1. Write the SHM equations

Since both particles have the same amplitude $A$ and time period $T$, they also have the same angular frequency $\omega = \frac{2\pi}{T}.$

Let the displacements be $x_1 = A\cos(\omega t + \phi_1), \qquad x_2 = A\cos(\omega t + \phi_2).$

2. Use initial conditions at $t=0$

For the first particle, at $t=0$ its displacement is $A$: $x_1(0)=A\cos\phi_1=A \implies \cos\phi_1=1.$ So we can take $\phi_1=0,$ and hence $x_1=A\cos\omega t.$

For the second particle, at $t=0$ its displacement is $-\dfrac{A}{2}$: $x_2(0)=A\cos\phi_2=-\frac{A}{2} \implies \cos\phi_2=-\frac12.$ Thus, $\phi_2=\frac{2\pi}{3} \quad \text{or} \quad \frac{4\pi}{3}.$

3. Use the condition “moving towards each other”

Velocity is $v = \frac{dx}{dt} = -A\omega \sin(\omega t+\phi).$

At $t=0$:

• For particle 1, $v_1(0)=-A\omega\sin 0=0.$ It is at extreme position $x=A$, so immediately after $t=0$ it moves leftward.

• For particle 2: $v_2(0)=-A\omega\sin\phi_2.$ To move towards particle 1 initially, since particle 2 starts at $x=-A/2$ (to the left of particle 1), it must move rightward, so we need $v_2(0)>0 \implies -\sin\phi_2>0 \implies \sin\phi_2<0.$ Among the two possible phases, this happens for $\phi_2=\frac{4\pi}{3}.$

Therefore, $x_2=A\cos\left(\omega t+\frac{4\pi}{3}\right).$

4. Condition for crossing each other

They cross when their displacements are equal: $x_1=x_2.$ So, $A\cos\omega t = A\cos\left(\omega t+\frac{4\pi}{3}\right).$ Cancelling $A$, $\cos\omega t = \cos\left(\omega t+\frac{4\pi}{3}\right).$

Using $\cos \alpha = \cos \beta \implies \alpha = 2n\pi \pm \beta,$ we take the nontrivial case: $\omega t = -\left(\omega t+\frac{4\pi}{3}\right)+2n\pi.$ Thus, $2\omega t = 2n\pi - \frac{4\pi}{3}.$ $\omega t = n\pi - \frac{2\pi}{3}.$

For the first positive time, take $n=1$: $\omega t = \pi - \frac{2\pi}{3} = \frac{\pi}{3}.$ Hence, $t=\frac{\pi/3}{\omega} = \frac{\pi/3}{2\pi/T} = \frac{T}{6}.$

5. Check with options

$t=\frac{T}{6}$ which corresponds to Option A.

6. Comparison with stored answer

Stored correct answer: A

Our derived answer: A

So the answer agrees with the stored correct answer.