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Linked to this question: Comparing Static Frictions

Suppose there is a cuboidal vertical tunnel, and a cubical block in it such that all surface of the block are in contact with the four walls of the cuboidal tunnel respectively. All walls of the tunnel are rough with different $\mu_s$'s. And the weight of the block is $m$ and it is a point mass such that it has only 3 translational degree of freedoms.

The block is in mechanical-equilibrium.

Now the block is being pulled vertically downwards by gravity, whose force is $mg$ and upwards by the four walls of the tunnel which apply static friction.

How to measure static friction provided by four walls? Can we even measure?

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This is a completely unrealistic situation as you have to consider a) The clearances b) Any tilt of the block c) non-flatness of the walls. The force to unstick the block varies greatly with these factors mentioned. –  ja72 May 17 '13 at 14:11
    
Can you tilt the tunnel and allow only one wall to be in contact at the time? –  ja72 May 17 '13 at 14:13
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@ja72 No, and which law of physics is being violated if such a situation is possible ? –  nonagon May 17 '13 at 14:59
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How are you going to make perfectly flat walls and a perfectly fitting cube. What happens if the temperature drops and now you have a press-fit situation, or the temp rises and you have too much clearance? How are you going to control the interface geometry to make a robust measurement? –  ja72 May 17 '13 at 15:21
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These are not physical laws . Just real-life situations . Is it even possible to have a friction-less surface ? NO , but you have those in physics questions . –  nonagon May 17 '13 at 15:23
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3 Answers

If this is purely academic exercise then the answer is no, you cannot discern between the different friction forces because they are all acting along the same line of action (through the cg of the part).

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A short and academic answer would be to simply say that the friction is equal to the gravity force in case of equilibrium. So you just need to know or measure the gravitational force exerted on your cube, which you already know as $mg$ if you know the mass of the cube $m$ and the acceleration of gravity $g$.

Now for something more practical. People who work on rock bolts apply an external force onto bolts (in your case, gravity), which consist basically in a metallic bar inserted in a rock. This is somewhat analogous to your case.

Some procedure is given in this article. The applied force is generally measured using a load cell. The displacement can be measured using LVDT sensors.

In a static, or quasi-static regime, the friction force equals the pull force. In fact, people usually consider the stress tensor. For a given force, you can measure the displacement, thus obtain a force-displacement relationship. This, in turn, can be used to derive constitutive laws of the anchored bolt, which describe how friction evolves as the joint between the bolt and the rock gets decoupled.

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No, you cannot measure this, because the friction would be negligible in comparison to gravity. Anyway, what would hold the block back? In a normal situation, the block would just fall, and friction would play almost no effect, less than even air resistance.

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