Calculating the Spring Constant of Rubber Bands: An Application of Hooke's Law

Thermodynamics and mechanics are inherently linked, and one fundamental concept in these fields is Hooke's Law. This law describes the relationship between the force applied to a spring and its resulting displacement. Let's explore this concept through a practical example involving rubber bands.

Understanding Hooke's Law

Hooke's Law

Hooke's Law states that the force required to extend or compress a spring is directly proportional to the displacement caused by that force. Mathematically, it can be represented as:

F kx

Where:

F is the force applied to the spring. x is the displacement of the spring. k is the spring constant, a measure of the stiffness of the spring.

This relationship means that a stiffer spring will have a higher spring constant, requiring more force to achieve a given displacement.

The Problem: Stretching Rubber Bands

A practical scenario involves two identical rubber bands. When a force of 30 N is applied, the rubber bands are stretched to a distance of 40 cm. The question is: what is the spring constant of each rubber band?

Series Configuration

For the rubber bands tied end-to-end in series, the total displacement is the sum of the displacements of each rubber band. Each rubber band will experience the same force but will contribute half of the total displacement.

Here, each rubber band is stretched by x 0.20 m. Using Hooke's Law:

F kx

30 N k(0.20 m)

Solving for k:

k 30 N / 0.20 m 150 N/m

So, the spring constant of each rubber band in the series configuration is 150 N/m.

Parallel Configuration

Now, consider the scenario where the rubber bands are side-by-side in parallel. Here, each rubber band experiences half of the total force, but the displacement remains the same as in the series configuration.

Each rubber band is subjected to F/2 15 N, and the total displacement is still 0.40 m. Using Hooke's Law again:

F kx

15 N k(0.40 m)

Solving for k:

k 15 N / 0.40 m 37.5 N/m

So, the spring constant of each rubber band in the parallel configuration is 37.5 N/m.

Conclusion

Understanding the spring constant and how it behaves under different configurations is crucial in mechanical and engineering applications. This simple problem demonstrates the practical application of Hooke's Law in analyzing the behavior of springs, whether in series or parallel configurations.

Further Reading

Hooke's Law - Wikipedia Hooke's Law - Khan Academy Hooke's Law and Mass-Spring Systems - ThoughtCo

By applying Hooke's Law, we can better understand and predict the behavior of elastic materials, which is essential for engineers and physicists alike.