課程名稱︰力學上 課程性質︰必修 課程教師︰龐寧寧 開課學院：理學院 開課系所︰物理系 考試日期（年月日）︰2018/11/12 考試時限（分鐘）：210 試題 : 1.[10%] The height of a hill in meters is given by z=2xy-3xx-4yy-18x+28y+12, where x is the distance east and y is the distance north of the origin. (a) Where is the top of the hill and how high is it? (b) How steep is the hill at x=y=1, that is, what is the angle between a vector perpendicular to the hill and the z-axis? (c) In which compass direction is the slope at x=y=1 steepest? (答案不必化簡) 2. [15%] Consider the motion of a charged particle of mass m and charge q in a uniform electromagnetic field E and B. Choose the z-axis to lie in the direction of B and let the plane containing E and B to be the yz-plane. With the initial conditions x(0)=-A/(qB/m), x'(0)=E_y/B, y(0)=0, y'(0)=A, z(0)=z_0, and z'(0)=z'_0. Find x(t), y(t), and z(t). Sketch the trajectory on the xyz-space for the cases (i) A>E_y/B>0 (ii) 0<A<E_y/B (iii) A=E_y/B>0 3. [15%] Given the equation of motion mx''+bx'+kx=F_0 cos(wt), obtaining the complementary solution x_c(t) and the particular solution x_p(t). 4. [15%] A pendulum is suspended from the cusp of a cycloid cut in a rigid support, where the length of the pendulum l equals the half of the cycloid arc length. Show that the path of the pendulum bob is cycloidal and obtain the oscillation frequency. 5. [12%] A point mass m slides without friction on a horizontal table at one end of a massless spring of natural length a and apring constant k. The spring is attached to the table so it can rotate freely without friction. The net force on the mass is the central force F(r)=-k(r-a). (a) Find and sketch both the potential energy U(r) and the effective potential U_eff(r). (b) What angular velocity w_0 is required for a circular orbit with radius r_0? (c) Derive the frequency of small oscillations w about the circular orbit with radius r_0. 6. [10%] Explain the reasons why the Green's function for the linear oscillation equation is independent of the initial conditions x(t_0)=x_0, and x'(t_0)=x'_0. 7. [11%] Given the van der Pol equation x''+μ(xx-aa)x'+x(w_0)^2=0 with 0<μ<<1, use the averaging theory to obtain the positions of the source and the stable limit cycle in the w_0x-y plane with y=μ(aax-x^3/3)-x' 8. [12%] Consider the logistic map x_(n+1)=μx_n(1-x_n) with 0<=x_n<1 and 0<μ<4. Explain the reasons why a superstable cycle must contain the point x=1/2. What are the difinitions of Feigenbaum's constant? --

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