Circular Motion: Period And Frequency Calculation

Hey guys! Let's dive into a physics problem that deals with circular motion. We have an object moving in a circle with a radius, and we need to figure out its period and frequency. Sounds interesting, right? Buckle up, and let's get started. Understanding circular motion is fundamental in physics, as it appears in numerous real-world scenarios, from the spinning of a merry-go-round to the orbit of planets around the sun. The period and frequency are key properties that describe how the object moves, providing insights into its rotational speed and the time it takes to complete a full cycle. In this discussion, we'll break down the concepts, equations, and steps required to solve this problem, ensuring you grasp the core principles. It's like a fun puzzle that uses math to explain how things move in circles. This knowledge is important, as it forms the basis for understanding more complex physics concepts, such as centripetal force, angular momentum, and rotational kinetic energy. We'll start with the basics, define the parameters, and work through the calculations step by step. We'll explore how the radius of the circular path, the linear velocity of the object, and the relationships between the period and frequency. By the end of this, you should be able to solve similar problems. So, let’s get into the world of physics! Ready? Then let's start!

Understanding the Problem: Parameters and Definitions

Okay, before we start solving, let’s define all the parameters and what they mean. Our problem states that an object is moving in a circular path. We're given a radius (${r}$) of 3 radians and a linear velocity (${v}$) of 4 m/s. Our goal is to calculate the period (${T}$) and the frequency (${f}$). In circular motion, an object follows a path around a central point, and there are several key parameters we need to understand. The radius (r) is the distance from the center of the circle to the object's position, defining the size of the circular path. The linear velocity (v), represents how fast the object is moving along the circular path. The period (T) is the time it takes for the object to complete one full revolution or cycle around the circle, usually measured in seconds. This essentially indicates the time required to complete one full rotation. The frequency (f), is the number of complete revolutions the object makes per unit of time, typically measured in Hertz (Hz). This shows how quickly the object is spinning. Now, since we have the radius, the linear velocity, and we are looking for the period and frequency. Knowing these values and their definitions sets the foundation for our calculations, which will follow. So, it's like having all the ingredients before starting to cook your favorite dish. We are ready to begin, so let’s get started with our calculations, where we can apply the equations to solve for the unknown values. Let's start with the period calculation.

Calculating the Period (T)

Now, let's find the period. The period (${T}$) is the time it takes for the object to complete one full revolution. To calculate the period, we can use the following formula, which relates the linear velocity, and the radius:
${ T = \frac{2\pi r}{v} }$

where:

• ${T}$ = period (in seconds)
• ${r}$ = radius (in meters)
• ${v}$ = linear velocity (in meters per second)

In our case, the radius, ${r}$, is 3 meters, and the linear velocity, ${v}$, is 4 m/s. Let’s plug these values into the formula:
${ T = \frac{2 \pi \times 3}{4} }$

Let's get the math working:
${ T = \frac{6\pi}{4} \approx \frac{18.85}{4} \approx 4.71 \text{ seconds} }$

So, the period ${T}$ is approximately 4.71 seconds. This means the object takes about 4.71 seconds to complete one full cycle around the circle. That's how simple it is to calculate the period when we know the radius and the linear velocity. Using the right formula, we can quickly determine how long one complete rotation takes. Now that we have calculated the period, let's proceed to determine the frequency.

Calculating the Frequency (f)

Alright, now that we have the period, let's calculate the frequency. The frequency (${f}$) is the number of complete revolutions the object makes per unit of time, usually measured in Hertz (Hz). Frequency is the inverse of the period. This means that if you know the period, you can easily find the frequency, and vice versa. The relationship between frequency and period is given by the following equation: ${ f = \frac{1}{T} }$

where:

• ${f}$ = frequency (in Hertz, Hz)
• ${T}$ = period (in seconds)

We've already calculated the period ${T}$ to be approximately 4.71 seconds. Let's substitute this value into the formula to find the frequency: ${ f = \frac{1}{4.71} \approx 0.21 \text{ Hz} }$

So, the frequency ${f}$ is approximately 0.21 Hz. This means the object completes about 0.21 cycles every second. Since the frequency is the inverse of the period, we can see that as the period increases, the frequency decreases. The object is going around a bit slower. So, there you have it: The frequency is the flip side of the period! Having calculated both, we've successfully solved the problem. You can now determine the period and frequency of an object in circular motion. It's like having a complete picture of the object's movement.

Conclusion: Summary of Results and Key Takeaways

Alright, let’s wrap things up and summarize what we've found and what it all means! In this problem, we analyzed an object moving in circular motion. We were given a radius of 3 meters and a linear velocity of 4 m/s. Our goal was to calculate the period and frequency of the object's motion. To recap, we started by identifying the important parameters: the radius (${r}$), the linear velocity (${v}$), the period (${T}$), and the frequency (${f}$). We then used these parameters in two key formulas.

• Period (T): We found that the period (T) is approximately 4.71 seconds, meaning the object takes about 4.71 seconds to complete one full revolution.
• Frequency (f): The calculated frequency (f) is approximately 0.21 Hz, indicating that the object completes about 0.21 cycles per second.

The key takeaways from this exercise are that:

1. Understanding the relationship between linear velocity, radius, period, and frequency is critical for solving circular motion problems.
2. The period and frequency are inversely related, with the frequency being the reciprocal of the period.
3. Using the right formulas is key to performing these calculations effectively and accurately.

Knowing how to apply these formulas allows us to analyze and predict the motion of objects in circular paths. This knowledge is important in various fields, like engineering, astronomy, and everyday life. So, whether you are trying to understand the motion of planets or the spinning of a wheel, the concepts we've discussed today are important. Keep practicing, and you'll become more familiar with these concepts.

Well, that’s all for today. Hope you enjoyed it! Bye! And happy learning, guys!