Calculating Tension In A Pulley System

Hey, folks! Let's dive into a classic physics problem: figuring out the tension in a wire within a pulley system. We'll break down the question step-by-step, making it super clear and easy to follow. Ready? Let's go!

The Problem: Unpacking the Setup

So, here's the deal: we have a system with blocks B and C connected by a wire that runs over a pulley. Block B weighs 3 kg, and block C weighs 2 kg. We're also given the acceleration due to gravity (g) as 10 m/s², and some trigonometric values: sin(q) = 0.5 and cos(q) = 0.9. The big question is: What's the tension in the wire? The goal is to calculate the tension force in the wire connecting block B (3 kg) to block C (2 kg) in a pulley system, considering gravity (10 m/s²), sin(q) = 0.5, and cos(q) = 0.9. Now, it might sound like a mouthful, but trust me, it's not as scary as it seems. We'll use Newton's Second Law to solve this, which is all about forces, mass, and acceleration (F=ma). This law is fundamental to understanding how objects move in response to forces, and it's key to solving this type of problem. The system involves pulleys, which change the direction of forces without altering their magnitude. The angle (q) is important because it helps us account for forces acting at an incline. Now let's break down each part of the equation to help find the solution.

To properly understand this, we need to begin with drawing a free-body diagram for each block. The diagram will help us visualize all the forces acting upon each of the blocks. For Block B, the forces acting on it are tension (T) from the wire, the weight of the block (which acts downwards and is calculated by multiplying the mass by the acceleration due to gravity), and the component of the weight that acts along the incline plane. For Block C, the forces involved are the tension (T) from the wire, and the block's weight. With these diagrams in place, we can now set up the equations of motion for each block. We'll solve for the tension using the known values of mass, gravity, sine, and cosine.

Step-by-Step Solution: Breaking it Down

Okay, guys, let's break down how to solve this. We're going to use Newton's Second Law (F=ma) for each block. This law tells us that the net force acting on an object is equal to its mass times its acceleration. We'll also need to consider the angle, which influences the forces acting on the blocks. The angle given affects how the weight of Block B interacts with the system due to its position on an inclined plane. Now, since the blocks are connected by a wire, they will both experience the same tension force.

First, let's look at Block C. The forces acting on it are the tension (T) pulling upwards and its weight (Wc) pulling downwards. Because we are considering the force of gravity, the weight of Block C can be calculated using the formula W = mg (mass times gravity). Because the system is accelerating, the tension force will be more than the weight of Block C.

For Block B, we have tension (T) pulling upwards, its weight component pulling downwards along the inclined plane, and the normal force perpendicular to the inclined plane (which we won't need for this calculation). The weight of Block B also needs to be calculated (Wb = m * g), and it can be broken down into components, one along the incline (Wb * sin(q)) and one perpendicular to the incline (Wb * cos(q)). The component along the incline contributes to the motion of the block. This component helps us determine how much the weight of the block is pulling it down the inclined plane. Because Block B's weight is pulling it down the plane and Block C's weight is pulling it up, it is important to consider both blocks when calculating the final tension.

Now, let's write down the equations of motion for each block. We're going to do this by analyzing the forces acting on each block. Since the blocks are connected by a wire, they'll have the same acceleration (a). For Block C, the net force is (T - Wc = m_c * a), and for Block B, we have (T - Wb * sin(q) = m_b * a). The next step is to isolate T, so that we can calculate the tension. This is where we bring in the values we already know, and calculate the weight of each block. Once the weight has been calculated, all the pieces should be in place to calculate the tension.

Calculation Time: Plugging in the Numbers

Alright, let's get down to business and plug in the numbers. We've got all the information we need, so let's get started. Mass of Block C is 2 kg, and its weight will be (2 kg * 10 m/s² = 20 N). Mass of Block B is 3 kg, and its weight will be (3 kg * 10 m/s² = 30 N). We can now write out the equations of motion for each block and set up the calculations to find the tension. The equation for Block C is (T - 20 N = 2 kg * a), and for Block B, it will be (T - 30 N * 0.5 = 3 kg * a). To solve, we can rearrange these equations and solve for the acceleration.

First, let's write the equations for the net force on each block. For block C, the net force is the tension minus the weight: T - 20 N = 2a. For block B, the net force is the tension minus the component of the weight along the incline: T - (30 N * 0.5) = 3a. This gives us: T - 15 N = 3a.

Now, let's solve these equations simultaneously. Rearrange the equation for block C: T = 2a + 20. Substitute this into the equation for block B: (2a + 20) - 15 = 3a. Simplify: 2a + 5 = 3a. Subtract 2a from both sides: a = 5 m/s².

Now that we have the acceleration, we can plug it back into one of the equations to find the tension. Let's use the equation for block C: T - 20 N = 2 * 5. Solve for T: T = 10 + 20. Thus, the tension, T = 30 N. Now, to find the answer, we have to select the one that matches the solution that we calculated.

The Answer: Finding the Right Choice

So, after all that hard work, what's the answer? The tension in the wire is 30 N. Given the options, there's no direct match among the provided answers (a) 10 N, (b) 15 N, (c) 20 N, and (d) 25 N. It looks like there may be a typo. However, in case there wasn't a typo, and we had to select from the provided choices, you'd need to re-evaluate your steps and consider whether any assumptions or calculations were off. Because the tension force is the same throughout the wire, the correct choice should have matched our calculated tension, which is 30 N.

In this case, the correct answer isn't listed in the choices, suggesting there might be an error in the options provided, the problem statement, or the calculations. That's okay! It happens. The important thing is that you've followed the process and understood how to solve the problem. Always double-check your work, and if you suspect an error, don't hesitate to re-evaluate your steps. This process helps cement your understanding and boosts your problem-solving skills.

Conclusion: Key Takeaways

• Always start with a clear diagram. Drawing a free-body diagram helps you visualize the forces at play. Always label the forces and the direction that they are applied in.
• Apply Newton's Second Law. It's your go-to tool for these problems. Make sure to use the appropriate equations for the situation you're in.
• Consider the angles. Remember how the angle affects the components of the weight. Make sure you account for it in the equations.
• Check your work. Always take a moment to make sure your answer makes sense. Look out for any possible errors.

So, there you have it! You've successfully solved a pulley system problem. Now you know how to calculate tension, and you're ready to tackle more physics challenges. Keep practicing, keep learning, and you'll become a physics pro in no time! Congrats on taking the time to learn, and always be eager to explore the wonders of physics!