The displacement of the spring is given by the function:

$x(t) = 12 \cos(t)$

where $x(t)$ is the displacement in centimeters and $t$ is the time in seconds.

For part (a), we are asked to find the displacement when $t = 0$. To do this, we substitute $t = 0$ into the equation:

$x(0) = 12 \cos(0)$

Since $\cos(0) = 1$, we have:

$x(0) = 12 \times 1 = 12 \, \text{cm}$

So, the displacement of the spring when $t = 0$ is 12 centimeters.

Would you like further details or explanations? Here are some related questions you might find interesting:

1. What is the period of the oscillation described by $x(t) = 12 \cos(t)$?
2. How would the displacement change if the equation was $x(t) = 12 \sin(t)$?
3. How do we find the velocity of the weight at a given time $t$?
4. How can we find the acceleration of the weight from the displacement function?
5. What is the relationship between the amplitude of the oscillation and the coefficient in front of the cosine function?

Tip: To analyze spring motion problems, remember that the general form of displacement in simple harmonic motion is $x(t) = A \cos(\omega t + \phi)$, where $A$ is the amplitude, $\omega$ is the angular frequency, and $\phi$ is the phase angle.