Two holes of unequal diameters d₁ and d₂ (d₁ > d₂) are cut in a metal sheet. If the sheet is heated

Correct Option: Both d₁ and d₂ will increase

Explanation:

When a metal sheet is heated, it undergoes thermal expansion. This process can be understood using the following principles:

1. Isotropic Expansion (Photographic Zoom Analogy):
Thermal expansion of a solid object is isotropic, meaning it expands uniformly in all directions. It behaves exactly like a photographic enlargement. When you enlarge a photograph of a metal sheet containing holes, the entire image scales up. Consequently, both the solid metal parts and the empty spaces (the holes) expand. Thus, the distance between any two points in the sheet increases.

2. Molecular Behavior:
At the microscopic level, heating the metal sheet increases the thermal energy of the atoms, causing them to vibrate with larger amplitudes. As a result, the average distance between all neighboring atoms increases. Since the boundary of each hole is made up of these expanding atoms, the circumference of each hole must increase, which in turn increases its diameter.

3. Mathematical Derivation:
The change in any linear dimension of a solid due to a temperature change is given by the formula for linear thermal expansion:

$L_f=L_i(1+\alpha \Delta T)$

where:
• $L_i$ is the initial linear dimension (such as diameter),
• $L_f$ is the final linear dimension after heating,
• $\alpha$ is the coefficient of linear expansion of the metal (which is positive), and
• $\Delta T$ is the change in temperature (positive when heated).

Since both the diameters _d₁ and _d₂ are linear dimensions of the holes, they will follow this relationship:

$d_{1f}=d_{1i}(1+\alpha \Delta T)$

and

$d_{2f}=d_{2i}(1+\alpha \Delta T)$

Since $\Delta T>0$, both final diameters will be greater than their respective initial values. Therefore, both _d₁ and _d₂ will increase regardless of their initial sizes.