Introduction:
In this problem, we are given the equation of motion for a point undergoing simple harmonic oscillation. We need to find the time at which the velocity of the point will be equal to half of its maximum velocity.

Given:
Equation of motion: x = a * sin(ωt + π/6)

Analysis:
To solve this problem, we will analyze the equation of motion and use phasor diagram to understand the behavior of the point.

Equation of Motion:
The equation of motion given is x = a * sin(ωt + π/6), where x is the displacement of the point, a is the amplitude of oscillation, ω is the angular frequency, and t is the time.

Phasor Diagram:
A phasor diagram is a graphical representation of a harmonic motion that shows the relationship between displacement, velocity, and acceleration of the point. It is a useful tool to understand the behavior of the point at different points in time.

Key Points:
1. Velocity is given by the derivative of displacement with respect to time.
2. The maximum velocity occurs when the displacement is maximum (a).
3. The velocity is zero at the extreme points of motion (when displacement is zero).
4. The direction of velocity is opposite to the direction of displacement.

Solution:
1. Let's start by finding the maximum velocity of the point. The maximum velocity occurs when the displacement is maximum (a).
- Maximum velocity (v_max) = ω * a

2. We need to find the time at which the velocity is equal to half of its maximum velocity.
- Half of the maximum velocity (v_half) = v_max / 2 = (ω * a) / 2

3. To find the time, we need to find the value of t when the displacement x is equal to zero.
- x = a * sin(ωt + π/6) = 0

4. We can solve this equation to find the values of t when the displacement is zero. The general solution for sin(ωt + π/6) = 0 is:
- ωt + π/6 = nπ, where n is an integer

5. We can rewrite the equation as:
- ωt = nπ - π/6
- t = (nπ - π/6) / ω

6. Now, we substitute the value of t in the equation for velocity:
- v = dx/dt = d(a * sin(ωt + π/6))/dt = ω * a * cos(ωt + π/6)

7. Substituting the value of t in the velocity equation, we get:
- v = ω * a * cos[(ω * (nπ - π/6)) / ω + π/6]
- v = ω * a * cos(nπ - π/6 + π/6)
- v = ω * a * cos(nπ)

8. From the equation, we can see that the velocity will be equal to zero when nπ is an odd multiple of π. In other words, when n is an odd number.

9. Now, we need to find the value of t when