A \(35-\mathrm{N}\) force is applied to a spring with spring constant \(k=220 \mathrm{N} / \mathrm{m} .\) How much does the spring stretch?

The spring constant, denoted as k, is a crucial factor in defining the behavior of a spring when it comes to stretching or compressing. It quantifies the stiffness of a spring and is defined as the amount of force required to stretch or compress the spring by a unit of length. Mathematically, we can express it as k = F / x, where F is the force applied to the spring, and x is the resulting displacement.

For instance, in our exercise, the given spring has a spring constant of 220 N/m, which means that it takes a force of 220 Newtons to extend or compress the spring by one meter. The higher the spring constant, the stiffer the spring, which implies that a larger force is necessary to produce the same amount of extension or compression for springs with higher k values. Conversely, a low spring constant indicates a less stiff spring that requires less force to be deformed.

Hooke's Law provides us with a direct proportional relationship between the force applied to a spring and the extension or compression it undergoes. According to this principle, F = k * x, the force (F) is equal to the spring constant (k) times the displacement, or extension (x). This means that the extension of a spring is directly proportional to the force applied to it.

In other words, if you double the force applied to a spring, its extension will also double, provided the spring's elastic limit hasn't been exceeded. The exercise posed provides a perfect example: an applied force of 35N on a spring with a constant of 220 N/m. Following the rearranged formula, x = F / k, we determine the stretch (extension) caused by the force. This relationship is linear as long as the force doesn't surpass the spring's elastic limit, which would lead to permanent deformation.

Elasticity in physics refers to the property of a material that allows it to return to its original shape after being deformed. Springs, rubber bands, and even biological structures like tendons exhibit elastic behavior. The key to elasticity is that the deformations are temporary: once the force causing the deformation is removed, the material goes back to its initial shape. This phenomenon is governed by a set of rules known as elastic potential energy.

Elastic potential energy is stored in a material when it is stretched or compressed. In the case of a spring, when it is pulled or pushed beyond its natural length, it stores this type of energy, which is released when the force is removed, allowing the spring to return to its original state. However, if the material is stretched beyond a certain limit, known as the elastic limit, it can become permanently deformed, losing its elastic properties. This is why understanding the elasticity of materials is so crucial in applications that involve repetitive stretching and compressing, to ensure that the material can withstand the forces without suffering permanent deformation.