The GATE Grind

GATE 2024 PH – Question 41

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

$\ddot x+\omega^2x=F\cos(\omega t)$, with $x(0)=\dot x(0)=0$. Which is correct?

  1. $x\propto t\sin(\omega t)$
  2. $x\propto t\cos(\omega t)$
  3. $x=\infty$
  4. $x\propto e^{\omega t}$

Practise this question in The GATE Grind →

Show answer and explanation

Correct answer: (A) $x\propto t\sin(\omega t)$

Explanation

This is a driven oscillator at **resonance**: the driving frequency equals the natural frequency $\omega$.

**Solution.** Using Laplace transforms (or a particular solution of the form $x_p=At\sin\omega t$), the particular solution is
$$x_p=\frac{F}{2\omega}\,t\sin\omega t .$$

**Check the initial conditions** for the complete solution $x=x_p$ (no complementary part is needed):
- $x(0)=0$ ✓
- $\dot x=\dfrac{F}{2\omega}(\sin\omega t+\omega t\cos\omega t)$, so $\dot x(0)=0$ ✓

$$x(t)=\frac{F}{2\omega}\,t\sin(\omega t)\;\propto\;t\sin(\omega t)\quad(\text{option A}).$$

The amplitude grows linearly with time.