Classic Incline problem with cylinder

AI Thread Summary
The discussion focuses on analyzing the motion of a cylinder on an incline using an alternate coordinate system. The kinetic energy is expressed in terms of the new coordinates, with potential energy adjusted for the center of mass, while the transformation back to the original x-y system raises questions about the definitions of coordinates. Participants clarify the relationship between the coordinates and their derivatives, particularly distinguishing between x and x_p, and discuss the implications of setting the incline angle beta to zero for verifying kinetic energy expressions. The conversation also touches on deriving equations of motion and establishing initial conditions for the system. Overall, the thread emphasizes the complexities of coordinate transformations and their impact on the analysis of motion.
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Homework Statement
Please seee below
Relevant Equations
Please seee below
FOr this,
Xg0AOpjUG5kSkda5tE2YaRRH9qpUvQmMNSROeGoy-iZg5eltVk.png

Use alternate coordinate system
ld6Bcxc8QBtLLX-byrt_J2HKSZLKZpLujuHHeo6PDe-CSUC4Pc.png

With ##ȳ##-axis parallel to incline and ##x̄##-axis parallel to the x-axis. Kinetic energy using this alternate coordinate system is ##T = \frac{1}{2}M\dot x_p^2 + \frac{1}{2}mR^2\dot \phi^2 + \frac{1}{2}m(\dot x̄^2 + \dot ȳ^2 + 2\dot x̄ \dot ȳ \cos\beta)## and potential energy is ##V = -mgȳ\sin \beta##. Note how potential energy of block was not included as we don’t know where the height of the COM is, however, it is a constant, so we can just all the high c, and it will add a ##+C## to the lagrangian

Now transform to original x-y coordinate system using transformations ##ȳ = -\frac{y}{sin \beta}## and ##x̄ = x + y\cot \beta## from decomposing ##ȳ## along the x and y axes. I think that ##\hat ȳ = \hat i - \hat j##. Then one would substitute the these transformations into the kinetic energy and potential energy expressions, however, how does one the ##x## and ##y## in these transformations in terms of ##x_p## and ##\phi##.

Thanks!
 
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The cylinfer has velocity of
x direction : ##R\dot{\phi} \cos \beta + \dot{x}_P##
y direction : -##R\dot{\phi} \sin \beta ##
and potential energy of ##-mgR\phi \sin \beta## which is 0 for ##\phi##= 0
 
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ChiralSuperfields said:
##\frac{1}{2}m(\dot {\bar x}^2 + \dot {\bar y}^2 + 2\dot {\bar x} \dot {\bar y} \cos\beta)##
Check: does that give the right answer when ##\beta=0##?
Btw, the latex works better using \dot{\bar x}.
ChiralSuperfields said:
I think that ##\hat ȳ = \hat i - \hat j##.
Very unlikely that ## \hat i - \hat j## would happen to have unit magnitude, but I'm not sure how you are defining them.
 
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anuttarasammyak said:
The cylinfer has velocity of
x direction : ##R\dot{\phi} \cos \beta + \dot{x}_P##
y direction : -##R\dot{\phi} \sin \beta ##
and potential energy of ##-mgR\phi \sin \beta## which is 0 for ##\phi##= 0
haruspex said:
Check: does that give the right answer when ##\beta=0##?
Btw, the latex works better using \dot{\bar x}.

Very unlikely that ## \hat i - \hat j## would happen to have unit magnitude, but I'm not sure how you are defining them.
@anuttarasammyak @haruspex

Thank you both for your replies!

I see how you got the relation for y by using the rolling condition vector decomposition. However, sorry I am confused how you got the relation for x. I get everything apart from the + time derivative of x_p term. Do you please know why it is the time derivative of x_p not the time derivative of x (since it is along the x-axis)?

Also yes, I think that the expression I found the the cylinder velocity works for the special case of ##\beta = 0##. However, it is strange to think about since I already defined a non-orthogonal coordinate system to be parallel to the incline for the current ##\beta## shown in the diagram, so I assume the coordinate system would remain fixed if we change the beta. However, there is a potential option that we could redefine the coordinate system again. Do you please know whether you intended for the coordinate system to also be rotated too as ##\beta## approaches zero or is it just meant to be fixed?

Thanks!
 
ChiralSuperfields said:
Also yes, I think that the expression I found the the cylinder velocity works for the special case of β=0. However, it is strange to think about since I already defined a non-orthogonal coordinate system to be parallel to the incline for the current β shown in the diagram, so I assume the coordinate system would remain fixed if we change the beta. However, there is a potential option that we could redefine the coordinate system again. Do you please know whether you intended for the coordinate system to also be rotated too as β approaches zero or is it just meant to be fixed?
Setting ##\beta =0## is just a way to verify that you have the correct form for the kinetic energy. If it is correct, it should work for any ##\beta.## Of course, the task in this case becomes a bit of a challenge to the unwary reader.. Of all the choices of symbols at your disposal you had to pick ##y## for the direction down the incline and ignore that ##x## and ##y## are already defined in the drawing as the standard Cartesian coordinates. So when ##\beta =0##, your ##y## becomes the already defined ##x## and is orthogonal to the already defined ##y##. That's confusing.

If I were doing this, I would pick ##s## as the generalized coordinate along the incline, with ##s=0##, say, at the upper corner of the wedge and proceed from there. Note that when ##\beta =0##, ##s\rightarrow x##.
 
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You still have not said how you define ##\hat i, \hat j##. If those are ##\hat x, \hat y## then I get ##\hat{\bar y}=\cos(\beta)\hat i-\sin(\beta)\hat j##.
 
