Solved: In Exercises 73 and 74, find the moments of inertia for the wire of density p A

Chapter 15, Problem 74

(choose chapter or problem)

Consider a wire of density \(\rho(x, y)\) given by the space curve

\(C: \mathbf{r}(t)=x(t) \mathbf{i}+y(t) \mathbf{j}, \quad 0 \leq t \leq b\).

The moments of inertia about the x - and y-axes are given by

\(I_{x}=\int_{C} y^{2} \rho(x, y) d s\) and \(I_{y}=\int_{C} x^{2} \rho(x, y) d s\).

In Exercises 73 and 74, find the moments of inertia for the wire of density \(\rho\).

A wire lies along \(\mathbf{r}(t)=a \cos t \mathbf{i}+a \sin t \mathbf{j}\), where \(0 \leq t \leq 2 \pi\) and a > 0, with density \(\rho(x, y)=y\).

Text Transcription:

rho (x, y)

C: r(t) = x(t)i + y(t)j,     0 leq t leq b

I_x = int_C y^2 rho(x, y) ds

I_y = int_C x^2 rho(x, y) ds

rho

r(t) = a cos ti + a sin tj

0 leq t leq 2 pi

rho(x, y) = y