Class 9 > Unit # 05: Forces and Matter > Stretching Springs and Hooke's Law

Talha's Physics Academy

Forces and Matter - Spring Extension & Hooke's Law

Explain extension in the springs with the help of graph.

Extension of a Spring

Consider a spring hung from a rigid support so that its top end is fixed. Weights are hung on the other end of the spring; these applied weights are called the load. As the load is increased, the spring is stretched and its length increases.

"The length of a spring increases as the force (load) increases. This increase in length of a spring is known as extension."

Figure: Load-extension graph showing linear proportional region and the elastic limit.

Graph Analysis

Carrying out an experiment to stretch a spring of original length $20\text{ cm}$, the resulting load-extension graph consists of two main parts:

1. Linear Region (Steady Slope): At first, the graph slopes up in a straight line. This shows that the extension increases in equal steps as the load increases uniformly.
2. Non-Linear Region (Bending): Eventually, the graph bends. This happens when the applied load is large enough to exceed the elastic limit, damaging the spring permanently so that it fails to return to its original length.

State and explain Hooke’s Law and derive formula $F = kx$.

Statement

"Within elastic limit, the displacement (extension) produced in a spring is directly proportional to the force applied."

Mathematical Derivation & Explanation

Let $F$ be the applied force (load) and $x$ be the displacement (extension) produced in the spring. According to Hooke's law:

$F \propto x$

Removing the proportionality sign by introducing a constant $k$:

$F = kx$

Where $k$ is the spring constant (a measure of the stiffness of the spring).

The Elastic Limit

Hooke's law is applicable to all kinds of deformation and all types of matter (solids, liquids, or gases) within a specific boundary known as the elastic limit.

The elastic limit defines the maximum force or stress that can be safely applied to a body without causing permanent deformation in its length, volume, or shape. Within this limit, the body fully recovers its original dimensions once the deforming force is removed. Beyond this limit, the spring deforms permanently.

Figure: Load-extension graph showing Hooke's Law.

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