The GATE Grind

GATE 2025 PH – Question 38

Classical Mechanics · small oscillations: coupled oscillations and normal modes · 2 marks · Multiple choice

The figure shows a system of two equal masses m and three massless horizontal springs with spring constants k1 , k2 , k1. Ignore gravity. The masses can move only in the horizontal direction and there is no dissipation. If m= 1, k1 = 2 and k2 = 3 (all in appropriate units), the frequencies of the normal modes of the system in the same system of units are

Two masses between springs k1, k2, k1.
  1. $\sqrt2,\sqrt8$
  2. $\sqrt2,\sqrt6$
  3. $\sqrt3,\sqrt{10}$
  4. $\sqrt3,\sqrt8$

Practise this question in The GATE Grind →

Show answer and explanation

Correct answer: (A) $\sqrt2,\sqrt8$

Explanation

Two equal masses $m$ are connected by springs $k_1$ (wall to mass 1), $k_2$ (between the masses) and $k_1$ (mass 2 to wall).

**Equations of motion:**
$$m\ddot x_1=-k_1x_1-k_2(x_1-x_2),\qquad m\ddot x_2=-k_1x_2-k_2(x_2-x_1).$$

**Normal modes** (by symmetry):
- **In phase** ($x_1=x_2$): the middle spring is not stretched, so $m\ddot x=-k_1x$ and $\omega^2=k_1/m=2$.
- **Out of phase** ($x_1=-x_2$): the middle spring is stretched by $2x$ and $m\ddot x=-(k_1+2k_2)x$, so $\omega^2=(k_1+2k_2)/m=2+6=8$.

**Frequencies:** $\omega=\sqrt2$ and $\sqrt8$ (option A).