JEE Advanced 1989

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Animated Solution for Physics - Properties of Solids and Liquids: Two rods of different materials having coefficients of thermal expansion $α_{1}, α_{2}$ and Young's moduli $Y_{1}, Y_{2}$ respectively are fixed between two rigid massive walls. The rods are heated such that they undergo the same increase in temperature. There is no bending of the rods. If $α_{1} : α_{2} = 2 : 3$ , the thermal stresses developed in the two rods are equal provided $Y_{1} : Y_{2}$ is equal to

Q: Why does the length of the rod not appear in the final formula for thermal stress?

The free expansion $\Delta L$ is proportional to the length $L$, but strain is defined as $\Delta L / L$. Therefore, the length cancels out when calculating strain, making thermal stress independent of the rod's length.

Q: What would happen if the walls were not perfectly rigid?

If the walls were flexible (with some stiffness $k$), they would yield slightly under the force. This would allow some expansion, reducing the strain and consequently lowering the thermal stress developed in the rods.

Select Answer:

Visualized Solution

Visualizing the Setup

We have two rods of different materials, each fixed firmly between two rigid, unyielding walls.
Let's represent the rods with their respective Young's moduli ( $Y_{1}, Y_{2}$ ) and coefficients of linear expansion ( $α_{1}, α_{2}$ ).

Understanding Free Thermal Expansion

If the rods were free to expand, a temperature increase of $Δ T$ would cause a change in length $Δ L_{free}$ given by:
$Δ L_{free} = Lα Δ T$

Defining Thermal Strain

Since the walls prevent this expansion, the compressive strain (thermal strain) $ϵ$ forced upon the rods is:
$ϵ = \frac{Δ L _{free}}{L} = α Δ T$

Relating Stress to Strain via Hooke's Law

According to Hooke's Law, stress $σ$ is related to strain $ϵ$ by Young's Modulus $Y$ :
$σ = Y ϵ$
Substituting $ϵ = α Δ T$ gives the thermal stress equation:
$σ = Y α Δ T$

Setting Up the Equality of Stresses

We are given that the thermal stresses developed in both rods are equal:
$σ_{1} = σ_{2}$
Substituting the formula for each rod:
$Y_{1} α_{1} Δ T = Y_{2} α_{2} Δ T$

Canceling Common Terms

Since both rods undergo the same increase in temperature, we can cancel $Δ T$ from both sides:
$Y_{1} α_{1} = Y_{2} α_{2}$

Expressing the Ratio of Young's Moduli

Rearranging the equation to find the ratio $Y_{1} : Y_{2}$ :
$\frac{Y _{1}}{Y _{2}} = \frac{α _{2}}{α _{1}}$

Calculating the Final Ratio

We are given $α_{1} : α_{2} = 2 : 3$ , which means $\frac{α _{1}}{α _{2}} = \frac{2}{3}$ or $\frac{α _{2}}{α _{1}} = \frac{3}{2}$ .
Therefore:
$\frac{Y _{1}}{Y _{2}} = \frac{3}{2}$
Thus, $Y_{1} : Y_{2} = 3 : 2$ .

Exploring Further Scenarios

What if the walls were not perfectly rigid but had a finite stiffness $k$ ?
How would that affect the thermal stress? Think about how the strain would be shared between the rod and the walls.

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The Sigma Insight: Young's Modulus, Bulk Modulus and Modulus of Rigidity

Solution Diagram

Analyzing the Setup

Imagine two rods of different materials, clamped tightly between two massive, unyielding walls.

When we heat these rods, they naturally want to expand and grow longer.

But because the walls are completely rigid, they prevent any actual expansion.

This constraint creates a massive internal struggle—what we call thermal stress.

Let's represent the rods with their respective Young's moduli (

Y_{1}, Y_{2}

) and coefficients of linear expansion (

α_{1}, α_{2}

---

The Physics of Thermal Stress

If the rods were free to expand, a temperature increase of

Δ T

would cause a change in length

Δ L_{free}

given by:

Δ L_{free} = Lα Δ T

Since the walls prevent this expansion, they effectively compress the rods by the exact amount they wanted to expand.

