The boundary value problem Ty"+ ρ(ω^2)y' = 0 , y(0) = 0, y(L) = 0 is the model of the shape
Of a rotating string. Suppose T and ρ are constants. Find the critical angular speed ω = ωn for
which there is non-trivial solutions.
To find the critical angular speed ω = ωn for which there are non-trivial solutions to the given boundary value problem, we can solve the differential equation by assuming that the solution is in the form of a sinusoidal function. Let's proceed with the solution step-by-step:
Step 1: Substitute the assumed solution y(x) = A sin(ωx) into the differential equation, where A is the amplitude of the string's displacement.
Step 2: Differentiate y(x) twice with respect to x to obtain y'(x) and y''(x).
Step 3: Substitute y(x), y'(x), and y''(x) into the differential equation Ty"+ ρ(ω^2)y' = 0.
Step 4: Simplify the equation by canceling common terms and rearranging.
Step 5: Solve the resulting algebraic equation for ω^2.
Step 6: Find the values of ω for which there are non-trivial solutions, which correspond to the roots of the algebraic equation obtained in the previous step.
Let's go through these steps in detail:
Step 1: Substitute the assumed solution y(x) = A sin(ωx) into the differential equation:
Ty"+ ρ(ω^2)y' = 0
T(Aω^2 sin(ωx)) + ρ(ω^2)(Aω cos(ωx)) = 0
Step 2: Differentiate y(x) twice:
y'(x) = Aω cos(ωx)
y''(x) = -Aω^2 sin(ωx)
Step 3: Substitute y(x), y'(x), and y''(x) into the differential equation:
T(-Aω^2 sin(ωx)) + ρ(ω^2)(Aω cos(ωx)) = 0
Step 4: Simplify the equation:
-ATω^2 sin(ωx) + ρAω^3 cos(ωx) = 0
Step 5: Cancel the common factor A and rearrange the equation:
-ATω^2 sin(ωx) + ρAω^3 cos(ωx) = 0
-ATω^2 sin(ωx) = -ρAω^3 cos(ωx)
tan(ωx) = -ATω/ρ
Step 6: Solve for ω^2:
tan(ωx) = -ATω/ρ
ω^2 = -ρAT
Since T and ρ are constants, the critical angular speed ω = ωn for non-trivial solutions is given by:
ωn = sqrt(-ρAT)
Note that for non-trivial solutions to exist, the value of ω^2 must be negative, which means that ρAT must be positive.