Spring Weight: Find Unknown Block Weight!

by Admin 42 views
Spring Weight: Find Unknown Block Weight!

Hey everyone! Today, we're diving into a fun physics problem involving springs and weights. Imagine you've got three identical springs hanging from the ceiling. The first one is just chilling, doing nothing. The second one has a 4.50 N block hanging from it. And the third one? Well, it's got a mystery block with an unknown weight. Our mission, should we choose to accept it, is to figure out the weight of that mystery block using the visual clues from the drawing.

Understanding Spring Behavior

Before we jump into solving the problem, let's quickly recap how springs behave. Springs follow Hooke's Law, which states that the force exerted by a spring is proportional to its extension or compression. In simpler terms, the more you stretch or compress a spring, the more force it exerts in response. Mathematically, Hooke's Law is expressed as:

F = k * x

Where:

  • F is the force exerted by the spring.
  • k is the spring constant (a measure of the spring's stiffness).
  • x is the displacement (the amount the spring is stretched or compressed).

In our case, the springs are hanging vertically, so the force stretching the spring is the weight of the block attached to it. The weight, W, is simply the force due to gravity acting on the mass of the block, given by:

W = m * g

Where:

  • m is the mass of the block.
  • g is the acceleration due to gravity (approximately 9.8 m/s²).

Since we're given the weight directly in Newtons (N), we don't need to worry about calculating it from mass and gravity. This simplifies things quite a bit!

Analyzing the Springs

The key to solving this problem lies in comparing the extensions of the three springs. Since the springs are identical, they all have the same spring constant, k. This means that the amount each spring stretches is directly proportional to the weight hanging from it. Let's denote the extensions of the three springs as x1, x2, and x3, corresponding to the first, second, and third springs, respectively.

  • Spring 1: Nothing is attached, so x1 = 0. This is our reference point.

  • Spring 2: A 4.50 N block is attached, so x2 is the extension due to this weight. We can write this as:

    4.50 N = k * x2

  • Spring 3: A block of unknown weight W is attached, so x3 is the extension due to this weight. We can write this as:

    W = k * x3

From the drawing, we need to visually determine the relationship between x2 and x3. For example, if x3 is twice as long as x2, then the weight W must be twice the weight of the block hanging from the second spring. Similarly, if x3 is 1.5 times x2, then the weight W is 1.5 times 4.50 N.

Determining the Weight of the Unknown Block

Okay, guys, let's assume that, upon careful examination of the drawing, we observe that the extension of the third spring, x3, is 1.8 times the extension of the second spring, x2. In other words:

x3 = 1.8 * x2

Now we can use this information to find the weight W of the block hanging from the third spring. We know that:

W = k * x3

And we also know that:

4.50 N = k * x2

We can rearrange the second equation to solve for k:

k = 4.50 N / x2

Now substitute this expression for k into the equation for W:

W = (4.50 N / x2) * x3

Since we know that x3 = 1.8 * x2, we can substitute this into the equation:

W = (4.50 N / x2) * (1.8 * x2)

The x2 terms cancel out, leaving us with:

W = 4.50 N * 1.8

W = 8.10 N

Therefore, the weight of the block hanging from the third spring is 8.10 N. Cool, right?

General Approach

To generalize, here’s the step-by-step approach you can use for similar problems:

  1. Understand Hooke's Law: Make sure you know the relationship between force, spring constant, and displacement (F = k * x).
  2. Analyze the Given Information: Identify the known weights and the corresponding extensions of the springs.
  3. Compare Extensions: Determine the ratio between the extension of the spring with the unknown weight and the extension of the spring with the known weight.
  4. Calculate the Unknown Weight: Multiply the known weight by the ratio of the extensions to find the unknown weight.
  5. Double-Check Your Work: Ensure your answer makes sense in the context of the problem. If the spring is stretched more, the weight should be greater.

Additional Considerations

  • Ideal Springs: We're assuming that the springs are ideal, meaning they obey Hooke's Law perfectly and have no mass. In real-world scenarios, springs may have some mass, which can affect their behavior.
  • Elastic Limit: Springs have an elastic limit, which is the maximum amount they can be stretched or compressed before they become permanently deformed. We're assuming that the springs are not stretched beyond their elastic limit.
  • Damping: In reality, springs can exhibit damping, which is the dissipation of energy due to friction. We're ignoring damping in this problem for simplicity.

Why This Matters

Understanding spring behavior is crucial in many areas of physics and engineering. Springs are used in a wide variety of applications, such as:

  • Suspension Systems: Springs are used in vehicle suspension systems to absorb shocks and provide a smooth ride.
  • Scales: Springs are used in scales to measure weight.
  • Clocks: Springs are used in mechanical clocks to store energy and regulate the movement of the hands.
  • Vibration Isolation: Springs are used to isolate sensitive equipment from vibrations.

By understanding the principles behind Hooke's Law and spring behavior, you can analyze and design these systems more effectively.

Practice Problem

Want to test your understanding? Try this practice problem:

Suppose you have three identical springs hanging from the ceiling. The first spring has nothing attached, the second spring has a 6.0 N block attached, and the third spring has an unknown weight attached. If the extension of the third spring is 2.5 times the extension of the second spring, what is the weight of the block hanging from the third spring?

(Answer: 15.0 N)

Conclusion

So there you have it! By carefully analyzing the extensions of the springs and applying Hooke's Law, we were able to determine the weight of the mystery block. Remember, physics problems often require a combination of understanding the underlying principles and careful observation. Keep practicing, and you'll become a spring-solving pro in no time! Hope this helps, and happy problem-solving, guys! Remember to always double-check your work and have fun exploring the world of physics!