Three Cylinders On Top of a Rough Surface and the Force

[. Arctic.]

Homework Statement

Three cylinders with the same size, density, and structure are piled on each other and on top of a rough surface. Find the minimum angle which the direction of the force acting between the cylinders and the rough surface makes with the vertical? The cylinders are stacked like this. There are two on the bottom and one at the top making a triangle shape. I'm not entirely sure if they want a number since no other information is given. I want to say it's at 45° angle, but I keep thinking I have to prove it using various equations. I wouldn't mind a hint. Thanks in advance for your help.

Homework Equations

The Attempt at a Solution

 

[haruspex]

Two things aren't clear: is the surface horizontal? are the cylinders rough enough to prevent slipping between themselves?
If yes to both then I would think you can calculate the angle exactly, so it's not a matter of there being a minimum angle. OTOH, if no to both then I would think there's not enough information.

 

[. Arctic.]

Yeah. The surface they are on top definitely is horizontal and from looking at the drawing provided, the cylinders are in fact rough enough to stay on top of each other without slipping. I may be over thinking it.

 

[haruspex]

Ok, I think I now see why it's still a matter of finding a minimum. So, put in some unknowns for magnitudes and directions of forces (using the symmetry) and write down free body equations for static equilbrium.

 

[. Arctic.]

Here is some additional information.

Can be solved using F=ma in the x and the y direction. Study, for example, the bottom right cylinder. In the y direction, you have the weight of the cylinder, the y-component of the friction, between the top cylinder and the bottom right cylinder, the y-component of the push caused by the top cylinder on the bottom right cylinder. The sum of the y-component of all these 3 forces is equal to 3W/2. In the x-direction you have the push coming in from the bottom left cylinder and acting on the bottom right cylinder, you have the x-component of the friction between the top cylinder and the bottom right cylinder, the friction from the ground acting on the bottom cylinder and another force. To get a minimum angle set the horizontal force acting by the left cylinder on the right cylinder equal to zero.

 

[haruspex]

Quite so. So go ahead and write out the free body equations for one of the lower cylinders. Create symbols for unknown forces as necessary (using a different symbol for each). Look at the sum of vertical forces, the sum of horizontal forces, and moments about the centre of the cylinder.

 

[. Arctic.]

Thanks for all the help.