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It seems obvious that a wider box should have more friction because more surface touches the ground. But our notes say area doesn't matter. Why?
It does feel like more area should mean more friction, but it does not, and here is the reason. Kinetic friction is given by f = μN, where μ is the coefficient of friction and N is the normal force. It depends only on the nature of the two surfaces and how hard they are pressed together, not on the apparent contact area. When you spread the same weight over a larger area, the load is shared, so the pressure at each point drops. The actual microscopic contact between surfaces happens only at tiny high spots, and the real contact area depends on pressure, not on the apparent area. A larger apparent area with lower pressure ends up with the same real contact, so the total friction stays the same. This is why a brick slides with the same friction on any of its faces, as long as weight and surfaces are unchanged.
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