GATE Mechanical Engineering (ME) 2021 Shift 1 Solved Paper

Question : 40 of 65

Marks: +1, -0

Consider a two degree of freedom system as shown in the figure, where PQ is a rigid uniform rod of length, b and mass m.

Assume that the spring deflects only horizontally and force F is applied horizontally at Q. For this system, the Lagrangian, L is

$\;{1}/{2} (M + m) x↖{˙}↙{2} + \;{1}/{2}mbθ ↖{{˙}x↖{{˙}\cos ⁡θ + \;{1}/{6}mb}↙{2}θ ↖{˙}↙{2}−\;{1}/{2}kx}↙{2} + mg\;{b}/{2}\cos ⁡θ$
$\;{1}/{2}mx↖{˙}↙{2} + \;{1}/{2}mbθ ↖{{˙}x↖{{˙}\cos ⁡θ + \;{1}/{6}mb}↙{2}θ ↖{˙}↙{2}−\;{1}/{2}kx}↙{2} + mg\;{b}/{2}\cos ⁡θ + fb sin \;⁡θ$
$\;{1}/{2}mx↖{˙}↙{2} + \;{1}/{2}mbθ ↖{{˙}x↖{{˙}\cos ⁡θ + \;{1}/{6}mb}↙{2}θ ↖{˙}↙{2}−\;{1}/{2}kx}↙{2}$
$\;{1}/{2} (M + m) x↖{˙}↙{2} + \;{1}/{2}mb^{2}θ ↖{˙}↙{2}−\;{1}/{2}kx^{2} + mg\;{b}/{2}\cos ⁡θ$

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