I see you're eager to understand the options provided for the question on uniform circular motion. Let's take a closer look at each option and think about the underlying concepts.

We have four options: A) $\textbf{F}_c = \frac{mv^2}{r}$, B) $\textbf{F}_c = \frac{mr}{v^2}$, C) $\textbf{F}_c = mv^2r$, and D) $\textbf{F}_c = \frac{m}{rv^2}$. To evaluate these options, let's consider the relationships between the variables involved.

As we discussed earlier, centripetal force is directly proportional to the mass of the object and the square of its velocity. It's also inversely proportional to the radius of the circle. With these relationships in mind, let's analyze each option.

Option A suggests that the centripetal force is equal to the mass times the velocity squared, divided by the radius. Option B proposes that the centripetal force is equal to the mass times the radius, divided by the velocity squared. Option C states that the centripetal force is equal to the mass times the velocity squared times the radius. Option D suggests that the centripetal force is equal to the mass divided by the radius times the velocity squared.

Now, let's think about the physical meaning of each option. How do the units of each variable relate to the unit of force? What happens to the centripetal force when the mass of the object increases or decreases? What happens when the velocity increases or decreases?

To help you evaluate these options, let's consider a few questions:

 What is the relationship between the mass of an object and its centripetal force?
 How does the velocity of an object affect its centripetal force?
* What role does the radius of the circle play in determining the centripetal force?

By thinking critically about these questions and considering the relationships between the variables, you can develop a deeper understanding of the concept of centripetal force and make an informed decision about which option is the most accurate.

Remember, the goal is not just to choose the correct option, but to understand the underlying principles and concepts that govern uniform circular motion. Take your time, and think carefully about each option. Do you want to discuss any of the specific options in more detail or explore more concepts related to uniform circular motion?