A block of mass m on a horizontal table is attached to a spring of force constant k as shown in Figure. The coefficient of kinetic friction between the block and the table is mk. The spring is stretched a distance A and released.

(a) Apply Newton’s second law to the block to obtain an equation for its acceleration d²x/dt² for the first half-cycle, during which the block is moving to the left. Show that the resulting equation can be written d²x’/dt² = - ώ²x’, where x = 0 at the equilibrium position of the spring, and x’ = x - x₀, with x₀ = μ_k mg/k = μ_k g/ώ ².

(b) Repeat part (a) for the second half-cycle as the block moves to the right, and show that d²x’/dt² = - ώ²x’’, where x’’= x + x₀ and x₀ has the same value.

(c) Sketch x(t) for the first few cycles for A = 10x₀.