(b) For convenience, factor out a common factor of Ma2/12 from the inertia tensor. Then find the eigenvalues, λi, of the remaining matrix, which are related to the moments of inertia by Ii = Ma2/12 λi.
Make a table to summarize the results... you will use them in questions 2 and 3.
(b) It will be okay to assume that m << M, but be careful not to assume that mv = 0!
(c) This problem is similar to examples in Section 8.7 of the text, or the one described here. After the putty hits the lamina, the center-of-mass will be translating with the velocity determined in (b), but the whole system will be rotating about the center-of-mass. Hence, the velocity of the putty after the collision will be due to both these motions. The change in momentum of the putty can be determined from the initial and final velocities.
To apply Lagrange's equations with more than one constraint, you will need to use Equation (10.7.9) in the text, which is a slight generalization of the equation we derived in class for one constraint.
In the end you will have six equations (four from Lagrange's equations and two from the constraints) and six unknowns (second derivatives of x1, x2,x3,xp and λ1 and λ2). Each unknown will be multiplied by a constant so you can write the system of equations as a matrix equation. You can stop there... don't bother to invert the 6x6 matrix to get the solution. That is only a technical exercise that is best done using a computer.
Next, use Lagrange's equations to determine the differential equation that θ(t) must satisfy.
If the bead was in a state of equilibrium then the second time derivative of θ(t) would be zero. Use that to calculate the equilibrium position, θ0.
Next, a small oscillation about the equilbrium position can be written using θ(t) = θ0 + φ(t) where φ(t) is small enough that you can assume that cos φ(t) = 1 and sin φ(t) = φ(t). Substitute these into the differential equation you got from Lagrange's equations and use trigonometric identies like sin(θ(t) = θ0 + φ(t)) = sin θ0 cos φ(t) + cos θ0 sin φ(t), etc. The result should be a differential equation for φ(t) that is of the same form as the ones we studied for the harmonic oscillator. Work out the frequency from inspection and comparison of these equations.