The distance r of the block at time t is:

mω² = ma = rω²
vdv/dr = rω²
∫v₀v dv = ω²∫rR/2 r dr
v² - v₀² = ω²(r² - R²/4)
∫R/2r dr/√(r² - R²/4) = ∫₀ᵗ ω dt
Let r = R/2 secθ
dr = R/2 secθ tanθ dθ
∫secθ dθ = ∫ω dt
ωt = ln(2r/R + √4r² - R²/R)
r = R/4(eωt + e⁻ωt)
Answer is option B.