Ncert Exercises & Solutions

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 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:

1. $2 πRα ∆ T$

2. ${πR}^{2} α ∆ T$

3. $4 {πR}^{3} α ∆ T / 3$

4. $4 {πR}^{3} α ∆ T$

(4) Hint: There will a volumetric expansion in the sphere.

Step 1: Find the volume of the sphere.

Let the radius of the sphere be R. As the temperature increases, the volume increases as shown.

Original volume,

$V_{o} = \frac{4}{3} {πR}^{3}$

Step 2: Find the increase in the volume of the sphere.

Coefficient of linear expansion =

$α$

$∴$ Coefficient of volume expansion = 3

$α$

$∴$

$\frac{1}{V} \frac{d V}{d T} = 3 α \Rightarrow d V = 3 V α d t ≃ 4 {πR}^{3} α ∆ T$

= Increase in the volume

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