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Figure Q4b shows a 2DOF system comprising: a block of mass M, supported in a friction-less guide by a pair of linear springs of stiffness k₁; and a mass m, suspended from the block by linear spring of stiffness k2. Coordinates X₁ and X2 describe, from their equilibrium positions, the respective vertical displacements of the block, and the mass m.

i) Write down the system kinetic energy T, and potential energy V in terms of coordinates X₁ and X2. ii) Construct, in terms of coordinates X1 and X2, a coupled pair of equations of motion,by making use of the Lagrange equations in the usual form: \frac{\mathrm{d}}{\mathrm{dt}}\left(\frac{\partial \mathrm{T}}{\partial \mathrm{q}_{\mathrm{t}}}\right)-\frac{\partial \mathrm{T}}{\partial \mathrm{q}_{\mathrm{r}}}+\frac{\partial \mathrm{V}}{\partial \mathrm{q}_{\mathrm{r}}}=\mathrm{Q}_{\mathrm{r}} \quad(\mathrm{r}=1,2, \ldots, \mathrm{m})

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