Work: Elastic Force

Relevance of the Topic

The elastic force is a fundamental concept in physics, playing an essential role in our understanding of the physical world around us. It is closely linked to matter and energy. By studying the elastic force, we delve into the world of potentials and conservative forces, concepts that permeate everything from the movement of a pendulum to the operation of complex electronic devices.

A deep understanding of the elastic force contributes to the development of mathematical skills and enhances the ability to solve problems, making this a key content in the physics curriculum. Moreover, studying the elastic force is a solid preparation for advanced mechanics, such as the study of oscillators, which appear in areas such as engineering, particle physics, and astronomy.

Contextualization

The elastic force is a topic that naturally arises after studying Newton's Laws, more specifically, Newton's Second Law, which defines the concept of force. However, the elastic force differs in nature from forces that originate from gravitational or electromagnetic interaction. While forces like gravity and friction are called field forces, the elastic force is an internal binding force, arising from the molecular structure of a body.

Within the broader scope of physics, matter and energy are central pillars. The study of the elastic force is closely linked to these pillars, since elastic energy is a form of potential energy. Thus, understanding the elastic force is to comprehend how energy can be stored and released within a physical system.

The elastic force finds practical application in our daily lives. It is thanks to it that springs can absorb impacts, elastic bands can be stretched and return to their original shape, the strings of musical instruments can vibrate, among many other applications. Mastery of this topic is, therefore, a crucial step towards understanding the physical world and the operation of numerous technologies.

Theoretical Development

Components

• Elastic Force: The elastic force is the restoring force that acts on a body when it is deformed (stretched, compressed, or twisted). It always acts towards the original equilibrium point of the body. According to Hooke's Law, the force is directly proportional to the distortion and opposite to the direction of its application.

• Hooke's Law: Hooke's Law, formulated by Robert Hooke in 1660, states that the force required to stretch (or compress) a spring is proportional to its extension (or compression). Mathematically, F = -kx, where F is the applied force, k is the spring's elastic constant, and x is the distortion.

• Work: In physics, the term "work" has a specific meaning. When a force is applied to an object and causes a displacement, it is said that work is done. Work is given by the product of the applied force by the displacement produced, in the direction of the force. For the elastic force, work is calculated as W = (1/2)kx².

Key Terms

• Elasticity: It is the property of a material or body to return to its original shape after the application of a force that deforms it. It is the force required to cause this deformation that results in the elastic force.

• Equilibrium Point: It is the state in which a body does not undergo forces that make it move or deform. In the case of the elastic force, the equilibrium point is the original, undeformed state of the spring.

• Elastic Constant: It is a measure of an object's resistance to elastic deformation. The higher the elastic constant, the more difficult it is to deform the object and the stronger the elastic force it exerts.

Examples and Cases

• Springs: Springs are examples of objects that exhibit elastic force. If a spring is stretched or compressed a certain distance from its equilibrium position, it will exert a force towards its original position. This force is the elastic force, and its magnitude is given by Hooke's Law.

• Work on a Spring: If a force is applied to a spring and causes a displacement, the work done by the elastic force can be calculated by the general work formula, W = Fs, where F is the applied force and s is the displacement. In the case of the elastic force, the force is constant and opposite to the displacement, so the work is given by W = (1/2)kx², where x is the displacement from the equilibrium position.

• Elastic Potential Energy: The work done by the elastic force is stored as elastic potential energy. The elastic potential energy is given by U = (1/2)kx², where U is the potential energy, k is the spring's elastic constant, and x is the displacement.

Detailed Summary

Relevant Points

• The Elastic Force, defined by Hooke's Law, is a restoring force that acts in the opposite direction to a deformation and always towards the equilibrium point. The magnitude of this force is directly proportional to the magnitude of the distortion.

• Hooke's Law establishes the mathematical relationship between the elastic force (F), the spring's elastic constant (k), and the distortion or deformation of the spring (x). F = -kx.

• The Work done by the elastic force is given by the product of the force by the distance in the direction of the force. In the case of the elastic force, the work is calculated by W = (1/2)kx².

• Elasticity is the property of materials to return to their original shape after the removal of the deforming force. The force required for this deformation is the elastic force.

• The Equilibrium Point of an elastic system is the state where the elastic force is zero, that is, the system does not undergo forces that make it move or deform.

• The Elastic Constant is a measure of the hardness of an object to elastic deformation and is directly related to the elastic force that the object exerts.

Conclusions

• The elastic force is one of the most common forms of force in nature, found in springs, rubber bands, tendons, among others.

• The elastic force is directly proportional to the distortion of the system. This means that the more the spring is stretched or compressed (distortion), the greater the elastic force.

• The elastic force plays a crucial role in energy conservation. When a spring is stretched, the energy is converted into elastic potential energy, which is stored until the spring is released, returning to its original shape and converting the potential energy back into kinetic energy.

• The elastic constant (k) is an intrinsic property of the material that makes up the elastic system. The higher the value of k, the greater the elastic force for a given distortion, making the object more resistant to deformation.

Exercises

1. (Easy) If the distortion of a spring is doubled, how does the elastic force change according to Hooke's Law?

2. (Medium) If the elastic force in a spring with an elastic constant of 100 N/m is 50 N, what is the distortion of the spring?

3. (Difficult) If a spring with an elastic constant of 200 N/m is stretched by 0.5 m, what is the work done by the elastic force?