A wedge of mass M and inclination θ can move freely on a smooth horizontal floor. A block of mass m is connected to the top of the wedge by a spring of force constant k lying along the incline. The angular frequency of small oscillations of the block relative to the wedge is:
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Question

A wedge of mass M and inclination θ can move freely on a smooth horizontal floor. A block of mass m is connected to the top of the wedge by a spring of force constant k lying along the incline. The angular frequency of small oscillations of the block relative to the wedge is:

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G1 Physics · Accepted Answer

Let x be the displacement of the block relative to the wedge along the incline. Horizontal momentum conservation couples the wedge recoil to x. The effective inertia for the relative coordinate is m_eff = m(M+m sin²θ)/(M+m). Hence ω² = k/m_eff = k(M+m)/[m(M+m sin²θ)], so ω = √[k(M+m)/{m(M+m sin²θ)}].