Pulley Frequency Calculation: A Vunesp Physics Problem

Hey there, physics enthusiasts! Today, we're diving into a classic problem from the Vunesp (SP) entrance exam, a staple for those aiming to ace the test. This problem involves a system of pulleys and belts, and it's all about understanding how their sizes and rotations relate to each other. Don't worry, it's not as scary as it sounds! We'll break it down step by step, making sure you grasp the concepts and can solve similar problems with confidence. So, grab your coffee, get comfortable, and let's unravel this physics puzzle together! This will be a great refresher and help you remember how to solve similar problems for other tests too. By the end, you'll not only have the answer to this specific question, but also a solid understanding of the principles at play.

Understanding the Problem: The Pulley System

The scenario: We're dealing with three pulleys of different sizes connected by belts. The first pulley has a radius of 10 cm, the second 20 cm, and the third 40 cm. The key here is that the belts are taut, meaning there's no slipping between the pulleys and the belts. The largest pulley (40 cm radius) is rotating at a frequency of 5 Hz (Hertz), which means it completes 5 full rotations every second. The question is: what is the frequency of rotation of the middle-sized pulley (20 cm radius)? This seems confusing, but we can totally break it down!

Why this matters: This kind of problem tests your understanding of rotational motion, specifically the relationship between the radius of a rotating object, its frequency (how fast it spins), and the linear speed of a point on its edge. This isn't just theoretical stuff; understanding this helps in real-world scenarios, like how gears work in a car, or how different-sized wheels on a bicycle affect your pedaling effort. We're going to use the relationship between angular speed (how fast something rotates) and the linear speed of a point on the circumference of the pulley. We need to remember that the linear speed of a point on the edge of the pulley is directly related to the distance it travels in one rotation, which is the circumference (2πr), and how many rotations it makes per second (frequency).

Key concepts: This problem revolves around two main ideas: (1) The linear speed of the belt is the same for all pulleys connected by it (since the belt doesn't stretch or slip). (2) The relationship between linear speed (v), radius (r), and frequency (f) is given by v = 2πrf. The belts ensure the linear speed remains constant, which allows us to find the unknown frequency. So, by understanding these fundamental principles, we'll solve this problem. Ready to dive in? Let's go!

Setting Up the Solution: Identifying the Relationships

Okay, before we start crunching numbers, let's get our strategy straight. The core idea here is that the linear speed of the belt is constant throughout the system. Imagine the belt as a long, flexible rope. As one pulley turns, it pulls the rope, and that same rope then causes the other pulleys to turn. Since the rope (belt) doesn't stretch or slip, the speed at which any point on the rope moves is the same. This crucial point lets us relate the motion of the different pulleys. So, the linear speed of the belt must be the same whether it's in contact with the large pulley or the intermediate one. If we can calculate the linear speed of a point on the edge of the big pulley, then we know the linear speed of a point on the edge of the intermediate pulley. Then, we can use that to figure out the intermediate pulley's frequency!

Step 1: Focus on the Belt's Linear Speed: The belt's linear speed is the critical link between the pulleys. Consider the largest pulley (radius r3 = 40 cm) and the intermediate pulley (radius r2 = 20 cm). If the belt moves at speed 'v', then a point on the edge of each pulley must move at the same speed. That's because the belt doesn't stretch and the pulley-belt system is without slipping, meaning the belt moves at the same speed as the point of contact on the pulley.

Step 2: Connecting the Dots with the Formula: The formula v = 2πrf is your best friend here. v represents linear speed, r is the radius, and f is the frequency (in Hertz). Knowing the radius and frequency of the large pulley, we can find the linear speed v of the belt. Then, knowing the linear speed v and the radius of the intermediate pulley, we can calculate its frequency. We'll use this formula to connect the linear speed, radius, and frequency of both the large and intermediate pulleys.

Step 3: Organize Your Data and Know Your Unknowns: Let's write down what we know. For the large pulley: radius (r3) = 40 cm = 0.4 m, frequency (f3) = 5 Hz. For the intermediate pulley: radius (r2) = 20 cm = 0.2 m, and we are trying to find f2 (its frequency). We have to remember to convert cm into meters. This is a very important step to not make a mistake later.

Calculating the Frequency: Crunching the Numbers

Alright, it's time to put our plan into action. Let's start with the big pulley and use the v = 2πrf formula to find the belt's linear speed. This is step one of the calculations, and we must do it correctly! Since the big pulley has a radius of 0.4 m and a frequency of 5 Hz, its linear speed (v) is: v = 2π * 0.4 m * 5 Hz ≈ 12.57 m/s. This is the linear speed of the belt.

Step 1: Determine the Linear Speed: We found that the linear speed of the belt is approximately 12.57 m/s. This is the speed at which any point on the belt moves. Because the belt is connected to both pulleys without slipping, this is also the linear speed of a point on the edge of the intermediate pulley. With that linear speed, we'll calculate the frequency of the intermediate pulley.

Step 2: Calculate the Frequency of the Intermediate Pulley: Now that we know the linear speed (v ≈ 12.57 m/s) and the radius of the intermediate pulley (r2 = 0.2 m), we can use the formula v = 2πrf again to find its frequency. Rearranging the formula to solve for f, we get: f = v / (2πr). Then, let's plug in the numbers for the intermediate pulley: f = 12.57 m/s / (2π * 0.2 m) ≈ 10 Hz. So, the intermediate pulley spins at approximately 10 Hz. See? Not too tough, right? This is the core of the problem, so make sure you understand it!

Step 3: State Your Answer: The frequency of the intermediate pulley is approximately 10 Hz. That's our final answer! So, the intermediate pulley rotates at twice the speed of the largest pulley. This makes sense because it's half the radius, and the linear speed of the belt is constant. This also helps you understand how different sizes of pulleys impact the final speed of rotation. So, knowing this kind of calculation can really help you understand other kinds of physics problems.

Conclusion: Wrapping Things Up and Key Takeaways

Recap: We started with a Vunesp physics problem involving pulleys, belts, and frequencies. We understood the problem, identified key concepts (linear speed, frequency, and the relationship v = 2πrf), set up a plan to find the linear speed, and then use it to find the unknown frequency. Finally, we crunched the numbers and found the frequency of the intermediate pulley.

Key takeaways:

• Understanding the relationship between linear speed, radius, and frequency is fundamental for these problems. This helps you break down the problem quickly.
• Recognizing that the linear speed of the belt is the same for all pulleys connected to it is crucial. This is the key piece of information that ties the entire system together. Without this, it's impossible to solve the problem.
• Master the formula v = 2πrf. Know how to use it and rearrange it to solve for different variables. This formula is one of the most important in physics.

What's next? Now that you understand this problem, try solving similar ones with different radii and frequencies. Practice is key to mastering these concepts. You can also explore problems involving gears, which work on the same principles. Next time you encounter a pulley system, you'll be able to tackle it with confidence. Keep practicing, and you'll be an expert in no time. You got this, guys! Don't be afraid to take your time and review any questions that still seem unclear, and don't hesitate to work together with friends. Good luck! Keep up the good work. The physics world is all yours!