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 Question:
We are provided with the percentage increase in volume due to thermal expansion. We must work backward to find the metal's coefficient of linear expansion ($\alpha$).

Step 2: Key Formula or Approach:
1. Volumetric expansion formula: $\frac{\Delta V}{V} = \gamma \Delta T$, where $\gamma$ is the coefficient of volumetric expansion.
2. Relationship between coefficients: For an isotropic solid, volumetric expansion is three times linear expansion: $\gamma = 3\alpha$.

Step 3: Detailed Explanation:
We are given a fractional volume increase of 0.33%. Convert this to a pure decimal:
$$\frac{\Delta V}{V} = 0.33% = \frac{0.33}{100} = 0.0033 = 33 \times 10^{-4}$$
We are given the temperature change $\Delta T = 50^\circ$C.
Substitute these into the volumetric expansion formula:
$$33 \times 10^{-4} = \gamma \times 50$$
Solve for $\gamma$:
$$\gamma = \frac{33 \times 10^{-4}}{50}$$
Multiply numerator and denominator by 2 to make division easier:
$$\gamma = \frac{66 \times 10^{-4}}{100} = 66 \times 10^{-6} = 6.6 \times 10^{-5} / ^\circ\text{C}$$
Now, relate this to the coefficient of linear expansion ($\alpha$):
$$\gamma = 3\alpha \implies \alpha = \frac{\gamma}{3}$$
$$\alpha = \frac{6.6 \times 10^{-5}}{3}$$
$$\alpha = 2.2 \times 10^{-5} / ^\circ\text{C}$$

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