普物期中考考古題（林敏聰） 14.Nov.2002

[Steggie (p2新板Tread_softly)]

我知道現在才拿到有點晚 = ="
不過之前板上 90 年的那一份題目並沒有考到 Oscillation，
所以應該還是有點參考價值的。

1. Please write down the three Newton's laws of motion in both translational
   and angular forms, including their "equivalent" expressions of Newton's
   second law. What are the assumptions for their application in classic
   mechanics? Are those "equivalent" forms really equivalent? Why? (20%)

2. Two particles, each of amss m and speed v, travel in opposite directions
   along parallel lines separated by a distance d.
   (a) In term of m, v and d find an expression for the magnitude L of
       angular momentum of the two-particle system around a point midway
       between the two lines.
   (b) Does the expression change if we change the point about which L is
       calculated?
   (c) Now reverse the direction of travel for one of the particles and
       repeat (a) and (b). (20%)

3. Please prove the parallel-axis theorem and calculate the rotational
   inertial of body (a), (b), (c), and (d) (see Fig.1). (20%)

   Fig.1
   (a) Solid cylinder about central axis
       圓柱，半徑 R，長 L，轉軸過柱面圓心，平行於 L
   (b) Solid cylinder about central diameter
       圓柱，半徑 R，長 L，轉軸過柱面直徑，緊貼著圓柱的其中一個底面
   (c) Solid sphere about any diameter
       圓球，半徑 R，轉軸過球心
   (d) Slab about perpendicular axis through center
       矩形板子，長 a，寬 b，轉軸垂直於板面，過其中一個頂點

4. Please prove the TWO equivalent expressions of Newton's second law in
   angular forms from those in translational form. (10%)

5. Two blocks of masses m and 4m are connected by a spring and rest on a
   frictionless surface. They are given velocities toward each other such
   that the block with mass m travels initially at v1 toward the center of
   mass, which remains at rest. What is the initial velocity of the other
   block? (10%)

6. (10%) Consider a damped simple harmonic motion with the total force
   ΣF = - kx - bv, where k is force constant of the spring, x the
   displacement, v the velocity, b the damping constant. Please write down
   and solve the (differential) equation of motion from the Newton's second
   law.

7. (a) (5%) Two identical springs are attached to a block of mass m and to
       fixed supports as shown in Fig.2. Show that the frequency of
       oscillation on the frictionless surface is
                        -1/2 <--上標
          f = 1/2π(2k/m)
   (b) (5%) Suppose that the two springs in Fig.2 have different spring
       constants k1 and k2. Please state the frequency f of oscillation of the
       block in f1 and f2, where f1 and f2 are the frequencies at which the
       block would oscillate if connected only to spring 1 or only spring 2.

   Fig.2  |                    |     ~~~~~ <--這是彈簧...|||
          |   k1    m     k2   |
          |~~~~~~~~~□~~~~~~~~~|   <--這邊有個底面，有些 telnet 軟體顯示不出來

8. Good luck!!!!

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感謝化學系 Raines 同學和她的學姊！

--
請推 Steggie 好心會有好報 ￣︶￣ 謝謝大家！！

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推 knightwww:感謝啊  不過樓上的推文是怎麼弄出來的???
推 davidlue:先在計事本打好再貼上(還要附上X y那些的)
推 xiphoarch:這是一定要推的啊啊啊啊啊~~~XD
推 funkastic:好心會有好報
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推 dQoQb:推的好
推 LiHuN:推~話說這還是我在批萬板上的首推 |||