Question 1

(a) Just apply the formulas for the components of the inertia tensor. Be careful about the limits of integration. Examples for integrating over a triangle be found here and here. Of course, these examples were ones in which it was the center of mass the was being calculated, not the components of the inertia tensor, but the way to treat the limits of integration is similar.

(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.

Question 2

(a) You will need to calculate the position of the center-of-mass of the lamina before the collision. Then, use momentum conservation applied to the lamina-putty system.

(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.

Question 3

Remember that to apply Euler's equations you have to write the components of the angular velocity vector along the principal axes of the body. So, you will need to calculate ω1=ω.n1, ω2=ω.n2, and ω3=ω.n3. before you can use Euler's equations. You should find that the torque that is needed is only in the k direction...

Question 4

Just follow the hint and do the algebra. It won't be that difficult.

Question 5

First, introduce another generalized coordinate, xp, to represent the distance between the two pulleys. That means that there will be four generalized coordinates along with two equations of constraint: f(x1,xp) = x1+xp-l1 = 0 and f(x2,x3) = x2+x3-l2 = 0.

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.

Question 6

First, calculate the kinetic and potential energies of the bead. Remember that the kinetic energy comes from the motion around the z-axis plus any motion due to changes in θ(t).

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.