Problem 1: Olive Euler

Part A

The principal axes of the system are the x-axis, the y-axis along the toothpick and the z-axis perpendicular to both the x and y axes.

For the olive, we have Ixx = ∫x²dm = ∫x²ρ0dV where dV is an infinitesimal volume 

Ixx = ∫x²dm = ∫x²ρ0dV

x=0

Ixx=0

This leads to Ixx = ρ0∫(z² + y²)dV

I = [Ixx 0 0; 0 Iyy 0; 0 0 Izz] = [ρ₀∫(z² + y²)dV 0 0; 0 (ρ₀∫(z² + x²)dV +1/2mr^2) 0; 0 0 (ρ₀∫(x² + y²)dV +1/2mr^2)]

where Ixx = ρ₀∫(z² + y²)dV for the entire system, Iyy ≈ ρ₀∫(z² + x²)dV +1/2mr^2 for the entire system and Izz ≈ ρ₀∫(x² + y²)dV +1/2mr^2 for the entire system.

Part B

Multiplying this total rotation matrix with our initial basis vectors...