Q.Consider a long steel bar of uniform cross-sectional area $A$ (measured perpendicular to its length) placed under a tensile stress by two equal and opposite forces $\mathbf{F}$ applied at its two ends, each directed outward along the length (axis) of the bar. Consider an internal plane that cuts across the bar making an angle $\theta$ with the length (axis) of the bar. Find the tensile (normal) and shearing (tangential) stresses acting on this plane, and hence determine:

Young’s Modulus: The Stretchiness of a Solid

When you pull on a rubber band, it stretches easily. When you pull on a steel rod of the same size, it barely moves. Both are elastic — they return to their original shape when you let go — but they resist stretching very differently. Young’s modulus is the number that tells you exactly how much a material resists being stretched or compressed lengthwise.

The Intuition: Stiffness per Unit Size

Think of a spring. A stiff spring requires a large force to stretch it a little. A soft spring stretches a lot with a small force. Young’s modulus is like the “stiffness” of a material, but it’s cleverly designed to be independent of the object’s shape and size.

If you take a thick steel rod and a thin steel wire of the same length, the rod is harder to stretch. That’s because you’re pulling on more material. Young’s modulus removes this size effect — it tells you the stiffness of the material itself, not the particular piece you’re holding.

The Precise Definition

Young’s modulus ($E$ or $Y$) is defined as the ratio of tensile stress to tensile strain, as long as the material obeys Hooke’s law (the deformation is reversible and proportional to the force).

Let’s break down the two parts.

Tensile Stress ($\sigma$) is the force per unit area. If you pull with a force $F$ on a rod of cross-sectional area $A$, the stress is:

Stress has units of pressure — pascals (Pa) or $\text{N/m}^2$. It tells you how “intense” the pulling is, regardless of the rod’s thickness.

Tensile Strain ($\varepsilon$) is the fractional change in length. If the original length is $L_0$ and it stretches by $\Delta L$, the strain is:

Strain is a pure number — it has no units. A strain of 0.01 means the rod stretched by 1% of its original length.

Putting it together:

What the Number Tells You

A high Young’s modulus means the material is very stiff — it takes a huge stress to produce even a tiny strain. Steel has $Y \approx 200 \times 10^9$ Pa. A low Young’s modulus means the material is easily stretched. Rubber has $Y \approx 0.01 \times 10^9$ Pa — about 20,000 times smaller than steel.

A Worked Example

A steel wire of length 2.0 m and cross-sectional area $1.0 \times 10^{-6} \text{ m}^2$ is pulled by a force of 100 N. How much does it stretch? (Young’s modulus of steel = $2.0 \times 10^{11}$ Pa)

From $Y = \frac{F L_0}{A \Delta L}$, rearrange:

$\Delta L = \frac{F L_0}{A Y} = \frac{100 \times 2.0}{(1.0 \times 10^{-6}) \times (2.0 \times 10^{11})} = \frac{200}{2.0 \times 10^5} = 1.0 \times 10^{-3} \text{ m} = 1.0 \text{ mm}$ …