Classical Mechanics (II)

Final

1. Prove the following statements about the inertia tensor. Please use rigorous

mathematical formulas instead of hand-waving arguments.

(a) (5%) If the rigid body has a mirror symmetry, and the origin of the coordinate lies in the plane of the mirror symmetry, then any axis perpendicular

to the mirror-symmetry plane is a principle axis.

(b) (5%) If the rigid body has a a Cn rotational symmetry, then the Cn-symmetric

axis is a principle axis. Cn means rotation of 2π/n radians.

2. (5%) Determine the principal moments of inertia for a sphere of radius R with a

cavity of radius r (Fig. 1).

R

r

3. Prove the following statements about the solutions of the wave equation.

(a) (5%) The solution of the wave equation:

∂

2Ψ2

∂x2

−

1

v

2

∂

2Ψ2

∂t2

= 0 (1)

with the initial conditions Ψ(x, t = 0) = h(x),

∂Ψ

∂t (x, t = 0) = g(x) is:

Ψ(x, t) = 1

2

[h(x + vt) + h(x − vt)] + 1

2v

Z x+vt

x−vt

g(w)dw (2)

(b) (5%) If Ψ(x, t, u) is the solution of the wave equation with the initial conditions Ψ(x, t = 0, u) = 0 and ∂Ψ

∂t (x, t = 0, u) = f(x, u), where u is a

parameter. Then ψ(x, t) ≡

R t

0 Ψ(x, t − u, u)du is the solution of the forced

wave equation:

∂

2ψ

2

∂x2

−

1

v

2

∂

2ψ

2

∂t2

= −

1

v

2

f(x, t) (3)

1

subject to the initial conditions ψ(x, t = 0) = 0 and ∂ψ

∂t (x, t = 0) =

0. Solutions with general initial conditions can be found by adding the

homogeneous solution Eq. (2).

4. Consider a rotating planar body with no thickness in the z direction. The angular

vector is ω = ω1xˆ+ω2yˆ+ω3zˆ, expanding in the basis of the principle axes. The

corresponding principle moments are I1, I2 and I3. If there is no external torque

on the body,

(a) (5%) show that the quantity ω

2

c ≡ ω

2

1 + ω

2

2

is conserved.

(b) (5%) Show that the equations of motion of ω can be simplified into one

single equation with the form of the plane pendulum:

¨θ +

g

l

sin θ = 0 (4)

where g/l ≡ ω

2

0 > 0. What are ω

2

0

and θ in terms of Ii and ωi?

(c) (5%) Under what conditions, ω3 can be zero at some time t? Is it possible

for ω3 to change the sign?

(d) (5%) Following (c), what’s the period of ω in this case (expressed as a

definite integral)? If ω is mostly pointing toward the principle axis with the

smallest principle moment, show that the result is consistent with (11.184).

5. Consider a 2D noninertial reference frame with coordinates x and y. The frame

is spinning around the z-axis with angular frequency ω3, where z-axis is perpendicular to the 2D noninertial frame. Meanwhile, the frame is also flipping around

the x-axis with angular frequency ω1. A particle with mass m is constrained to

moving in this 2D plane without friction. In addition, a massless spring with

spring constant k and zero natural length is attached to the particle. The other

end of the spring is fixed at the origin.

(a) (5%) What’s dω/dt ?

(b) (5%) Write down the equations of motion of the particle in this 2D noninertial reference frame.

(c) (5%) What are the four characteristic frequencies? Note that the frequencies are in general complex numbers.

(d) (5%) Under what conditions, the four frequencies are all real and the particle is always oscillating?

(e) (5%) Under what conditions, there exist solutions that are neither oscillating nor exponentially running away? Write down the form of the general

solutions under these conditions.

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6. Consider a 1D chain of molecules with the same mass m for each atom. The

elastic forces between atoms are represented by springs with the same spring

constant k. In this problem, you only need to consider the longitudinal vibrations.

(a) (5%) Write down the Lagrangian of the system and show that it is invariant

under the inversion symmetry P, where P means ⃗x → −⃗x around the

inversion center.

(b) (5%) Show that P

j Pijan,j = ηnan,i for each n, where Pij is the matrix representation of the inversion symmetry P, and the vectors ⃗an =

(an,1, an,2, …, an,N )

T

are the eigenvectors of the normal modes n = 0, 1, …, N−

1. What are the possible values of ηn?

(c) (5%) Use the trick in (b) to find out the eigenvectors and the corresponding

eigenfrequencies of the normal modes with N = 4.

(d) (10%) Find the eigenvectors and the eigenfrequencies as a function of the

particle numbers N. What is ηn? (Hint: substitute aj = aei(jγ−δ)

as in the

textbook (12.142) and take the real part, where i =

√

−1.)

(e) (5%) By taking the continuum limit, κ ≡ dk/dl and ρ ≡ dm/dl, what are

an(x) and the characteristic frequencies ωn?

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