First slide

Thermal expansion

Question

The radius of a metal sphere at room temperature T is R and the coefficient of linear expansion of the metal is $α$ . The sphere is heated a little by a temperature $∆$ T, so that its new temperature is T + $∆$ T. The increase in the volume of the sphere is approximately, {NCERT Exemplar]

Moderate

A

$2 πRα ∆ T$
B

${πR}^{2} α ∆ T$
C

$4 {πR}^{3} α ∆ T / 3$
D

$4 {πR}^{3} α ∆ T$

Solution

Let the radius of the sphere be R. As the temperature increases, radius of the sphere increases as shown in figure.
Original volume,
Coefficient of linear expansion = $α$
$∴$ Coefficient of volume expansion = 3 $α$
$∴ \frac{1}{V} \frac{d V}{d T} = 3 α \Rightarrow d V = 3 V α d T \approx 4 π R^{3} α Δ T$
= Increase in the volume

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