A block of mass m lies on a horizontal table. The coefficient of static friction between the block and the table is mu_s. The coefficient of kinetic friction is mu_k, with \mu_{\rm k} < \mu_{\rm s}.
1. If the block is at rest (and the only forces acting on the block are the force due to gravity and the normal force from the table), what is the magnitude of the force due to friction?
2. Suppose you want to move the block, but you want to push it with the least force possible to get it moving. With what force F must you be pushing the block just before the block begins to move?
3. Suppose you push horizontally with half the force needed to just make the block move. What is the magnitude of the friction force?
Express your answer in terms of some or all of the variables mu_s, mu_k, and m, as well as the acceleration due to gravity g.
4. Suppose you push horizontally with precisely enough force to make the block start to move, and you continue to apply the same amount of force even after it starts moving. Find the acceleration a of the block after it begins to move.