Introduction:
The given problem involves two particles executing Simple Harmonic Motion (SHM) with different displacement equations. We need to find the value of w for which the energy of both particles is the same.

Understanding the problem:
To solve this problem, we first need to understand the concept of energy in SHM. The total mechanical energy of a particle undergoing SHM is the sum of its kinetic energy (KE) and potential energy (PE). In general, the energy of a particle in SHM can be expressed as:

E = KE + PE

The kinetic energy of a particle in SHM is given by:

KE = (1/2) * m * v^2

where m is the mass of the particle and v is its velocity.

The potential energy of a particle in SHM is given by:

PE = (1/2) * k * x^2

where k is the spring constant and x is the displacement of the particle from its equilibrium position.

Finding the value of w:
To find the value of w for which the energy of both particles is the same, we need to equate their energies and solve for w.

Particle 1:
The displacement equation for particle 1 is given as X1 = 8 sin(10t π/6).

To find the velocity equation, we differentiate the displacement equation with respect to time:

V1 = dx1/dt = 8 * (10π/6) * cos(10t π/6)

The kinetic energy equation for particle 1 becomes:

KE1 = (1/2) * m * (8 * (10π/6) * cos(10t π/6))^2

The potential energy equation for particle 1 becomes:

PE1 = (1/2) * k * (8 sin(10t π/6))^2

Particle 2:
The displacement equation for particle 2 is given as X2 = 5 cos wt.

To find the velocity equation, we differentiate the displacement equation with respect to time:

V2 = dx2/dt = -5w * sin wt

The kinetic energy equation for particle 2 becomes:

KE2 = (1/2) * m * (-5w * sin wt)^2

The potential energy equation for particle 2 becomes:

PE2 = (1/2) * k * (5 cos wt)^2

Equating the energies:
To find the value of w for which the energy of both particles is the same, we equate the kinetic and potential energies for both particles:

KE1 + PE1 = KE2 + PE2

Substituting the respective equations, we get:

(1/2) * m * (8 * (10π/6) * cos(10t π/6))^2 + (1/2) * k * (8 sin(10t π/6))^2 = (1/2) * m * (-5w * sin wt)^2 + (1/2) * k * (5 cos wt)^2

Simplifying the equation will give us the value of w for which the energy of both particles is the same.