A rigid lamina (i.e. a 2 dimensional object) has principal moments of inertia about the centre of mass of I1 = (μ² - 1), I2 = (μ² + 1), I3 = 2μ². (a) Show, using Euler's equations, that in the body-fixed frame, the component of the angular velocity in the plane of the lamina (i.e. √(Ω₁² + Ω₂²)) is constant in time. (b) Choose the initial angular velocity to be Ω⃗ = μN x̂₁ + N x̂₃. Define tan α = Ω₂/Ω₁, which is the angle the component of Ω in the plane of the lamina makes with x̂₁. Show that it satisfies α̇ = Ω₃ and from this show that α̈ = -1/μ² (Ω₁² + Ω₂²) sin α cos α = -N² cos α sin α. Show that the solution to the motion is Ω⃗(t) = μN(cosh Nt)⁻¹ x̂₁ + μN tanh Nt x̂₂ + N(cosh Nt)⁻¹ x̂₃. (NB It is enough to check that this is the solution; you do not need to solve the differential equation).