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Question

(A) The bulk modulus $K$ is defined as the ratio of the change in pressure to the volumetric strain: $K = \frac{P}{\left( \frac{-\Delta V}{V} \right)}

Vedclass · Accepted Answer

$\frac{P}{3\alpha K}$

Vedclass · Answer

(A) The bulk modulus $K$ is defined as the ratio of the change in pressure to the volumetric strain: $K = \frac{P}{\left( \frac{-\Delta V}{V} \right)} \Rightarrow \frac{\Delta V}{V} = \frac{P}{K}$ where $\Delta V$ is the decrease in volume due to pressure $P$. When the cube is heated by a temperature $\Delta t$,its volume increases due to thermal expansion: $\Delta V = V_0 \gamma \Delta t$ where $\gamma$ is the coefficient of volume expansion and $V_0$ is the initial volume. For a solid,the coefficient of volume expansion $\gamma$ is related to the coefficient of linear expansion $\alpha$ by $\gamma = 3\alpha$. To bring the cube back to its original size,the increase in volume due to heating must equal the decrease in volume due to pressure: $\frac{\Delta V}{V_0} = \gamma \Delta t = 3\alpha \Delta t$ Equating the two expressions for volumetric strain: $\frac{P}{K} = 3\alpha \Delta t$ Solving for the temperature change $\Delta t$: $\Delta t = \frac{P}{3\alpha K}$

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