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It's been a long time since I did problems on friction but here's one...

Consider a cuboidal block of dimensions 123 meters with a total mass of 1 kg. There are 3 cases that we need to look at:

  1. Its base of area 1*2 sq. meters is in contact with the surface
  2. Its base of area 1*3 sq. meters is in contact with the surface
  3. Its base of area 2*3 sq. meters is in contact with the surface

for all the above cases, the coefficient of friction ($\mu$) is a constant. the normal force ($N$) is also the same since the weight of the block is the same and $N$ has to equal the downward acting force mg

since $f = \mu N$, that means the friction force is the same in each of the three cases.

That means the friction force is independent of the orientation of the block!? But that doesn't make sense... with a larger contact area friction should intuitively increase. If friction doesn't, what does? And does the contact area even matter then?

Maddy
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1 Answers1

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The normal force is the same, regardless of which side of the box is contacting the ground. It will always be equal and opposite to $mg$ for an earthly system in equilibrium - but I'm certain you understand that.

On a more intuitive level, as the weight force is equally distributed on the bottom surface of the box, think of dividing this surface into a collection of minuscule areas $da$. Each of these little areas exerts some normal force - in the case of the contact surface with the largest area, this normal force will be of course be smaller, but there are more $da$'s composing the whole surface. For the smallest surface, there will be fewer $da$'s, but each $da$ will exert a greater normal force. All of these normal forces necessarily sum to the same net force ($-mg$), and all of their respective tiny frictional forces will therefore necessarily sum to the same net frictional force.

Editing: Something else that may be tripping you up. The geometry and quirkiness of the contact surface does matter, but $N$ is completely independent of this - $\mu$, however, is absolutely not.

dnxtn
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