A variety of phenomena can be captured by simplistic 1-dimensional models. Learn the essentials of mechanical motion in this stripped-down setting.

Paul wants to hang a picture frame on his wall. To mark out the dimensions, he applies a horizontal force to it in the middle of the frame. If the wall is perfectly smooth, then what is the minimum force that Paul has to exert to keep the frame in equilibrium?

**Details**

- The coefficient of friction \(\mu\) between Paul's hand and the frame is equal to 1.

Paul keeps his luggage box (denoted by B in the image) on frictionless ground and goes to the restroom. When he returns, he finds that someone has put his luggage box (denoted by A) on top of his luggage such that his luggage box is tied with a horizontal rope to a wall. The surface between boxes A and B is rough. Paul applies a force \(F\) in the horizontal direction to take out his luggage.

How does the minimum horizontal force \(F_\textrm{min}\) he needs to apply to remove box B depend on the mass of the boxes?

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