The equation of motion:
The given equation represents the displacement of a particle undergoing oscillatory motion. The equation is X = 7cos(0.5πt), where X represents the displacement of the particle at time t.

Understanding the equation:
To understand the equation, we need to know the properties of the cosine function. The cosine function oscillates between -1 and 1, and its value is maximum at 0 and π. Hence, the coefficient 7 in front of the cosine function represents the amplitude of the oscillation.

The time period:
The time period of an oscillation is the time taken for one complete cycle of the motion. In this case, the equation is X = 7cos(0.5πt), where the coefficient of t is 0.5π. Comparing this with the general equation of cosine function A*cos(ωt), we can see that the angular frequency ω is equal to 0.5π.

The time period T of the oscillation can be calculated using the formula T = (2π)/ω. Plugging in the value of ω, we get:
T = (2π)/(0.5π) = 4 seconds

Displacement at maximum:
The maximum displacement of the particle occurs when the cosine function takes its maximum value of 1. In this case, the maximum displacement is given by:
X_max = 7cos(0.5πt) = 7*1 = 7

Finding the time A:
To find the time taken for the particle to move from the position of equilibrium to the maximum displacement, we need to solve the equation X = 7cos(0.5πt) for X = 7. Substituting X = 7 in the equation, we get:
7 = 7cos(0.5πt)

Solving for t:
Dividing both sides of the equation by 7, we get:
1 = cos(0.5πt)

Taking the inverse cosine on both sides, we get:
0.5πt = cos^(-1)(1)

Since the cosine function is maximum at 0 and π, we have:
0.5πt = 0

Solving for t, we get:
t = 0/0.5π = 0

The time taken:
From the above calculation, we can conclude that the particle moves from the position of equilibrium to the maximum displacement in zero seconds (option D).