To find the $x$-coordinate of the particle in terms of $A \cos(\omega t - \alpha)$ and determine when it first hits the positive $x$-axis, let’s break down the information provided:

1. Radius ($A$): The particle is moving in a circle with radius $A = 3$.
2. Circular Frequency ($f$): The circular frequency is 60 Hz. This means $\omega = 2\pi f = 2\pi \times 60 = 120\pi$ radians per second.
3. Initial Angle: The initial angle $\alpha$ of the particle’s position is 60° or $\pi/3$ radians.

Step 1: Write the Equation for $x(t)$

The general equation for the $x$-coordinate of a particle moving in a circle is: $x(t) = A \cos(\omega t - \alpha)$ Substituting $A = 3$, $\omega = 120\pi$, and $\alpha = \pi/3$: $x(t) = 3 \cos(120\pi t - \pi/3)$

Step 2: Determine When the Particle Hits the Positive $x$-Axis

The particle hits the positive $x$-axis when the angle $\theta = \omega t - \alpha$ is an integer multiple of $2\pi$. This means that: $120\pi t - \frac{\pi}{3} = 2n\pi$ for $n$ being an integer.

Solving for $t$: $120\pi t = 2n\pi + \frac{\pi}{3}$ $t = \frac{2n\pi + \frac{\pi}{3}}{120\pi}$ $t = \frac{2n + \frac{1}{3}}{120}$

The smallest positive $t$ occurs when $n = 0$: $t = \frac{\frac{1}{3}}{120} = \frac{1}{360} \text{ seconds}$

So, the particle first hits the positive $x$-axis at $t = \frac{1}{360}$ seconds.

Would you like more details on any part of this solution, or have any further questions?

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Here are some related questions to expand understanding:

1. How do we find the $y$-coordinate of the particle at any given time $t$?
2. What is the complete parametric form of the particle’s motion?
3. How would this solution change if the particle were moving clockwise?
4. What is the particle's angular velocity, and how is it related to circular frequency?
5. How does changing the radius affect the time it first hits the positive $x$-axis?

Tip: When solving circular motion problems, always identify the radius, angular frequency, and initial angle for setting up equations efficiently.