课程名称︰力学下
课程性质︰物理系必修
课程教师︰蔡尔成
开课学院：理学院
开课系所︰物理系
考试日期（年月日）︰2018/6/25
考试时限（分钟）：110
试题 :
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?
图片：
三个共起点的向量，最左边为x_3、中间是L以及x'_3(两者方向相同)、最右边是ω。
L与x_3夹角θ、x_3与ω夹角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.
（本题有装置图但由于仅用文字叙述不致无法理解故省略）
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.