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(C) The length of the cylinder remains unchanged,which means the thermal expansion is exactly compensated by the longitudinal compression.The thermal

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$3F / (\pi r^2 YT)$

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(C) The length of the cylinder remains unchanged,which means the thermal expansion is exactly compensated by the longitudinal compression.The thermal expansion strain is given by $\Delta L / L = \alpha T$,where $\alpha$ is the coefficient of linear expansion.The compressive strain due to force $F$ is given by $\Delta L / L = 	ext{Stress} / Y = F / (A Y) = F / (\pi r^2 Y)$.Equating the two strains: $\alpha T = F / (\pi r^2 Y)$.Therefore,$\alpha = F / (\pi r^2 YT)$.The coefficient of volume expansion $\gamma$ is related to the coefficient of linear expansion $\alpha$ by the relation $\gamma = 3\alpha$.Substituting the value of $\alpha$,we get $\gamma = 3F / (\pi r^2 YT)$.

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