### 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 Ma^{2}/12 from the
inertia tensor. Then find the eigenvalues, λ_{i}, of the
remaining matrix, which are related to the moments of inertia by I_{i} = Ma^{2}/12 λ_{i}.

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
ω_{1}=**ω**.**n**_{1},
ω_{2}=**ω**.**n**_{2}, and
ω_{3}=**ω**.**n**_{3}.
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, x_{p}, to represent
the distance between the two pulleys. That means that there will be four
generalized coordinates along with two equations of constraint:
f(x_{1},x_{p}) = x_{1}+x_{p}-l_{1} = 0
and
f(x_{2},x_{3}) = x_{2}+x_{3}-l_{2} = 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
x_{1}, x_{2},x_{3},x_{p} and λ_{1} and
λ_{2}). 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,
θ_{0}.

Next, a small oscillation about the equilbrium position can be written using
θ(t) = θ_{0} + 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) = θ_{0} + φ(t)) = sin θ_{0} cos φ(t) + cos θ_{0} 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.