Given below are two statements: one is labelled as Assertion A and the other is labelled as Reason R

Q: If the rod is expanding, isn't there an internal force pushing the atoms apart? Why isn't that considered stress?

That's a great question! The force pushing the atoms apart is due to thermal kinetic energy, but 'Stress' in mechanics is specifically defined as the *restoring* force that opposes deformation. If the material deforms freely, it doesn't need to 'fight back' against any external constraint, so the macroscopic mechanical stress is zero.

Q: What if the rod is lying on a rough surface with friction? Will there be stress then?

Yes! If the surface has significant friction, the friction will act as an external force opposing the expansion. In that case, the rod is no longer 'lying freely', and some thermal stress will indeed develop, especially near the center of the rod.

Q: Why does Reason R not explain Assertion A? It seems related.

They are related to the same topic (heating a rod), but R doesn't provide the *logical cause* for A. A rod can be heated and want to expand (Reason R is true), but if it's clamped, it will have massive stress. The only reason stress is zero in Assertion A is the word 'freely'. Reason R completely misses mentioning the 'free' aspect, which is the actual cause.

Q: Is there any situation where a freely lying object develops thermal stress when heated?

Generally, no, if heated uniformly. However, if you heat a freely lying object *non-uniformly* (e.g., heating only the top surface of a thick block), the top wants to expand while the bottom doesn't. This internal mismatch creates internal constraints, leading to thermal stress even without external walls!

Q: What is the formula for thermal stress if the rod was fixed?

If the rod of length $L$ is fixed between rigid walls, the prevented expansion is $\Delta L = L \alpha \Delta T$. The strain is $\frac{\Delta L}{L} = \alpha \Delta T$. Using Young's Modulus ($Y = \frac{\text{Stress}}{\text{Strain}}$), the Thermal Stress becomes $Y \alpha \Delta T$.

Question

Given below are two statements: one is labelled as Assertion A and the other is labelled as Reason R.
Assertion (A) When a rod lying freely is heated, no thermal stress is developed in it.
Reason (R) On heating, the length of the rod increases.
In the light of the above statements, choose the correct answer from the options given below

Options:

(A) Both A and R are true but R is not the correct explanation of A.

(B) A is false but R is true.

(C) A is true but R is false.

(D) Both A and R are true and R is the correct explanation of A.

Sigma Percentile · Accepted Answer

### The Illusion of Expansion and Stress

Imagine you are standing in a completely empty, massive field. You decide to stretch your arms out as wide as you can. Do you feel any resistance from the environment? Do you feel 'stressed' by the air around you? Of course not. You are free to expand your personal space.

Now, imagine you are packed tightly in a crowded subway train during rush hour. You try to stretch your arms out exactly the same way. What happens? You immediately push against the people next to you, and they push back! You feel a massive amount of resistance. You feel *stressed*.

This exact analogy perfectly explains the physics of **Thermal Stress**.

### Analyzing the Assertion (A)

The assertion states: *"When a rod lying freely is heated, no thermal stress is developed in it."*

When we heat a metal rod, we are essentially giving its atoms more kinetic energy. They vibrate more violently and demand more space, causing the macroscopic length of the rod to increase. This is thermal expansion.

However, **stress** in physics is strictly defined as the internal restoring force per unit area ($Stress = \frac{F}{A}$). A restoring force only arises when a material's natural tendency to deform (in this case, expand) is physically restricted or opposed by an external boundary.

Since our rod is lying *freely* on a surface, there are no walls, clamps, or rigid supports stopping it from growing. It expands just like you stretching in the empty field. Because there is zero restriction, there is zero restoring force. Consequently, the thermal stress is exactly zero.

Therefore, **Assertion (A) is absolutely true.**

### Analyzing the Reason (R)

The reason states: *"On heating, the length of the rod increases."*

This is a fundamental law of thermodynamics and material science. For most solid materials, an increase in temperature ($\Delta T$) leads to a proportional increase in length ($\Delta L$), governed by the equation:

$$\Delta L = L \alpha \Delta T$$

Where $\alpha$ is the coefficient of linear expansion. This statement is a universally accepted fact.

Therefore, **Reason (R) is also true.**

### The Crucial Disconnect

Now comes the ultimate test of an Assertion-Reason question: Does (R) correctly explain (A)?

Let's ask the question: *Why* is there no thermal stress in the rod?

Is it *because* the rod increases in length? No!

If we took the exact same rod, clamped it rigidly between two immovable concrete walls, and heated it, it would still *want* to increase in length. The thermal energy still dictates expansion. But because the walls prevent the actual physical expansion, a massive restoring force is generated inside the rod, leading to immense thermal stress (calculated as $Stress = Y \alpha \Delta T$).

The true reason the stress is zero in our original scenario is strictly because the rod is **free and unrestricted**, not simply because it expands. The expansion is a symptom of heating, but the *freedom* is the cause of the zero stress.

Thus, while both statements are factually correct independent of each other, the Reason fails to explain the Assertion.

**Correct Option: (a) Both A and R are true but R is not the correct explanation of A.**

Select Answer:

Visualized Solution

\text{Visualizing the Setup}

\text{Understanding Thermal Stress}

\text{Evaluating Assertion (A)}

\text{Evaluating Reason (R)}

\text{Checking the Explanation}

\text{Final Conclusion}

The Sigma Insight: Thermal Expansion

The Illusion of Expansion and Stress

Analyzing the Assertion (A)

Analyzing the Reason (R)

The Crucial Disconnect

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