Question

An iron sphere having diameter \(D\) and mass \(M\) is immersed in hot water so that the temperature of the sphere increases by \(\delta T\). If \(\alpha\) is the coefficient of linear expansion of the iron then the change in the surface area of the sphere is

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

Hint:

For thermal expansion, \[ L'=L(1+\alpha\Delta T) \] and area expansion can be obtained by squaring the linear dimension.

Answer 1

The Correct Option is D

Step 1: Write the initial surface area of the sphere.
The surface area of a sphere with diameter $D$ is: \[ A = \pi D^2 \]
Step 2: Find the new diameter after thermal expansion.
When the temperature increases by $\delta T$ and $\alpha$ is the coefficient of linear expansion, every linear dimension increases by a factor $(1 + \alpha \delta T)$: \[ D' = D(1 + \alpha \delta T) \]
Step 3: Find the new surface area.
\[ A' = \pi (D')^2 = \pi D^2 (1 + \alpha \delta T)^2 \] Expanding: \[ A' = \pi D^2 (1 + 2\alpha\delta T + \alpha^2 \delta T^2) \]
Step 4: Calculate the change in surface area.
\[ \Delta A = A' - A = \pi D^2(1 + 2\alpha\delta T + \alpha^2\delta T^2) - \pi D^2 \] \[ \Delta A = \pi D^2(2\alpha\delta T + \alpha^2\delta T^2) \] \[ \Delta A = \pi D^2 \alpha\delta T(2 + \alpha\delta T) \]
Step 5: Match with the given options.
We have $\Delta A = \pi D^2 \alpha \delta T(\alpha\delta T + 2)$. This matches option 4: $\pi D^2 \alpha \delta T(\alpha\delta T + 2)$.
Step 6: State the final answer.
The change in surface area of the iron sphere is $\pi D^2 \alpha \delta T(\alpha \delta T + 2)$. \[ \boxed{\pi D^2 \alpha \delta T(\alpha \delta T + 2)} \]