A uniform disc of mass 4 kg rolls without slipping. A horizontal spring of force constant 24 N/m is attached to the material point that is uppermost in the equilibrium configuration. For small oscillations the spring extension is approximately twice the displacement of the centre. If ω² = N s⁻², find N.
Physics · Simple Harmonic Motion · time period of linear shm
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Question

A uniform disc of mass 4 kg rolls without slipping. A horizontal spring of force constant 24 N/m is attached to the material point that is uppermost in the equilibrium configuration. For small oscillations the spring extension is approximately twice the displacement of the centre. If ω² = N s⁻², find N.

Physics · Simple Harmonic Motion · time period of linear shm

G1 Physics · Accepted Answer

For a rolling disc, Mₑ = 3M/2. The top-point spring has effective spring constant kₑ = 4k with respect to centre displacement. Therefore, ω² = kₑ/Mₑ = 4k/(3M/2) = 8k/(3M). Substitute k = 24, M = 4. Thus, ω² = 8×24/(12) = 16 s⁻². Hence, N = 16.