Question

A particle executes simple harmonic motion between \( x = -A \) and \( x = +A \). If the time taken by the particle to go from \( x = 0 \) to \( \frac{A}{2} \) is 2 s, then the time taken by the particle in going from \( x = \frac{A}{2} \) to \( A \) is:

• A. 3 s
• B. 2 s
• C. 1.5 s
• D. 4 s

Hint:

In SHM, the time taken to travel between points is not uniform and depends on the amplitude and angular frequency.

Answer 1

The Correct Option is D

Step 1: {Using the standard equation of SHM}
\[ \frac{A}{2} = A \sin(\omega t_1) \] \[ \omega t_1 = \sin^{-1}\left(\frac{1}{2}\right) = \frac{\pi}{6} \quad {(i)} \] Step 2: {Calculating total time to reach \( A \)}
\[ A = A \sin \omega (t_1 + t_2) \] \[ \omega (t_1 + t_2) = \sin^{-1}(1) = \frac{\pi}{2} \] \[ \omega t_2 = \frac{\pi}{2} - \frac{\pi}{6} = \frac{\pi}{3} \quad {(ii)} \] Step 3: {Finding the ratio of times}
\[ \frac{t_1}{t_2} = \frac{1}{2} \Rightarrow t_2 = 2t_1 = 2 \times 2 = 4 \, {s} \] Thus, the time taken to go from \( \frac{A}{2} \) to \( A \) is 4 s.