Question

The volume of a metal sphere increases by 0.33% when its temperature is raised by 50$^\circ$C. The coefficient of linear expansion of the metal is ______.

• A. $2.2 \times 10^{-5} / ^\circ$C
• B. $6.6 \times 10^{-5} / ^\circ$C
• C. $13.2 \times 10^{-5} / ^\circ$C
• D. $19.8 \times 10^{-5} / ^\circ$C

Hint:

Read carefully to see which coefficient the question is asking for! Option (b) deliberately contains the value of $\gamma$ ($6.6 \times 10^{-5}$) to trap students who forget to divide by 3 at the final step.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
The coefficient of cubical expansion $\gamma$ is related to the coefficient of linear expansion $\alpha$ by the relation $\gamma = 3\alpha$.

Step 2: Formula Application:
$\gamma = \frac{\Delta V}{V \cdot \Delta T}$. Given $\frac{\Delta V}{V} = 0.33% = \frac{0.33}{100} = 0.0033$. $\Delta T = 50^\circ$C.

Step 3: Explanation:
$\gamma = \frac{0.0033}{50} = \frac{3.3 \times 10^{-3}}{50} = 0.066 \times 10^{-3} = 6.6 \times 10^{-5} / ^\circ$C. Now, $\alpha = \gamma / 3 = (6.6 \times 10^{-5}) / 3 = 2.2 \times 10^{-5} / ^\circ$C.

Step 4: Final Answer:
The coefficient of linear expansion is $2.2 \times 10^{-5} / ^\circ$C.