Finding the Natural Length

Oct 13, 2025 by wbrum.org 36 views

Introduction

In this article, we will explore the concept of an elastic string and how it behaves under different forces. We will use the given information to find the natural length of the string, which is the length it would have if there were no external forces acting on it.

Hooke's Law

Hooke's Law states that the force required to stretch a spring or an elastic string by a distance x is proportional to x. Mathematically, this can be expressed as:

F = kx

where F is the force, k is the spring constant, and x is the distance of stretching.

Given Information

We are given that a force of 100 N stretches an elastic string to a total length of 20 cm. This means that the distance of stretching is 20 cm - L, where L is the natural length of the string. We are also given that an additional force of 100 N stretches the string 5 cm further. This means that the distance of stretching is now 25 cm - L.

Equations

We can use Hooke's Law to write two equations based on the given information:

100 = k(20 - L) ... (1) 100 = k(25 - L) ... (2)

Solving for k

We can solve for k by dividing equation (2) by equation (1):

(25 - L) / (20 - L) = k / k (25 - L) / (20 - L) = 1

Cross-multiplying, we get:

25 - L = 20 - L 5 = 20 - L - (20 - L) 5 = 0

This is a contradiction, which means that our initial assumption that the string is elastic is incorrect. However, we can still use the given information to find the natural length of the string.

Finding the Natural Length

We can use the fact that the force required to stretch the string by a distance x is proportional to x. This means that the force required to stretch the string by 5 cm is proportional to 5 cm. Since the force required to stretch the string by 5 cm is 100 N, we can write:

100 = k(5)

Dividing both sides by 5, we get:

20 = k

Now, we can use equation (1) to find the natural length of the string:

100 = k(20 - L) 100 = 20(20 - L) 100 = 400 - 20L 20L = 300 L = 15

Therefore, the natural length of the string is 15 cm.

Conclusion

In this article, we used the concept of Hooke's Law to find the natural length of an elastic string. We were given that a force of 100 N stretches the string to a total length of 20 cm, and an additional force of 100 N stretches the string 5 cm further. We used the fact that the force required to stretch the string by a distance x is proportional to x to find the natural length of the string, which is 15 cm.

References

Hooke, R. (1678). De potentia restitutiva, or of spring explaining the power of springs. London: J. Martyn.
Halliday, D., Resnick, R., & Walker, J. (2013). Fundamentals of physics. John Wiley & Sons.

Introduction

In our previous article, we explored the concept of an elastic string and how it behaves under different forces. We used the given information to find the natural length of the string, which is the length it would have if there were no external forces acting on it. In this article, we will answer some frequently asked questions related to the topic.

Q: What is Hooke's Law?

A: Hooke's Law states that the force required to stretch a spring or an elastic string by a distance x is proportional to x. Mathematically, this can be expressed as:

F = kx

where F is the force, k is the spring constant, and x is the distance of stretching.

Q: What is the spring constant (k)?

A: The spring constant (k) is a measure of the stiffness of a spring or an elastic string. It is defined as the force required to stretch the spring or string by a unit distance.

Q: How do you find the natural length of an elastic string?

A: To find the natural length of an elastic string, you need to know the force required to stretch the string by a certain distance and the distance of stretching. You can use Hooke's Law to write an equation and solve for the natural length.

Q: What is the difference between an elastic string and a spring?

A: An elastic string and a spring are both elastic objects, but they behave differently under different forces. A spring is a rigid object that can be stretched or compressed, while an elastic string is a flexible object that can be stretched or compressed.

Q: Can an elastic string be stretched beyond its natural length?

A: Yes, an elastic string can be stretched beyond its natural length, but it will not return to its original shape. The string will be permanently deformed and will not behave like an elastic object.

Q: What is the purpose of Hooke's Law?

A: Hooke's Law is used to describe the behavior of elastic objects under different forces. It is a fundamental concept in physics and engineering and is used to design and analyze elastic systems.

Q: Can Hooke's Law be applied to all elastic objects?

A: No, Hooke's Law can only be applied to elastic objects that behave linearly. Non-linear elastic objects do not follow Hooke's Law and require more complex mathematical models to describe their behavior.

Q: What are some real-world applications of Hooke's Law?

A: Hooke's Law has many real-world applications, including:

Designing springs and elastic systems
Analyzing the behavior of elastic materials under different forces
Developing mathematical models to describe the behavior of elastic objects
Understanding the behavior of elastic systems in different environments

Conclusion

In this article, we answered some frequently asked questions related to the topic of an elastic string and Hooke's Law. We hope that this article has provided you with a better understanding of the concept and its applications.

References

Hooke, R. (1678). De potentia restitutiva, or of spring explaining the power of springs. London: J. Martyn.
Halliday, D., Resnick, R., & Walker, J. (2013). Fundamentals of physics. John Wiley & Sons.