Explanation:

When two particles undergo SHM along parallel lines with the same time period and equal amplitudes, they continue to move in the same direction. At a particular instant, if one particle is at its extreme position while the other is at its mean position, they will cross each other after a further time.

Reasoning:

Let us suppose that two particles A and B are undergoing SHM along parallel lines with the same time period t and equal amplitudes. At a particular instant, let particle A be at its extreme position, and particle B be at its mean position.

From the equation of SHM, we know that the displacement of a particle undergoing SHM can be represented as:

x = A sin(ωt + φ)

where,

x - displacement of particle
A - amplitude of oscillation
ω - angular frequency of oscillation
t - time
φ - phase constant

As both particles are undergoing SHM with the same time period t and equal amplitudes, the equation of SHM for both particles can be represented as:

xA = A sin(ωt + φA)
xB = A sin(ωt + φB)

where,

xA - displacement of particle A
xB - displacement of particle B
φA, φB - phase constants for particle A and particle B, respectively

As particle A is at its extreme position, its displacement can be represented as:

xA = A sin(ωt + π/2)

As sin(π/2) = 1, we get:

xA = A

As particle B is at its mean position, its displacement can be represented as:

xB = A sin(ωt)

As sin(0) = 0, we get:

xB = 0

Now, to find the time at which both particles will cross each other, we need to equate their displacements, i.e.,

xA = xB

A sin(ωt + π/2) = A sin(ωt)

sin(ωt + π/2) = sin(ωt)

sin ωt cos π/2 + cos ωt sin π/2 = sin ωt

cos ωt = 0

ωt = nπ/2

where n is an odd integer.

As both particles are moving in the same direction, they will cross each other after completing half a cycle, i.e., after time period t/2.

Thus, the time at which both particles will cross each other can be represented as:

t = (nπ/2) / ω + t/2

As ω = 2π/t, we get:

t = (nπ/2) / (2π/t) + t/2

t = (nπ/4)t + t/2

t/2 = (nπ/4)t

nπ/4 = 1/2

n = 2

Thus, the time at which both particles will cross each other is:

t = (2π/2) / ω + t/2

t = π/ω + t/2

As both particles have the same time period t, we get:

t = π/ω + t/2 = πt + t/2

t/2 = πt

t = 2π

Hence, both particles will cross each