Question

The radius of a metal sphere at room temperature $T$ is $R$ and the coefficient of linear expansion is $\alpha$. The sphere heated a little by a temperature $△T$. So that its new temperature is $T+△T$. The increase in the volume of the sphere is approximate:

• A. $4\pi R^3\alpha △T$
• B. $2\pi R^3\alpha △T$
• C. $\pi R^3\alpha △T$
• D. $4\pi R^3\alpha △T/3$

Answer 1

The correct option is A $4\pi R^3\alpha △T$

Radius=$R$

Room temperature=$T$

Coefficient of linear expansion=$\alpha$

New temperature=$T+△T$

$R_{new}=R(1=\alpha △T)$

$V_{new}=\frac{4}{3}\pi R_{new}^3$

$△V=\frac{4}{3}\pi [R_{new}^3-R^3]$

$\frac{4}{3}\pi R^3\left[\right(1+\alpha △T{)}^3-1]$

$\sim \frac{4}{3}\pi R^3(1+3\alpha △T-1)$

$\sim 4\pi R^3\alpha △T$