Consider two cylindrical rods of identical dimensions, one of rubber and the other of steel. Both the rods are

NCERT Exemplar · Q8

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(a) Both the rods will elongate but there shall be no perceptible change in shape.

(b) The steel rod will elongate and change shape but the rubber rod will only elongate.

(c) The steel rod will elongate without any perceptible change in shape, but the rubber rod will elongate and the shape of the bottom edge will change to an ellipse.

(d) The steel rod will elongate, without any perceptible change in shape, but the rubber rod will elongate with the shape of the bottom edge tapered to a tip at the centre.

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Concept understanding — Youngs Modulus

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).

Y = \frac{\text{Tensile Stress}}{\text{Tensile Strain}}

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:

\sigma = \frac{F}{A}

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:

\varepsilon = \frac{\Delta L}{L_0}

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:

Y = \frac{F/A}{\Delta L / L_0} = \frac{F L_0}{A \Delta L}

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.

Note

Young’s modulus is only valid in the elastic region — where the material returns to its original shape after the force is removed. If you stretch too far (past the elastic limit), the material deforms permanently or breaks, and Young’s modulus no longer applies.

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:

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

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The key idea is that Young’s modulus governs how much a material resists stretching; steel’s huge modulus means negligible lateral contraction, while rubber’s tiny modulus allows significant lateral contraction (Poisson effect) that visibly changes the shape of the bottom edge — the correct option is (D).

Understanding the setup and the concept.

Both rods are identical in dimensions — same length $L$ , same cross-sectional area $A$ . Each is fixed at the top and has a mass $M$ hanging from the free end. The mass applies a tensile force $F = Mg$ along the axis of each rod.

The question is about two effects: elongation (axial strain) and lateral contraction (change in shape of the cross-section). The first is governed by Young’s modulus $Y$ , the second by Poisson’s ratio $\nu$ . But here the key difference is the magnitude of the axial strain, which determines whether the lateral contraction is perceptible.
Elongation: how much does each rod stretch?

For a rod under axial tension, the elongation $\Delta L$ is given by

\Delta L = \frac{F L}{A Y}.

Steel has $Y \approx 2 \times 10^{11} \, \text{Pa}$ , rubber has $Y \approx 5 \times 10^6 \, \text{Pa}$ — a factor of about $40\,000$ difference. For the same $F$ , $L$ , and $A$ , the rubber rod stretches roughly $40\,000$ times more than the steel rod.

So both elongate, but the rubber rod’s elongation is enormous and easily visible, while the steel rod’s elongation is microscopic.

Lateral contraction: the Poisson effect. When a rod stretches axially, it contracts laterally (gets thinner). The lateral strain $\epsilon_\text{lat}$ is related to the axial strain $\epsilon_\text{axial}$ by Poisson’s ratio $\nu$ :

\epsilon_\text{lat} = -\nu \, \epsilon_\text{axial}.

For steel, $\nu \approx 0.3$ ; for rubber, $\nu \approx 0.5$ (nearly incompressible).

The absolute lateral contraction is $\Delta d = -\nu \, \epsilon_\text{axial} \, d$ , where $d$ is the original diameter. Since $\epsilon_\text{axial} = \Delta L / L$ , and $\Delta L$ is tiny for steel, $\Delta d$ is utterly imperceptible. For rubber, $\Delta L$ is huge, so $\Delta d$ is also large — the rod becomes noticeably thinner.

Shape of the bottom edge: why does it change? The mass $M$ is attached at the centre of the free end, not uniformly across the cross-section. This means the load is concentrated at a point. For steel: the rod is so stiff that the point load is transmitted almost uniformly across the cross-section within a very short distance (Saint-Venant’s principle). The bottom face remains essentially flat — no perceptible shape change. …

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Q.Consider two cylindrical rods of identical dimensions, one of rubber and the other of steel. Both the rods are fixed rigidly at one end to the roof. A mass MMM is attached to each of the free ends at the centre of the rods.

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Q.Consider two cylindrical rods of identical dimensions, one of rubber and the other of steel. Both the rods are fixed rigidly at one end to the roof. A mass $M$ is attached to each of the free ends at the centre of the rods.

Unlock everything free for 14 days