课程名称︰力学上
课程性质︰物理系大二必带
课程教师︰蔡尔成
开课学院：理学院
开课系所︰物理学系
考试日期（年月日）︰108年11月4日
考试时限（分钟）：unknown
是否需发放奖励金：是
（如未明确表示，则不予发放）
试题 :
注：为方便表示数学符号，以下较难输入之数学符号以LaTeX码呈现。
Mechanics 11-4-2019
1.[10%] Let r=|\vec{x}| and \hat{n}=\vec{x}/r. At r≠0, show that (a)\del r=
\hat{n}, (b)\del\dot(\hat{n}/r^2)=0 (c)\del×(\hat{n}/r^2)=0 (d)\del^2(1/r)
=0, (e)\del^2(ln r)=1/r^2.
2.[15%] A solid cube of uniform density and sides of b is in equilibrium on
top of a cylinder of radius R as shown in Fig.1. The planes of four sides of
the cube are parallel to the axis of the cylinder. The contact between cube
and sphere is perfectly rough. Under what conditions is the equilibrium
stable or not stable?
3.A pendulum is suspended from the cusp of a cycloid cut in a rigid support as
shown in Fig.2. The upper cycloid can be parametrized as
x=a(φ+sinφ), y=a(1-cosφ).
The length of the pendulum is l=4a.
(a)[10%] Show that the path described by the pendulum bob is cycloidal and
can be given by
x_b=a(φ-sinφ), y_b=a(cosφ-1)
where φ is the angle of rotation of the circle generating the cycloid (0≦
φ≦2π with φ=π corresponding to the cusp of the upper cycloid).
(b)[15%] Show that the oscillations are exactly isochronous with a frequency
ω0=\sqrt{g/l}, independent of the amplitude.
4.(a)[2%] Define \vec{A}\cdot\vec{B}=\sum_iA_iB_i. Show that \vec{A}\cdot\vec{
B}=|\vec{A}||\vec{B}|cosθ where θ is the angle between the directions of
\vec{A} and \vec{B}.
(b)[3%] Define \vec{A}×\vec{B}=\vec{C} with C_i=\sum_{j,k}ε_{ijk}A_jB_k.
Show that |\vec{A}×\vec{B}|=|\vec{A}||\vec{B}|sinθ.
5.[10%] Explain why the gradient operator \del≡\frac{\partial}{\partial x}
\hat{i}+\frac{\partial}{\partial y}\hat{j}+\frac{\partial}{\partial z}\hat{k}
transform s like a vector under rotation.
6.[10%] Obtain the differantial equation for the phase path of a simple harmo-
nic oscillator and show that it represents a family of ellipses.
7.[10%] A damped oscillator under external sinusoidal driving force obeys the
equation \ddot{x}+2β\dot{x}+ω^2_0x=Acos(ωt). At what frequency does the
kinetic energy resonance or the maximum of <1/2m\dot{x}^2> occur?
8.Let \tilde{L} denote the linear operator d^2/dt^2+2βd/dt+ω^2_0. Assume
ω^2_0≧β^2 and ω≡\sqrt{ω^2_0-β^2}.
(a)[5%] For the homogeneous equation \tilde{L}(x)=d^2/dt^2+2βd/dt+ω^2_0=0,
find that the solution x(t) subject to the initial conditions x(t_0)=1 and
\dot{x}(t_0)=0.
(b)[5%} Find the Green function G(t,t') that satisfies
\tilde{L}G(t,t')=δ(t-t')
and G(t,t')=0 when t≦t'.
(c)[5%] Show that the response function x(t) of a linear damped oscillator,
satisfying \tilde{L}(x)=A(t) with A(t)=0 when t<0, subject to the initial
conditions x(0)=\dot{x}(0)=0, can be written as
+∞
x(t)=∫ G(t,t')A(t')dt'.
-∞