For a block of mass m attached to a spring k along the smooth face of a freely moving wedge of mass M and angle θ, which pair gives the equilibrium extension xₑ and the angular frequency ω of small oscillations relative to the wedge?
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Question

For a block of mass m attached to a spring k along the smooth face of a freely moving wedge of mass M and angle θ, which pair gives the equilibrium extension xₑ and the angular frequency ω of small oscillations relative to the wedge?

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G1 Physics · Accepted Answer

At static equilibrium all parts are at rest, so wedge recoil is absent and the spring balances the component of gravity along the incline: kxₑ=mg sinθ, hence xₑ=mg sinθ/k. For oscillations, wedge recoil changes the effective inertia to m e f f = m ( M + m m_eff=m(M+m m e ​ f f = m ( M + m sin²θ)/(M+m), giving ω²=k/ m e f f = k ( M + m ) / [ m ( M + m m_eff=k(M+m)/[m(M+m m e ​ f f = k ( M + m ) / [ m ( M + m sin²θ)].