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Step-by-Step Solution
Step 1: Understand the Physical Situation
The solid metal cube expands differently along each axis due to different coefficients of linear expansion. Given:
• Along x-axis: \alpha_x = 5 \times 10^{-5}\,/^\circ \mathrm{C}
• Along y-axis: \alpha_y = 5 \times 10^{-6}\,/^\circ \mathrm{C}
• Along z-axis: \alpha_z = 5 \times 10^{-6}\,/^\circ \mathrm{C}
The task is to find the coefficient of volumetric expansion, \gamma , which is given by:
\gamma = \alpha_x + \alpha_y + \alpha_z
We have \gamma = C \times 10^{-6}\,/^\circ \mathrm{C} and we need to determine the value of C .
Step 2: Sum the Individual Coefficients of Linear Expansion
According to the relation for small expansions:
\gamma = \alpha_x + \alpha_y + \alpha_z
Substitute the given values:
\gamma = 5 \times 10^{-5} + 5 \times 10^{-6} + 5 \times 10^{-6}
Combine the terms carefully, noticing that 5 \times 10^{-5} = 50 \times 10^{-6} :
5 \times 10^{-5} + 5 \times 10^{-6} + 5 \times 10^{-6} = 50 \times 10^{-6} + 10 \times 10^{-6} = 60 \times 10^{-6}\,/^\circ \mathrm{C}
Step 3: Identify the Value of C
We have:
\gamma = 60 \times 10^{-6}\,/^\circ \mathrm{C}
Since \gamma = C \times 10^{-6}\,/^\circ \mathrm{C} , it follows that:
C = 60
Final Answer
The coefficient of volumetric expansion can be written as 60 \times 10^{-6}\,/^\circ \mathrm{C} and hence the value of C is 60.
