The given problem involves a particle of mass 1 kg hanging from a spring with a force constant of 100 N/m. The particle is pulled slightly downward and released, leading to free simple harmonic motion with a time period T. We need to determine the time when the kinetic energy and potential energy of the system become equal, denoted as T/x.

To solve this problem, we can follow the steps outlined below:

1. Understand Simple Harmonic Motion:
Simple harmonic motion (SHM) occurs when a system experiences a restoring force proportional to its displacement from the equilibrium position. In this case, the spring provides the restoring force, and the particle oscillates around the equilibrium position.

2. Equation for Simple Harmonic Motion:
The equation for simple harmonic motion is given by:
x = A * cos(ωt + φ)
where x represents the displacement of the particle, A is the amplitude, ω is the angular frequency, t is the time, and φ is the phase constant.

3. Potential Energy in Simple Harmonic Motion:
The potential energy of a particle undergoing simple harmonic motion can be expressed as:
U = (1/2) * k * x^2
where U is the potential energy, k is the force constant of the spring, and x is the displacement of the particle.

4. Kinetic Energy in Simple Harmonic Motion:
The kinetic energy of the particle can be expressed as:
K = (1/2) * m * v^2
where K is the kinetic energy, m is the mass of the particle, and v is the velocity of the particle.

5. Equating Potential and Kinetic Energy:
To find the time when the kinetic energy and potential energy become equal, we can equate the expressions for potential energy and kinetic energy:
(1/2) * k * x^2 = (1/2) * m * v^2

6. Relationship between Displacement and Velocity:
In simple harmonic motion, there is a relationship between the displacement and velocity of the particle:
v = ω * A * sin(ωt + φ)

7. Substituting the Velocity Expression:
Substituting the expression for velocity into the equation for kinetic energy:
(1/2) * k * x^2 = (1/2) * m * (ω * A * sin(ωt + φ))^2

8. Simplifying the Equation:
Simplifying the equation further:
k * x^2 = m * (ω^2 * A^2 * sin^2(ωt + φ))
k * x^2 = m * ω^2 * A^2 * (1 - cos^2(ωt + φ))
k * x^2 = m * ω^2 * A^2 - m * ω^2 * A^2 * cos^2(ωt + φ)

9. Using Trigonometric Identity:
We can use the trigonometric identity sin^2θ + cos^2θ = 1 to rewrite the equation:
k * x^2 = m * ω^2 * A^2 * (1 - sin^2(ωt + φ))

10. Relationship between Displacement and Time:
In simple harmonic motion, there is a relationship between the displacement and time of the particle: