The Boundary Value Problem $T \frac { d ^ { 2 } y } { d x ^ { 2 } } + \rho \omega ^ { 2 } = 0 , y ( 0 ) = 0 , y ( L ) = 0$

Multiple Choice

The boundary value problem $T \frac { d ^ { 2 } y } { d x ^ { 2 } } + \rho \omega ^ { 2 } = 0 , y ( 0 ) = 0 , y ( L ) = 0$ is a model of the shape of a rotating string. Suppose $T$ and $\rho$ are constants. The critical angular rotation speed $\omega = \omega _ { n }$ , for which there exist non-trivial solutions are

A) $\omega _ { n } = ( T / \rho ) \left( \frac { n \pi } { L } \right) ^ { 2 }$
B) $\omega _ { n } = \sqrt { T / \rho } \frac { n \pi } { L }$
C) $\omega _ { n } = \sqrt { T / \rho } \frac { n \pi } { 2 L }$
D) $\omega _ { n } = ( T / \rho ) \left( \frac { n \pi } { 2 L } \right) ^ { 2 }$
E) $\omega _ { n } = \sqrt { \rho / T } \frac { n \pi } { L }$