausible interpretation of this problem .

Can you explain more ? Was there a diagram provided ?

There is a .jpg attached with the diagram. I think the picture is worth 10^3 words. Let me know if you can't see this, or still unclear. Thanks.

 

[Scronin267]

Here's the .jpg again.

 

[TSny]

So, just to be clear, there is no information given regarding the following:
(1) the speed of the large block relative to the floor at the instant the small block is placed on the large block.
(2) the initial position of the small block relative to the large block, other than it's somewhere on the horizontal section.

 

[Scronin267]

So, just to be clear, there is no information given regarding the following:
(1) the speed of the large block relative to the floor at the instant the small block is placed on the large block.
(2) the initial position of the small block relative to the large block, other than it's somewhere on the horizontal section.

correct!

 

[TSny]

correct!

OK. You'll just have to express the answer in terms of these unknown initial conditions.

Are you familiar with setting up a problem in an accelerating reference frame?

 

[Scronin267]

OK. You'll just have to express the answer in terms of these unknown initial conditions.

Are you familiar with setting up a problem in an accelerating reference frame?

errr. Not really. I am also working a problem involving rotational reference frames (i.e. coriolis, and centrifugal fictitious forces), but am confused by that too. I know about coordinate transforms... kinda. just things like x'=x+at, for an accelerating frame. I guess, I'm lost!

 

[TSny]

errr. Not really. I am also working a problem involving rotational reference frames (i.e. coriolis, and centrifugal fictitious forces), but am confused by that too. I know about coordinate transforms... kinda. just things like x'=x+at, for an accelerating frame. I guess, I'm lost!

I'm afraid this doesn't sound too promising. This is not the place to try to explain things from first principles. However, if you want to look at a brief discussion of working in a uniformly accelerated frame, then you can try http://www.iitg.ernet.in/asil/Lecture-14.pdf. If you understand well the first 4 or 5 pages of these notes, then you should be able to make a good attempt at this problem.

 

[Scronin267]

I understand that page up until tidal forces. I feel like after that is applied to rotation, which should not be relevantn (I should read the whole thing, but time is short!)? I don't know how to apply this to the given system. The acceleration of the large block in the inertial ref. frame causes the fictitious force in the non-inertial A. This A is the acceleration forcing the small block back and up the quarter pipe. The small block will have a velocity at the top of the curved portion, which will propel it against g, for some distance easily found with d=v_ot-1/2gt^2.

 

[TSny]

OK. So in the frame of the big block, the little block experiences a constant, horizontal, fictitious force to the right.

My next hint would be: "work-energy theorem".

 

[Scronin267]

I understand that page up until tidal forces. I feel like after that is applied to rotation, which should not be relevantn (I should read the whole thing, but time is short!)? I don't know how to apply this to the given system. The acceleration of the large block in the inertial ref. frame causes the fictitious force in the non-inertial A. This A is the acceleration forcing the small block back and up the quarter pipe. The small block will have a velocity at the top of the curved portion, which will propel it against g, for some distance easily found with d=v_ot-1/2gt^2.

So I guess, the question is how to find the velocity of the small block atop the curved portion of the ramp? There must be some way to find the work the large block is doing? then this work is simply w=K.E + U.E. We are looking for the max height, thus K.E=0. Then, w=mgh.
Then how do I explain that this system is a damped H.O, which is correct I believe?

 

[Scronin267]

Ohh, just w=∫F.dr?
which you calculate for the circular section of the ramp. Then find the speed at the top of curved portion K.E=w-mgR (we don't have R, Ohh well).

 

[Scronin267]

Ohh, just w=∫F.dr?
which you calculate for the circular section of the ramp. Then find the speed at the top of curved portion K.E=w-mgR (we don't have R, Ohh well).

where F = F_fictitious

 

[TSny]

The work done by the fictitious force is easy to find. The force is constant and always acts to the right (+x direction).

It is just like the real gravitational force that is constant and always acts vertically downward (-y direction). You should know how to easily find the work done by the force of gravity on an object even if the object moves along a curved path. Apply the same idea to the fictitious force.

 

[Scronin267]

Hmm, I think I got it
The max height is simply the horizontal distance traveled by the small block in the large block reference frame. This distance was unspecified, but call it x_o. Thus, h_max=x_o.
This will be a S.H.O, based won the equivilance princ, because the small block can't tell the difference between the grav acceleration g, and the referece fram acceleration also g.
Thank for the guidance!

 

[TSny]

Hmm, I think I got it
The max height is simply the horizontal distance traveled by the small block in the large block reference frame. This distance was unspecified, but call it x_o. Thus, h_max=x_o.

Can you show in detail how you got this? What about the initial velocity of the small block relative to the large block?

This will be a S.H.O,

It will not be a SHO. The net force does not obey Hooke's law. But the small block will oscillate. You mentioned damping previously. Do you still think it's damped?

based won the equivilance princ, because the small block can't tell the difference between the grav acceleration g, and the referece fram acceleration also g.

Yes, that's right.