A small bead can slide without friction in a radial groove of a horizontal disc rotating with constant angular speed ω. If r is the bead's distance from the centre, its radial equation of motion in an inertial frame is:
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Question

A small bead can slide without friction in a radial groove of a horizontal disc rotating with constant angular speed ω. If r is the bead's distance from the centre, its radial equation of motion in an inertial frame is:

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G1 Physics · Accepted Answer

The bead is constrained to remain in the radial groove, so its angular coordinate is θ=ωt. In polar coordinates, radial acceleration is a_r = d²r/dt² − rω². For a smooth radial groove, the groove force is perpendicular to the groove, so there is no real force along the radial direction. Therefore m[d²r/dt² − rω²]=0, giving d²r/dt² = ω²r.