A block of mass M is attached to a spring of negligible mass and can slide on a horizontal surface along the x-direction, as shown above. There is friction present between the block and the surface. The spring exerts no force on the block when the center of the block is located at x=0. At the instant shown above, the block is located at x=0 and moving toward the right with speed v=v0. (b) The above graph indicates the force exerted on the block by the spring as a function of position x, with the positive direction toward the right. On the same graph, draw a graph of the net force on the block, as a function of position x, for situations where the block is moving toward the right.

(c) Is the graph you drew in part (b) consistent with your prediction in part (a) ?

Yes No

Provide claims for why the answers to parts (a) and (b) are or are not consistent.
(d) In the figure above, the block of mass M has come to rest at a location to the right of the spring’s equilibrium position. The spring has a spring constant k, and the coefficients of kinetic and static friction between the block and the surface are μk and μs, respectively.

i. On the dot below, which represents the block at rest and to the right of x=0, draw and label the forces (not components) that act on the block. Each force must be represented by a distinct arrow starting on, and pointing away from, the dot.

ii. Derive an expression for the maximum displacement possible for the block that will allow the block to still remain at rest. Express your answer in terms of M, k, μk, μs, and physical constants, as appropriate.