Question

Consider the boundary-value problem introduced in the construction of the mathematical model for the shape of a rotating string: T d2y dx2 + rhoω2y = 0, y(0) = 0, y(L) = 0. For constants T and rho, define the critical speeds of angular rotation ωn as the values of ω for which the boundary-value problem has nontrivial solutions. Find the critical speeds ωn and the corresponding deflections yn(x). (Give your answers in terms of n, making sure that each value of n corresponds to a unique critical speed.)

Answer 1

`y_n(x) =C_n \sin \sqrt{(\rho)/(T) } w_nx=C_n \sin \sqrt{(\rho)/(T) } \sqrt{(T)/(\rho) } (n \pi)/(L) x`

`y_n(x) = C_n \sin (n \pi x)/(L)`

Step-by-step explanation:

The given differential equation is

`T(d^2y)/(dx^2) + \rho w ^2y=0` and y(0) = 0, y(L) =0

where T and ρ are constants

The given rewrite as

`(d^2y)/(dx^2) + (\rho w^2)/(T) y=0`

auxiliary equation is

`m^2+ (\rho w^2)/(T) =0\\\\m= \pm\sqrt{(\rho)/(T) } wi`

Solution of this de is

`y(x)=C_1 \cos \sqrt{(\rho)/(t) } wx + C_2 \sin \sqrt{(\rho)/(T) } wx`

y(0)=0 ⇒ C₁ = 0

`y(x) = C_2 \sin \sqrt{(\rho)/(T) } wx`

y(L) = 0 ⇒

`C_2 \sin \sqrt{(\rho)/(T) } wL=0`

we need non zero solution

⇒ C₂ ≠ 0 and

`\sin \sqrt{(\rho)/(T) } wL=0`

`\sin \sqrt{(\rho)/(T) } wL=0 \rightarrow \sqrt{(\rho)/(T) } wL=n \pi`

`w_n = \sqrt{(T)/(\rho) } (n \pi)/(L)`

solution corresponding these
`w_n` values

`y_n(x) =C_n \sin \sqrt{(\rho)/(T) } w_nx=C_n \sin \sqrt{(\rho)/(T) } \sqrt{(T)/(\rho) } (n \pi)/(L) x`

`y_n(x) = C_n \sin (n \pi x)/(L)`