课程名称︰力学上
课程性质︰必修
课程教师︰庞宁宁
开课学院：理学院
开课系所︰物理系
考试日期（年月日）︰2018/11/12
考试时限（分钟）：210
试题 :
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?
(答案不必化简)
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?