To solve this problem, we will use the following steps:

1. Determine the angular position of the particle at $t = 2.49 \, \text{s}$ using the angular velocity $\omega$ and the initial condition.

2. Convert the angular position into the x-coordinate of the particle, given the circle's radius and the initial position.

Step 1: Calculate the Angular Position

The angular position $\theta$ is given by: $\theta(t) = \omega t + \theta_0$ where:

• $\omega = 14 \, \text{rad/s}$ is the angular velocity,
• $t = 2.49 \, \text{s}$,
• $\theta_0$ is the initial angular position, determined from the initial condition.

At $t = 0$, the particle has $x = 0.73 \, \text{m}$. The relationship between $x$, $r$, and $\theta_0$ is: $x = r \cos \theta_0$ Given $r = 4.8 \, \text{m}$ and $x = 0.73 \, \text{m}$, solve for $\cos \theta_0$: $\cos \theta_0 = \frac{0.73}{4.8} \implies \cos \theta_0 \approx 0.15208$ This gives: $\theta_0 = \cos^{-1}(0.15208) \approx 1.418 \, \text{rad}$

Now, substitute into the angular position formula: $\theta(t) = 14 \cdot 2.49 + 1.418 \approx 36.276 \, \text{rad}$

Since angular motion is periodic, reduce $\theta$ modulo $2\pi$ to find the equivalent angle within one revolution: $\theta(t) = 36.276 \mod 2\pi \approx 5.497 \, \text{rad}$

Step 2: Calculate the x-coordinate

The x-coordinate is given by: $x = r \cos \theta$ Substitute $r = 4.8 \, \text{m}$ and $\theta = 5.497 \, \text{rad}$: $x = 4.8 \cos(5.497)$

Using trigonometric calculations: $\cos(5.497) \approx 0.707$ $x \approx 4.8 \cdot 0.707 \approx 3.39 \, \text{m}$

Final Answer:

The $x$-coordinate of the particle at $t = 2.49 \, \text{s}$ is approximately: $x \approx 3.39 \, \text{m}$

Would you like a step-by-step visualization or further clarification?

Here are 5 related questions to explore:

1. How can we determine the y-coordinate at $t = 2.49 \, \text{s}$?
2. How does the angular velocity affect the motion of the particle?
3. What happens if the angular velocity is not constant?
4. Can we generalize this for elliptical motion instead of circular?
5. How do we calculate the velocity vector of the particle at any given time?

Tip: Always reduce angles using modulo $2\pi$ for simpler calculations when dealing with circular motion!