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When solid is heated , its length chang...

When solid is heated , its length changes according to the relation `l=l_0(1+alphaDeltaT)` , where l is the final length , `l_0` is the initial length , `DeltaT` is the change in temperature , and `alpha` is the coefficient of linear is called super - facial expansion. the area changes according to the relation `A=A_0(1+betaDeltaT)`, where A is the tinal area , `A_0` is the initial area, and `beta` is the coefficient of areal expansion.
A metal disc having a circular hole at its center is heated . If the metal expands on heating , the diameter of the hole

A

increases.

B

decreases

C

remains unchanged

D

increases or decreases depending on the metal .

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The correct Answer is:
To solve the problem regarding the expansion of a metal disc with a circular hole at its center when heated, we can follow these steps: ### Step-by-Step Solution: 1. **Understanding the Problem**: We have a metal disc with a circular hole at its center. When the disc is heated, we need to determine how the diameter of the hole changes due to the thermal expansion of the metal. 2. **Initial Parameters**: - Let \( D_0 \) be the initial diameter of the hole. - Let \( \alpha \) be the coefficient of linear expansion of the metal. - Let \( \Delta T \) be the change in temperature when the metal is heated. 3. **Linear Expansion of the Metal**: According to the linear expansion formula, the final diameter \( D \) of the hole after heating can be expressed as: \[ D = D_0 (1 + \alpha \Delta T) \] Here, \( D_0 \) is the initial diameter of the hole, and \( \alpha \Delta T \) accounts for the change in size due to heating. 4. **Analyzing the Terms**: - \( D_0 \) is a positive value (the initial diameter). - \( \alpha \) (coefficient of linear expansion) is also a positive value. - \( \Delta T \) is positive since we are heating the metal (final temperature \( T_2 \) is greater than initial temperature \( T_1 \)). 5. **Conclusion**: Since all terms in the expression \( D = D_0 (1 + \alpha \Delta T) \) are positive, it follows that: \[ D > D_0 \] This means that the final diameter \( D \) of the hole will be greater than the initial diameter \( D_0 \). Therefore, the diameter of the hole increases when the metal disc is heated. 6. **Final Answer**: The diameter of the hole increases when the metal disc is heated.
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