question_answer

A uniform cylindrical rod of length L and radius r, is made from a material whose Young's modulus of Elasticity equals Y. When this rod is heated by temperature T and simultaneously subjected to a net longitudinal compressional force F, its length remains unchanged. The coefficient of volume expansion, of the material of the rod, is (nearly) equals to : [JEE Main 12-4-2019 Afternoon]

A) \[F/(3\pi {{r}^{2}}YT)\]

B)      \[3F/(\pi {{r}^{2}}YT)\]

C) \[6F/(\pi {{r}^{2}}YT)\] D)      \[9F/(\pi {{r}^{2}}YT)\] Correct Answer: B Solution : [b]

\[\therefore \] Length of cylinder remains unchanged

so\[{{\left( \frac{F}{A} \right)}_{Compressive}}={{\left( \frac{F}{A} \right)}_{Thermal}}\]

\[\frac{F}{\pi {{r}^{2}}}=Y\alpha T\]

(\[\alpha \]is linear coefficient of expansion)

\[\therefore \]\[\alpha =\frac{F}{YT\pi {{r}^{2}}}\]

\[\therefore \]The coefficient of volume expansion\[\gamma =3\alpha \]

\[\therefore \]\[\gamma =3\frac{F}{YT\pi {{r}^{2}}}\]