Classical Dynamics of Particles and Systems · Problem 11-19

Coupled Oscillations — Marion & Thornton

Physics — Mechanics · inertia-tensor-logarithmic-spiral-plate

Problem

Consider a plane homogeneous plate of density ρ\rho bounded by the logarithmic spiral r=keαθr = ke^{\alpha\theta} and the radii θ=0\theta = 0 and θ=π\theta = \pi. Obtain the inertia tensor for the origin at r=0r = 0 if the plate lies in the x1x_1-x2x_2 plane. Perform a rotation of the coordinate axes to obtain the principal moments of inertia, and use the results of the previous problem to show that they are I1′=ρk4P(Q−R),I2′=ρk4P(Q+R),I3′=I1′+I2′I_1' = \rho k^4 P(Q - R), \quad I_2' = \rho k^4 P(Q + R), \quad I_3' = I_1' + I_2' where P=e4πα−116(1+4α2),Q=1+4α22α,R=1+4α2P = \frac{e^{4\pi\alpha} - 1}{16(1 + 4\alpha^2)}, \quad Q = \frac{1 + 4\alpha^2}{2\alpha}, \quad R = \sqrt{1 + 4\alpha^2}

Browse the problem library