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I'm struggling to find the correct Force balance in the following situation: enter image description here

The black parts are hard stopers. The blue part is moving perfectly aligned with the Y-axis in the negative direction. The Orange parts will move perfectly aligned with the X-axis, and the blue part will cause the movement. The system is symmetric. There is friction between the orange parts and the blue parts.

The blue part will move until the orange parts hit the hard stoppers.

In that situation, imagine that two forces are applied to the orange parts like in the following situation: enter image description here

How could we describe the system considering:

  • F1, F2 and F3
  • alpha
  • friction between the surfaces
  • Weight of the blue part

My struggle is to find the correct equation to iterate the previous variables and understand what angle and friction would be needed for a given F1 to withstand F2 and F3 without the blue part going up.

Thanks for reading!

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2 Answers 2

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F2 and F3 would be equal if there is static equilibrium, However I am not sure if system is free to accelerate in x-axis, If that is the case then the normal force which I showed as F2 in diagram below would be minimum of F2 and F3 and the difference would be going to the acceleration of whole system

Seeing F.B.D it's evident the relationship you want would be

$kF2\sin\alpha + 0.5 F1 = F2\cos\alpha$

where $k$ is the coefficient of static friction. Also here F1 is the minimum force required to stop blue part from moving upwards assuming F2 is large enough to overcome friction, i.e without application of F1 the blue block will for sure move upwards. Also assuming blue block is massless

Since whole system is symmetrical I considered only one half while drawing F.B.D

F.B.D for the blue part

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I would start by using symmetry to split the problem in half. Then, consider what you're looking for: the value of F1 so that there is no movement. This is a static system. Since we've just split the problem in half, let's define F0 = 1/2 * F1.

Using F2, we can create a free body diagram of the half of the blue wedge. It will have F0 acting in the negative Y direction, F2 acting in the negative X direction. There will be a normal and friction force acting on the wedge as well. Each of these will have an X and Y component. Spend some time thinking about which direction each of these forces acts in.

Once you've calculated the X and Y components of each of these forces, create two equations of statics: the sum of the forces in the X direction = 0, and the sum of the forces in the Y direction = 0.

Solve these simultaneous equations for F0.

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