課程名稱︰力學
課程性質︰系必修
課程教師︰高英哲
開課學院：理學院
開課系所︰物理學系
考試日期（年月日）︰2009.1.14
考試時限（分鐘）：180
是否需發放獎勵金：是
試題 :
Total score: 110pts
Do all the problems. Please put down the details of your calculations/
derivation as well as your reasoning. Good luck !!
1.Please answer true (○) or false (╳) to the following questions (10pts):
(a) The Lagrangian formulation of the mechanical system is a new theory for
classical mechanics beyond Newtonian mechanics. (2pts)
(b) If a system is invariant with respect translation in the x-direction,
the linear momentum px is conserved. (2pts)
(c) All orbits in a central field are closed. (2pts)
(d) H = E implies energy conservation. (2pts)
(e) If a force Ｆ is conservative, ▽×Ｆ = 0. (2pts)
2.(30pts) The point of suspension of a simple pendulum of length l and mass m
is attached to a support which is driven horizontally with time.
(a) Find the Lagrangian of the system in terms of generalized coordinates
(y,θ), and equation of motion for θ. (10pts)
(b) For a sinusoidal motion of the support with y = y0cosωt, find pθ.
Derive the Hamiltonian. (10pts)
(c) Find the steady-state solution to the equation of motion for small
angular displacements. Under which condition is the motion unstable?
(10pts)
3.(30pts) A particle of mass m is constrained to move on the inside surface of
a smooth cone of half-angle α. The particle is subject to a gravitational
force.
(a) Find the Lagragian and equations of motion of the system in generalized
coordinates θ and r. (10pts)
(b) Find the total magnitude of the force of constraint for a circular orbit
by rewriting the Lagrangian in θ, r and z, and using the Lagrange
multipliers and constraints. (10pts)
(c) Show that a circular orbit is stable by finding the equation of the
motion of a small perturbation in the radial direction. Find the
oscillation frequency of the small perturbation. (10pts)
4.(10pts) A particle of mass m moves in an attractive spherical potential
U(r) = -V, r < R; U(r) = 0, r > R.
(a) Plot the effective potential for a given angular momentum L. (5pts)
(b) Discuss possible trajectories for different values of L and energy E.
(5pts)
5.(10pts) A satellite orbits around the Earth in a circular orbit at radius R
and velocity v. A rocket fires in the radial direction, giving the satellite
an outward radial velocity v in addition to its original velocity. Ignore
the difference in the satellite mass before and after the rocket burn.
(a) Find the ratio of the new energy and angular momentum to the old. (5pts)
(b) What is the new orbit of the satellite? (5pts)
6.(20pts) A particle of mass m moves under a conservative force with potential
energy V(x) = cx/(x^2+a^2), where c is a constant.
(a) Find the position of stable equilibrium, and the period of small
oscillations about it. (10pts)
(b) If the particle starts from this point with velocity v, find the range
of v for which it (1) oscillates, (2) escapes to +∞. (10pts)
Useful Formulae
‧Orbit Equation:
d^2 u μ
─── + u = - ─── F (1/u), u = 1/r
dθ^2 (ul)^2
‧Keplerian Orbits: 2
2El
α/r = 1 + εcosθ; α = l^2/μk; ε = √(1+───)
μk^2
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