The resulting compressive strain (thermal strain)

ϵ

forced upon the rods is:

ϵ = \frac{Δ L _{free}}{L} = α Δ T

According to Hooke's Law, stress

σ

is related to strain

ϵ

by Young's Modulus

Y

σ = Y ϵ

Substituting

ϵ = α Δ T

gives the thermal stress equation:

σ = Y α Δ T

---

Setting Up the Ratio

We are given that the thermal stresses developed in both rods are equal:

σ_{1} = σ_{2}

Substituting the formula for each rod:

Y_{1} α_{1} Δ T = Y_{2} α_{2} Δ T

Since both rods undergo the same increase in temperature, we can cancel

Δ T

from both sides:

Y_{1} α_{1} = Y_{2} α_{2}

Rearranging the equation to find the ratio

Y_{1} : Y_{2}

\frac{Y _{1}}{Y _{2}} = \frac{α _{2}}{α _{1}}

We are given

α_{1} : α_{2} = 2 : 3

, which means:

\frac{α _{1}}{α _{2}} = \frac{2}{3} ⟹ \frac{α _{2}}{α _{1}} = \frac{3}{2}

Therefore:

\frac{Y _{1}}{Y _{2}} = \frac{3}{2}

Thus, the ratio of Young's moduli

Y_{1} : Y_{2}

is $3 : 2$ , which corresponds to Option (c).

Common Traps & FAQs

Why does the length of the rod not appear in the final formula for thermal stress?

The free expansion

Δ L

is proportional to the length

L

, but strain is defined as

Δ L / L

. Therefore, the length cancels out when calculating strain, making thermal stress independent of the rod's length.

What would happen if the walls were not perfectly rigid?

If the walls were flexible (with some stiffness

k

), they would yield slightly under the force. This would allow some expansion, reducing the strain and consequently lowering the thermal stress developed in the rods.

Is thermal stress compressive or tensile when a rod is heated?

When heated, a rod wants to expand. Since the rigid walls prevent this expansion by pushing inward, the stress developed inside the rod is compressive.

Why is Young's Modulus inversely proportional to the coefficient of thermal expansion here?

Because the stresses are equal, a material that wants to expand more (higher

α

) must be easier to compress (lower

Y

) to keep the stress equal to that of a material that expands less.

Does this formula hold if the temperature is lowered instead of raised?

Yes, but if the temperature is lowered, the rod wants to contract. The walls would have to pull outward to prevent contraction, creating tensile stress instead of compressive stress.

Select Answer:

Visualized Solution

Visualizing the Setup

Understanding Free Thermal Expansion

Defining Thermal Strain

Relating Stress to Strain via Hooke's Law

Setting Up the Equality of Stresses

Canceling Common Terms

Expressing the Ratio of Young's Moduli

Calculating the Final Ratio

Exploring Further Scenarios

The Sigma Insight: Young's Modulus, Bulk Modulus and Modulus of Rigidity

Analyzing the Setup

The Physics of Thermal Stress

Setting Up the Ratio

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Select Answer:

Visualized Solution

Visualizing the Setup

Understanding Free Thermal Expansion

Defining Thermal Strain

Relating Stress to Strain via Hooke's Law

Setting Up the Equality of Stresses

Canceling Common Terms

Expressing the Ratio of Young's Moduli

Calculating the Final Ratio

Exploring Further Scenarios

The Sigma Insight: Young's Modulus, Bulk Modulus and Modulus of Rigidity

Analyzing the Setup

The Physics of Thermal Stress

Setting Up the Ratio

Similar Questions

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If Y,K and η are the values of Young's modulus, bulk modulus and modulus of rigidity of any material, respectively. Choose the correct relation for these parameters.

If $Y, K$ and $η$ are the values of Young's modulus, bulk modulus and modulus of rigidity of any material, respectively. Choose the correct relation for these parameters.