課程名稱︰力學
課程性質︰物理系必修
課程教師︰張慶瑞
開課學院：理學院
開課系所︰物理學系
考試時間︰11/13/2006
是否需發放獎勵金：否
（如未明確表示，則不予發放）
試題 :
(某些科學記號請以想像力補上謝謝)
Time restriction：10:20 to 12:10 ( 12:30 , actually...)
1. Two Dimensional Retard Projectile Motion:
Consider a particle moves in a two dimensional space with a gravitational
field and undergoing a retarding force. The equation of motion m (d^2)r/dt^2
= -mgy-mKdr/dt describes this motion with position vector r=(x,y), gravity
constant g, and retarding strength K. Selecting the initial conditions, x(0)
=y(0)=0, dx/dt=u, dy/dt=v at t=0.
(a)Find x(t) and y(t).(10pts.)
(b)Make a sketch for trajectories by increasing K.(5pts.)
(c)Show that for K→0, the trajectory is parabolic.(5pts.)
2. Forced Oscillation:
Applying the force beginning at time t=t_0, consider the following three
oscillations.
(a)The step-function-forced oscillation (see Figure(a)) for t_0<t is described
by the differential equation, (d^2)x/dt^2 + 2βdx/dt + (ω_0^2)x=a, with
constants β and a being the damping amplitude and force strength,
respectively. Determine x(t) by considering the initial condition x(t_0)=0
and dx/dt(t_0)=0.(10pts.)
(b)Follow from (a), but replace the step function force with the impulse force
(see Figure(b)), (d^2)x/dt^2 + 2βdx/dt + (ω_0^2)x=a for t_0<t<t_1. Find
x(t).(10pts.)
(c)For the case in which the oscillator is subject to a expotentially decaying
force with switching time at t=0, (d^2)x/dt^2 + 2βdx/dt + (ω_0^2)x=
F_0 e^(-η)θ(t)/m, where θ(t) is the step function. What is the response
x(t)? (Hint:The Green's function G(t,t')=θ(t-t')/m(ω^2-β^2)^(1/2) *
e^(-β(t-t'))sinω_1(t-t') satisfies the linear differential equation (d^2)
G(t,t')/dt^2 + 2βdG(t,t')/dt + (ω_0^2)G=δ(t-t')/m. Multiplying the
external force F_0 e^(-η)θ(t) and further integrating over t', on both
sides, would lead you to the answer.)(10pts.)
(Figure (a): a step function F(t)/m=a at t=t_0)
(Figure (b): a implulse function F(t)/m=a when t_0<t<t_1)
3. Variational Method:
A geodesic is a line that represents the shortest path between any two points
when the path is restricted to a particular surface. Find the geodesic on a
sphere represented by the Cartesian coordinates (x,y,z).(10pts.)
4. Conservative Quantities:
Considering a uniform disk rolling without slipping on an inclined plane, and
the inclined plane is on a smooth table, please find four constants during
this process. The radius of the disk is a and the mass of the disk is m, while
the mass for the inclined plane is M and the inclined angle is α.
You can use the generalized coordinates of x and s(sliping downward the plane).
(20pts.)
5. Recovering to The Newton's Equation:
Consider a particle of mass m moving freely in a conservative force field
whose potential function is U. Find the Hamiltonian function, and show that
the canonical equations of motion reduce to Newton's equation.(Use rectangular
coordinates.) What if U=ky, k is a constant?(20pts.)
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