GATE 2026 PH – Question 64
Two 1 kg blocks are connected to massless springs of stiffness 8 N/m and 4 N/m on a frictionless floor, as shown, with one spring end fixed to a wall. The normal frequencies are $\omega_H>\omega_L$. Their ratio $\omega_H/\omega_L$ (two decimal places) is _____
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Correct answer: 2.40 to 2.42
Explanation
Two blocks of mass 1 kg are connected in series to a wall by springs: $k_1=8$ N/m (wall to block 1) and $k_2=4$ N/m (block 1 to block 2).
**Equations of motion** ($m=1$):
$$\ddot x_1=-k_1x_1-k_2(x_1-x_2)=-12x_1+4x_2,\qquad\ddot x_2=-k_2(x_2-x_1)=4x_1-4x_2 .$$
**Eigenvalue problem** for $\omega^2$:
$$\begin{vmatrix}12-\omega^2&-4\\-4&4-\omega^2\end{vmatrix}=0\;\Rightarrow\;\omega^4-16\omega^2+32=0 .$$
$$\omega^2=\frac{16\pm\sqrt{256-128}}{2}=8\pm4\sqrt2 .$$
**Ratio of the frequencies:**
$$\frac{\omega_H}{\omega_L}=\sqrt{\frac{8+4\sqrt2}{8-4\sqrt2}}=\sqrt{\frac{13.657}{2.343}}=\sqrt{5.828}=\mathbf{2.41}.$$