Solution:

Given, time period T = 3 seconds

To find: Time taken to complete 3/8th oscillation

Frequency of the particle:

f = 1/T = 1/3 Hz

Amplitude of the particle:

The amplitude of the particle is not given in the question. Hence, we assume the amplitude to be A.

At mean position:

When the particle is at mean position, its displacement from the mean position is zero. Hence, the equation of motion of the particle at mean position is given by:

x = A cos(2πft)

where, x = 0 (displacement of the particle from mean position), f = 1/3 Hz and t = 0 (initial time)

Hence, A cos(0) = 0

This implies that A = 0. Hence, the particle is at rest at mean position.

At 3/8th oscillation:

To find the time taken to complete 3/8th oscillation, we need to find the displacement of the particle after completing 3/8th oscillation from the mean position.

The displacement of the particle from the mean position after n oscillations is given by:

x = A cos(2πft + nπ)

where, n is the number of oscillations completed by the particle.

To complete 3/8th oscillation, the particle needs to complete 3/8th of a full oscillation.

Hence, n = 3/8 × 1/f = 3/8 × 3 = 9/8

Substituting the values of f, A and n in the above equation, we get:

x = A cos(2π × 1/3 × t + 9π/8)

When the particle completes 3/8th oscillation, its displacement from the mean position is equal to the amplitude of the particle.

Hence, A cos(9π/8) = A

Solving for cos(9π/8), we get:

cos(9π/8) = cos(π/8) = (√2 + √6)/4

Substituting this value of cos(9π/8) in the equation of motion, we get:

x = A(√2 + √6)/4

Hence, the time taken to complete 3/8th oscillation is given by:

x = A(√2 + √6)/4 = A/2 × (√2 + √6)

T = 1/f = 3 seconds

Hence, the time taken to complete 3/8th oscillation is:

t = (3/8)T = (3/8) × 3 = 9/8 seconds

Therefore, the particle will complete 3/8th oscillation in 9/8 seconds.