Forces, Springs, And Extensions: A Physics Problem Solved

Hey guys! So, we've got a physics problem on our hands, and it sounds like you're a bit stuck. No worries, happens to the best of us! Let's break this down together. We're dealing with forces, springs, and extensions, which might sound complicated, but we'll get through it step by step. The scenario involves three forces – 10N, 30N, and 20N – applied to objects hung on X, Y, and Z dynamometers (spring scales), respectively. The kicker? The springs in these dynamometers stretch the same amount. Our goal is to figure out the relationship between these forces and the spring extensions. Ready to dive in? Let's go!

Understanding the Basics: Forces, Springs, and Hooke's Law

Alright, before we get to the nitty-gritty, let's refresh our memories on the key concepts here. First up, we have forces. Think of a force as a push or a pull – like when you lift a weight or when gravity pulls you towards the ground. We measure force in Newtons (N). The problem gives us three different forces: 10N, 30N, and 20N. Got it?

Next, we've got springs. Imagine a slinky. When you pull on it, it stretches, right? That's what springs do. They can be stretched or compressed, and they exert a force that tries to return them to their original shape. Now, the cool part: Hooke's Law. This is the superhero of this problem. Hooke's Law tells us how much a spring stretches when you apply a force. The law states that the force (F) exerted by the spring is directly proportional to the extension (x) of the spring. Mathematically, it's expressed as F = kx, where 'k' is the spring constant – a measure of how stiff the spring is. The higher the 'k', the harder it is to stretch the spring.

Now, let’s go back to our dynamometers. They are essentially spring scales. When you hang something on a dynamometer, the object's weight (which is a force) pulls on the spring inside. The spring stretches, and the scale measures the force based on how much the spring extends. The problem says the springs in X, Y, and Z dynamometers stretch the same amount, even though different forces are applied. Let's use this information to analyze the situation and find the relationship we're looking for. This is where the real fun begins!

The Key to the Problem: Equal Extensions

Okay, so the fact that the springs stretch the same amount is super important. That means, for each dynamometer, the force applied is related to the spring's extension in a specific way. Let’s denote the extension as 'x' for all three dynamometers since they are all the same. We know that F = kx, but we also know that the 'x' is constant, which means 'F' will vary with 'k'.

For dynamometer X, the force is 10N, so we can say 10N = kx (where k is the spring constant for X). Similarly, for dynamometer Y, 30N = kx (k is for Y), and for dynamometer Z, 20N = kx (k is for Z). Notice that since the extension 'x' is the same for all three, the spring constant 'k' must vary for each dynamometer to accommodate the different forces. Essentially, since the extension is the same, the springs are not necessarily identical; they could have different spring constants.

So, think about it like this: If the spring stretches the same amount with different forces, that means the springs themselves must have different stiffness. A stiffer spring (higher k value) would require a larger force to achieve the same extension as a less stiff spring. Now, we're ready to tackle the specifics of how the forces relate.

Solving for the Relationships: Putting it All Together

Now that we've set up the basics, let's dive into the core of the problem. We want to understand the relationships between the spring constants of the dynamometers (X, Y, and Z) given that their extensions are the same under different forces. Remember, Hooke's Law is our friend here: F = kx. Since all extensions (x) are equal, let's represent them simply as 'x'. We can write the following equations:

• For dynamometer X: 10N = kx x
• For dynamometer Y: 30N = ky x
• For dynamometer Z: 20N = kz x

Here, kx, ky, and kz represent the spring constants for dynamometers X, Y, and Z, respectively. To determine the relationship, we can divide the equations. For instance, dividing the equation for Y by the equation for X, we get:

(30N) / (10N) = (ky x) / (kx x) 3 = ky / kx

This means ky = 3kx. The spring constant for Y is three times the spring constant for X. You can perform similar operations using the equation for Z. Dividing the equation for Z by the equation for X gives:

(20N) / (10N) = (kz x) / (kx x) 2 = kz / kx

This means kz = 2kx. The spring constant for Z is twice the spring constant for X. Therefore, we can deduce a few important relationships. We are comparing spring constants, but the extension is constant. So when the extension is constant, the spring constant is directly proportional to the forces. This means the dynamometer with the greatest force will have the greatest spring constant, and so on.

The Final Answer and Further Considerations

Based on the analysis, we have the following relationships between the spring constants:

• ky = 3kx
• kz = 2kx

So, we can say that the spring constant for Y is the largest, followed by Z, and then X. This makes sense because the force applied to Y (30N) is the greatest, and to X (10N) is the smallest. Since the springs all extend the same amount, the spring constant needs to be higher to accommodate a greater force. Remember, Hooke's law is at play here! If the extension is the same, a stronger force (like that on Y) necessitates a stiffer spring (higher k value). A less strong force (like the one on X) implies a less stiff spring (lower k value).

Now, let’s consider what would happen if the extensions were not the same. If the springs had different extensions, then the relationship between force and the spring constant would change. The problem would get more complex, and we would have to account for the different extensions.

Finally, this problem illustrates a fundamental principle in physics: the relationship between force, displacement, and the properties of the spring. It's a great example of how we can use Hooke's Law to understand the behavior of springs and how they respond to applied forces. Keep in mind that real-world springs might have complexities not included here, but the basic principle remains the same. You're doing great – keep up the amazing work!

Let's Recap: Key Takeaways

Okay, let's quickly recap what we’ve covered, guys. We've tackled a physics problem involving forces, springs, and extensions. Here's a quick summary:

• Understanding Forces: We know forces are pushes or pulls, measured in Newtons (N).
• Hooke's Law: This is our go-to law, which states that the force exerted by a spring is directly proportional to its extension (F = kx).
• Equal Extensions: The key was that all springs in the dynamometers extended the same amount.
• Spring Constants: The spring constants (k) were different for each dynamometer due to the varying forces applied.
• Relationships: We found that the spring constant for Y was the greatest and for X was the least.

Great job sticking with it! Physics can be challenging, but breaking down problems into smaller parts makes it manageable. Remember to practice these concepts and look for more problems to solve – it’s a great way to solidify your understanding. You’ve got this!

So, next time you see a spring, think about Hooke's Law and the relationship between force and extension. You’re now one step closer to mastering physics! And if you get stuck again, remember you have a whole community ready to help you.