A bead is initially at distance r₀ from the centre of a smooth radial groove on a disc rotating with constant angular speed ω. Its initial radial speed is zero. Its radial position after time t is:
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A bead is initially at distance r₀ from the centre of a smooth radial groove on a disc rotating with constant angular speed ω. Its initial radial speed is zero. Its radial position after time t is:

G1 Physics · Accepted Answer

The radial equation is d²r/dt² = ω²r, or d²r/dt² − ω²r = 0. Its general solution is r = Ae^(ωt)+Be^(−ωt). Using r(0)=r₀ gives A+B=r₀. Also dr/dt = ωAe^(ωt)−ωBe^(−ωt), and dr/dt=0 at t=0 gives A=B. Thus A=B=r₀/2. Hence r = (r₀/2)[e^(ωt)+e^(−ωt)] = r₀ cosh(ωt).