課程名稱︰力學下
課程性質︰物理系必修
課程教師︰蔡爾成
開課學院：理學院
開課系所︰物理系
考試日期（年月日）︰2018/6/25
考試時限（分鐘）：110
試題 :
1 (20pt)
A symmetric body moves without the influence of forces or torques. Let x_3 be
the symmetry axis of the body and L be along x'_3. The angle between ωand
x_3 is α. What is the angular velocity dφ/dt of the symmetry axis about L
in terms of I_1, I_3, ω, and α?
圖片：
三個共起點的向量，最左邊為x_3、中間是L以及x'_3(兩者方向相同)、最右邊是ω。
L與x_3夾角θ、x_3與ω夾角θ。
[Hint] In the body frame, the angular velocity is related to the
Eulerian angles by
╭φ'sinθsinψ+θ'cosψ╮
│ │
ω=│φ'sinθcosψ-θ'sinψ│
│ │
╰ φ'cosθ+ψ' ╯, where A'=dA/dt.
2 (20pt)
A mass M moves horizontally along a smooth rail. A pendulum is hung from M
with a weightless rod and mass m at its end.
(a, 8pt) find the eigenfrequencies, and
(b, 12pt) describe the normal modes.
（本題有裝置圖但由於僅用文字敘述不致無法理解故省略）
3 (10pt)
Consider a relativistic rocket whose velocity with respect to a certain
inertial frame is v and whose exhaust gases are emitted with a constant
velocity V with respect to the rocket. Show that the equation of motion is
m dv/dt + V(1-β^2) dm/dt=0.
4 (15pt)
A rigid body with three distinct principal mements of inertia is undergoing
force-free motion. Show that rotation around the principal axis corresponding
to either the greatest or smallest moment of inertia is stable and that
rotation around the principal axis corresponding to the intermediate moment
of inertia is unstable.
5(25pt)
A system has n degrees of freedom and is described in terms of generalized
coordinates q_k, k=1,2,...,n with the Lagrangian L=T-U. In the vicinity of
the equilibrium configuration, we have
T=0.5 Σ(m_(jk) q'_j q'_k), U=0.5Σ(A_(jk) q_j q_k),
where the quantities A_(jk)=A_(kj) and m_(jk)=m_(kj) are numbers.
[Note that summations are for j and k, and x'=dx/dt].
We also have U≧0, T≧0, and T=0 only when all q'_j are zeros. In terms of
matrix notation,
T = 0.5 [q']^T {m} [q'], and
U = 0.5 [q]^T {A} [q],
where {A}_(ij) = A_(ij), {m}_(ij) = m_(ij), and [q]_i = q_i.
(a, 5pt)
Show that the Lagranges equations can be written as
{A}[q]+{m}[q'']=0
(b, 5pt)
It is known that a real symmetric matrix can always be diagonalized by an
orthogonal matrix. Thus, we have
{m} = {R_m}{D_m}{R_m}^T,
where {D_m} is diagonal and {R_m}{R_m}^T = {R_m}^T{R_m} = {I} with {I} being
the n×n identity matrix. Give the reason why all the diagonal elements of
{D_m} must be positive, an we may write {D_m} = {λ_m}^2 with {λ_m} being
real and diagonal.
(c, 5pt)
Show that {S_m}{m}{S_m} = {I},
where {S_m} = {R_m}{λ_m}^(-1).
(d, 5pt)
Define matrix {A'}={S_m}^T{A}{S_m} is also real and symmetric and therefore
can be diagonalized as
{A'} = {R_A}{ω^2}{R_A}^T,
where {ω^2} is diagonal. Let {B}={S_m}{R_A}. Show that {B} can simultaneously
diagonalize {m} and {A}, in the sense that
{B}^T {m}{B}={I}, {B}^T {A}{B}={ω^2}.
(e, 5pt)
Define the normal coordinates η_r(t)=[η(t)]_r via [q(t)]={B}[ η(t)].
Show that η_r satisfies
(η''_r)+(ω_r)^2 (η_r) = 0,
where (ω_r)^2 is the r-th diagonal element of {ω^2}.
6 (10pt)
Event one(t_1, x_1, y_1, z_1) and event two(t_2, x_2, y_2, z_2) are timelike,
i.e. (c(t_2-t_1))^2>(x_1-x_2)^2+(y_1-y_2)^2+(z_1-z_2)^2, in an inertia
reference frame K. If t_2>t_1, show that it is impossible to find another
inertial frame K' in which t'_2<t'_1.