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ChiralSuperfields said:
Do you please know why it is the time derivative of x_p not the time derivative of x (since it is along the x-axis)?
x_p is x coordinate of the rectangle corner (or any other part fixed point is OK) of the moving triangle. What is x you mentioned ?
 
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anuttarasammyak said:
The cylinfer has velocity of
x direction : ##R\dot{\phi} \cos \beta + \dot{x}_P##
@ChiralSuperfields is using ##x## as the same as ##x_P## and ##\bar x = x_P+R{\phi} \cos \beta##.
 
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kuruman said:
Setting ##\beta =0## is just a way to verify that you have the correct form for the kinetic energy. If it is correct, it should work for any ##\beta.## Of course, the task in this case becomes a bit of a challenge to the unwary reader.. Of all the choices of symbols at your disposal you had to pick ##y## for the direction down the incline and ignore that ##x## and ##y## are already defined in the drawing as the standard Cartesian coordinates. So when ##\beta =0##, your ##y## becomes the already defined ##x## and is orthogonal to the already defined ##y##. That's confusing.

If I were doing this, I would pick ##s## as the generalized coordinate along the incline, with ##s=0##, say, at the upper corner of the wedge and proceed from there. Note that when ##\beta =0##, ##s\rightarrow x##.
haruspex said:
You still have not said how you define ##\hat i, \hat j##. If those are ##\hat x, \hat y## then I get ##\hat{\bar y}=\cos(\beta)\hat i-\sin(\beta)\hat j##.
anuttarasammyak said:
x_p is x coordinate of the rectangle corner (or any other part fixed point is OK) of the moving triangle. What is x you mentioned ?
haruspex said:
@ChiralSuperfields is using ##x## as the same as ##x_P## and ##\bar x = x_P+R{\phi} \cos \beta##.
Thank you for your replies @kuruman, @haruspex and @anuttarasammyak!

Re: Coordinate system. I agree, sorry about that.

Re: Unit vector. Yes, sorry, that is my mistake, that makes sense the unit vector is that.

Re: x vs x_p coordinate. I don't think that x is the same as x_p since x_p ∈ x in the x-y coordinate system (Not x-bar, y-bar coordinate system). The x bar is in the different coordinate system with the x bar and y bar axes. The x that I for the x and y axes.

Thanks!
 
  • #10
Also does someone please know what the GPE of the block is? Is it ok if I just say the heigh of the COM is y = -k, where k is a constant?

For (c), I am trying to solve the equations of motion. In terms of integration constants, I solve the second order ODE coupled system and I have ##\phi = ht + k## and ##x_p = wt + c##. However, does someone please know what is good assumption to make to find Inital conditions?

I was thinking ##\phi(0) = \phi_{qi}##, ##\dot \phi(0) =\dot \phi_{qi}## ##x_p(0) = x_{2e}## ##\dot x_p(0) = \dot x_{2e}##

Thanks!
 
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  • #11
ChiralSuperfields said:
The x that I for the x and y axes.
Does your "x" or "time derivative of x" have something to do with the Lagrangian of
1714287201357.png

? I do not find it in parameter of the Lagrangian.
 
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  • #12
anuttarasammyak said:
Does your "x" have something to do with the Lagrangian of
View attachment 344176
? I do not find it in parameter of the Lagrangian.
Thank you for your reply @anuttarasammyak!

Sorry I am confused what you are saying. Do you please mean that I need ##\phi## and ##x_p## axes?

That is a interesting idea to consider. However, do you please know how one would draw them?
1714287591411.png

Not sure what one would call them but is is certainly not orthogonal. Maybe simi-cyclic coordinate system?
 
  • #13
Axes are x and y as I wrote x- y- directions of velocity in my post #2.
In xy coordinate, ##P(x_p,0)##. The contact point on the slope ##C(x_C,y_C)## in xy axes coordinate is
x_C=x_P+R\phi \cos \beta+x_{C0}
y_C=-R\phi \sin \beta+y_{C0}
where ##(x_{C0}, y_{C0})## is initial position of C when t=0, ##\phi=0##. Then the center of cylinder ##Q(x_Q,y_Q)## are given as
x_Q=x_C+ R\sin \beta
y_Q=y_C+R\cos \beta
by adding constants. All these things are in works of xy coordinates. You get Lagrangian from these Cartesian expressions. Once you got Lagrangian written by ##x_P## and ##\phi## , forget about Cartesian coordinates and go into analytics of ##x_P## and ##\phi## by Lagrange equation, which is the merit of analytical mechanics.
 
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  • #14
ChiralSuperfields said:
x vs x_p coordinate. I don't think that x is the same as x_p
You are right…
As I understand your working, (x, y) is the location of the centre of the cylinder in the coordinates as diagrammed.
If ##(\bar x, \bar y)## is the same point in your ##(\hat{\bar x}, \hat{\bar y})## coordinates then ##x\hat x+y\hat y=\bar x\hat{\bar x}+\bar y\hat{\bar y}##.
Also, ##\hat{\bar x}=\hat x##, ##\hat{\bar y}=\hat x\cos(\beta)-\hat y\sin(\beta)##, whence ##x=\bar x+\bar y\cos(\beta)##, ##y=-\bar y\sin(\beta)##.

This choice of coordinates effectively considers the motion of the cylinder as the same motion as the block (##\dot x_P=\dot{\bar x}##) plus a motion down the slope of the block (##\dot{\bar y}=R\dot\phi##).
 
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