Chapter 13, Problem 87A

To determine

Volume of sphere at $\text{12}°\text{C}$ .

Answer to Problem 87A

Initial volume of sphere will be $2.4{\text{cm}}^3$ .

Explanation of Solution

Given:

Final volume of sphere, $V_2\text{ = 2}\text{.56}{\text{cm}}^3$

Initial temperature $\text{= 12}°\text{C}$

Final temperature $\text{ = 984}°\text{C}$

Volume of coefficient of expansion, $\beta \text{ = 48}\times {\text{10}}^{-6}\text{per}°\text{C}$

Formula used:

Change in volume:

  $\text{Δ}V=\beta V\text{Δ}T$

Where, $\text{Δ}V$ is change in volume, $\beta$ is coefficient of volume expansion, $V$ is the volume, $\Delta T$ is the change in temperature.

Calculation:

Expand the formula of change in volume, $\text{Δ}V=\beta V\text{Δ}T$ .

New formed formula is

$\left(V_2-V_1\right)\text{=}\beta V\left(T_2-T_1\right)$ or $V_1=\frac{V_2}{\beta \text{V}\left(T_2-T_1\right)}$

Substitute all the given values in newly formed formula,

  $V_2-V_1=48\times {10}^{-6}°\text{C}\times V_1\times \left(984-12\right)°\text{C}$

  $V_2=48\times {10}^{-6}\times V_1\times 972+V_1$

  $V_1=\frac{V_2}{(48\times {10}^{-6}\times 972)+1}$

  $\text{=}\frac{2.56}{1.046}=2.4{\text{cm}}^3$

Conclusion:

At $\text{12}°\text{C}$ , volume of cooper sphere was $2.4{\text{cm}}^3$ .