Question

An iron sphere of diameter $ D $ and mass $ M $ is immersed in hot water, causing its temperature to rise by $ \Delta T $. If $ \alpha $ is the coefficient of linear expansion, the change in surface area of the sphere is:

• A. \( \pi D^2 \cdot \alpha \Delta T (\alpha \Delta T - 4) \)
• B. \( \pi D^2 \cdot \alpha \Delta T (\alpha \Delta T + 4) \)
• C. \( \pi D^2 \cdot \alpha \Delta T (\alpha \Delta T - 2) \)
• D. \( \pi D^2 \cdot \alpha \Delta T (\alpha \Delta T + 2) \)

Hint:

For surface area expansion, use \( \Delta A = A (2\alpha \Delta T + (\alpha \Delta T)^2) \)

Answer 1

The Correct Option is D

Let initial surface area: \[ A = \pi D^2 \] The surface area change \( \Delta A \) due to thermal expansion for a sphere: We know: \( \Delta A = A \cdot 2\alpha \Delta T + A \cdot (\alpha \Delta T)^2 \) \[ \Delta A = \pi D^2 \cdot \left( 2\alpha \Delta T + \alpha^2 \Delta T^2 \right) = \pi D^2 \cdot \alpha \Delta T (\alpha \Delta T + 2) \]