GATE Mechanical 2024 | Question: 45

Question

A vibratory system consists of mass $m$, a vertical spring of stiffness $2 k$ and a horizontal spring of stiffness $k$. The end $\mathrm{A}$ of the horizontal spring is given a horizontal motion $x_{A}=a \sin \omega t$. The other end of the spring is connected to an inextensible rope that passes over two massless pulleys as shown. Assume $m=10 \mathrm{~kg}, k=1.5 \mathrm{kN} / \mathrm{m}$, and neglect friction. The magnitude of critical driving frequency for which the oscillations of mass $m$ tend to become excessively large is ___________ $\mathrm{rad} / \mathrm{s}$ (answer in integer).