課程名稱︰力學上 課程性質︰物理系大二必帶 課程教師︰蔡爾成 開課學院：理學院 開課系所︰物理學系 考試日期（年月日）︰108年11月4日 考試時限（分鐘）：unknown 是否需發放獎勵金：是 （如未明確表示，則不予發放） 試題 : 註：為方便表示數學符號，以下較難輸入之數學符號以LaTeX碼呈現。 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'. -∞ --

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