推 answertw :矮英哲 04/25 18:02
課程名稱︰力學下
課程性質︰系必修
課程教師︰高英哲 教授
開課學院：理學院
開課系所︰物理學系
考試日期（年月日）︰2009/04/22
考試時限（分鐘）：160
是否需發放獎勵金：是！
（如未明確表示，則不予發放）
試題 :
Do All the problems. Please put down the details of your calculations/
derivation as well as your reasoning. Define your notations properly.
Good luck !!
1.(5%) State the postulates of special relavity.
2.(10%) Consider a system of N particles: M = Σ m , P = Σ m v , and
i i i i i
L = Σ r ×m v . If the internal force between particles is given by
i i i i
. .
f = k[(r -r )-b(r -r )], show that the equation Ｎ = dＬ/dt is not valid.
ij j i j i ext
3.(10%) A uniform heavy chain of length a hangs initially with a part of
length b hanging over the edge of a table. The remaining part, of length a-b
is coiled up at the edge of the table. If the chain is released, find the
speed of the chain when the last link leaves the end of the table.
4.(10%) The process of the production of an electron-positron pair by a photon
- +
γ → e + e is only possible if some other body is present to conserve
both energy and momentum. The rest mass of the electron is m. What is the
minimum energy that the photon must have, with the presence of a nucleus of
rest mass M, in order to produce an electron-positron pair?
5.(20%) A spaceship is moving away from the earth at a speed v/c = β. When
the ship is at a distance of D from earth as measured in the earth's
reference frame, a radio signal is sent out to the spaceship by an observer
on earth (event E1). The signal reaches the spaceship at event E2.
(a)(4%) Plot the spacetime diagram describing the process.
(b)(8%) Find the time it takes for the signal to reach the ship, and the
location of the spaceship when the signal is received, in the earth's
reference frame.
(c)(8%) Same as above, but in the ship's reference frame.
6.(20%) A thin flat rectangular plate of mass M and sides a×2a, rotates with
constant angular velocity ω about an axle through two diagonal corners. The
axle is supported at the corners of the plate by bearings which can exert
forces only on the axle. Ignore gravity and fricion. Consider the coordinate
systems fixed on the plate as in Fig. 1: x-y-z axes are along the principal
axes of the plate and z' is the axle, where α is the angle between z- and
z'-axes. (cosα = 2/√5, sinα = 1/√5).
(a)(6%) Find the moment of inertia about the x-, y- and z-axes.
(b)(6%) Find the angular momentum of the plate, and the torque required to
maintain the motion.
(c)(8%) Find the inertia tensor in the x'-y'-z' coordinate system.
7.(15%) A body is dropped from rest at latitude λ at a height h above the
ground.
(a)(12%) Find the magnitude of the horizontal delfection. Show that the
deflection is eastward.
(b)(3%) Since Earth turns to the east, common sense would seem to argue that
the body should drift westward. Please give an explanation.
8.(10%) Consider a homogeneous solid cube, of mass M and side of length b.
Find the ratio of the moments of inertia about an axis passing through
(1) the CM and one corner of the cube, and (2) the CM and the midpoint on an
edge.
Useful information:
．Vector in fixed frame expressed in terms of vector in rotating frame and
rotation velocity.
╭ dX ╮ ╭ dX ╮
│──│ =│──│ + ω×X
╰ dt ╯fixed ╰ dt ╯rot
．For a 2d coordinate rotation of angle θ,
╭ x'1 ╮ ╭ cosθ sinθ╮╭ x1 ╮
│ │ = │ ││ │
╰ x'2 ╯ ╰-sinθ cosθ╯╰ x2 ╯
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