Some questions about toppling of a set of two cylinders

[IgnacioPR]

Dear colleagues of Physics Forum:

I am trying to estimate the inclination (angle) at which the figure presented with the enclosed photo, composed by two cylinders of different densities (ρ₁=2650 kg/m³ and ρ₂=7000 kg/m³), will topple, when progressively inclined from a horizontal position. Let's assume that both cylinders are glued, that is, they cannot displace between them. Let's also assume that the set will not slide and only "fail" through toppling.

Length of the cylinder 1 is l₁=100 mm and length of the cylinder 2 is l₂=250 mm. Both pieces have a radius, r = 27 mm. The top one (c2) is displaced r/2 from the centre, in the way shown in the figure below.

Could you help me on calculating the point of toppling (angle)? I tried to estimate it, but I think I should consider momentums in two different axes...and I do not know how to proceed. Also of interest is to calculate the point (location) at which this set will topple around...

[I hope the photo be easy to understand. The set is rotating around the y-axis]

Ignacio

 

[anorlunda]

I think what you are seeking is center of gravity https://en.wikipedia.org/wiki/Center_of_mass#Center_of_gravity

But you must also assume no sliding or slipping.

 

[IgnacioPR]

Dear anorlunda:

Thanks for your response. It's right, but I have already estimated the centre of gravity. The problem I found is that this centre of gravity of the complete body (composed by the other two), falls out of a vertical symmetry plane, so I don't know how to calculate momentum from the rotation pivot.

Thanks,
Ignacio

 

[anorlunda]

I missed the part about glued in the OP. If they are glued together, then you have a single object. Are you asking how to find the C.O.G. for that object?

 

[gneill]

You're rotating around the y-axis, and from that point of view (looking towards the Z-X plane) the disks are aligned, not offset. So their centers of mass will fall along a straight line that is coincident with the axes of the cylinders.

Here are some sketches of various views of your setup:

The red dots represent the center of mass of the combined object.

 

[IgnacioPR]

Yes, you have a single object but with a different density distribution, since one is a steel cylinder and the other one is a rock cylinder. I have found the COG easily, but I am struggling to find a way for estimating the point of toppling of the entire set. You have momentums created in two dimensions, so I don't know how to estimate the angle of toppling by the composition of two momentums.Thanks for your help.
Ignacio

 

[gneill]

Are you saying that the offset cylinder will create a torque about the pivot point so that the combined object will turn/roll as the platform tilts?

 

[IgnacioPR]

Hi!

Thanks for your response.

I am saying that two momentums or torques will co-exist: one created by the offset cylinder in an axis and other one created by the normal and shear forces, corresponding to the weight of the set. What I don't know is how to analyse these two torques acting in different axes.

Thanks,
Ignacio

 

[IgnacioPR]

Are you saying that the offset cylinder will create a torque about the pivot point so that the combined object will turn/roll as the platform tilts?

Hi! Thanks for your response. I am saying that two momentums or torques will co-exist: one created by the offset cylinder in an axis and other one created by the normal and shear forces, corresponding to the weight of the set. What I don't know is how to analyse these two torques acting in different axes. Thanks, Ignacio

Reference https://www.physicsforums.com/threa...ut-toppling-of-a-set-of-two-cylinders.961575/

 

[gneill]

I'm not sure I can be of much help here. Perhaps if you work with vectors you would be able to find a net torque vector at a point, then find its components projected on to different axes. $\tau = r\times F_g$ where $r$ is the vector from the point of interest to the center of mass, and $F_g$ the vector force of gravity working on the center of mass. A point of interest, for example, might be the center of the base of the bottom cylinder.

You haven't described what forces are keeping the object from slipping or turning on the tilting platform.Is it just static friction? Do you a value for that?

 

[Fred Wright]

Make a drawing like the rectangles in gniell's drawing. Make the left hand side of the bottom rectangle (tilting point) at x=0. Draw a line from the center of mass of the composite system to x=0, z=0. Now draw a line from the center of mass to x= diameter of lower cylinder, z=0. The angle at the vertex of this triangle is the critical angle.