Three cylinders with the same size, density, and structure are piled on each oth

Question

Three cylinders with the same size, density, and structure are piled on each other and on top of a rough sruface. Find the minimuni angle which the direction of the force acting between the cylinders and the rough surface makes with the vertical?

NOTE: If there were no cylinder sitting on the two bottom cylinders, this force will be vertical, but since the top cylinder is pushing the two bottom cylinders to the side, the ground will react with an opposite force acting on the cylinders to the side.

Here's my interpretation of the system:

. O
O O
-------

NOTE: This 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. Three cylinders with the same size, density, and structure are piled on each other and on top of a rough sruface. Find the minimuni angle which the direction of the force acting between the cylinders and the rough surface makes with the vertical?

NOTE: If there were no cylinder sitting on the two bottom cylinders, this force will be vertical, but since the top cylinder is pushing the two bottom cylinders to the side, the ground will react with an opposite force acting on the cylinders to the side.

Here's my interpretation of the system:

. O
O O
-------

NOTE: This 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.

Explanation / Answer

if there is no motion then only normal reaction acts on the cylinders

in the vertical direction

so the min angle with the vertical will be zero degree

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