[試題] 99上 梁啟德 普通物理學甲 期中考
課程名稱︰普通物理學甲
課程性質︰必修
課程教師︰梁啟德
開課學院：工學院
開課系所︰機械系
考試日期（年月日）︰2010/11/11
考試時限（分鐘）：110 mins
是否需發放獎勵金：是 謝謝
（如未明確表示，則不予發放）
試題：
Mid-term examination(11/11/2010)
1.Three vectors (向量) are given by a=3.0i+3.0j-2.0k, b=-1.0i-4.0j+2.0k, and
c=2.0i+2.0j+1.0k. Please find (a) a‧bxc(5%) (b) a‧(b+c)(5%) (c) ax(b+c)(5%)
[Hint:right-hand rule, i, j, and k are the unit vectors in the x-, y- and z-
directions, respectively. ixj=k, jxk=i, ixk=-j]
2.Please prove (證明) that the rotational inertia of a solid cylinder about
1 2 1 2
central diameter (see Fig.1) is given by I=--- mR + --- mL , where m is
4 12
the mass of the cylinder.(10%)
Axis │ Solid cylinder
│ (or disk) about
│ central diameter
│
┌──────┴──────┐┬
│ │ R
│ │┴
│ │
└──────┬──────┘
├ L │ ┤
│
│
Figure 1 │
3.Figure 2 shows a stream of water flowing through a hole at depth h=10cm in a
tank holding water to height H=40cm. (a) t what distance x does the stream
strike the floor? (5%) (b) At what depth should a second hole be made to give
the same value? (5%) (c) At what depth should a hole be made to maximize x?
(5%)
║ ║
┬ ╟────┬╢
│ ║ h║ v
│ ║ │║ ---->
│ ║ ┴ ─╮
H ║ ║
│ ║ ║
│ ║ ║ (拋物線)
│ ║ ║
│ ║ ║
┴ ╚═════╝ ╰───
├───x───┤
4.A conical pendulum consists of a bob of mass m in motion in a circular path
in a horizontal plane as shown in Fig. 3. During the motion, the supporting
wire of length L maintains the constant angle θ with the vertical. Show
(證明) that the magnitude of the angular mometum of the bob about the
2 3 4
m gl sin θ 0.5
circle's center is given by L = (------------) ,where g is the free-fall
cosθ
acceleration.(10%)
████████
▕＼
▕θ＼ L
▕ ＼
▕ ＼
▕ ＼
╭-----------┼---------＼╮
│ └-----------○m
╰-----------┴-----------╯
5.The position of a particle is given by x(t)=4+ln(t)sin(2t), where t is the
time in units of second (t>0). Please find the velocity v(t)(5%) and the
acceleration a(t)(5%) [Hint: ln(t)=log (t), where e is a mathematical
e
constant, the base of the natural logarithm, which equals approximately
x
2.718281828, and is also known as Euler's number.e is the exponential
x
de x
function. ---- = e ]
dx
6.As you read this page on paper, a cosmic ray proton passes along the
left-right width of the page with a relative speed v and a total energy of
-8
1.424 x 10 J. According to your measurements, that left-right width is
0.21m (rest length). (a) What is the width according to the proton's
reference frame? (5%) How much time did the passage take according to
(b) your reference frame (5%) and (c) the proton's frame? (5%) [Hint: Rest
-10
(mass) energy of a proton is ~1.5 x 10 J].
7.A 55kg woman runs up a flight of stairs with a height of 4.5 m in 3.5 s.
What average power (平均功率) is required of her? (10%) [Hint: free-fall
acceleration g=9.8m/s^2]
8.One end of a uniform 4.00-m-long rod of weight Fg is supported ba a cable.
The other end rests against the wall, where it is held by friction, as shown
in Figure 4. The cofficient of static friction between the wall and the rod
is μs=0.500. Determine the minimum distance x from point A at which an
additional object, also with the same weight Fg, can be hung without causing
the rod to slip at point A.(15%) [sin(37)≒3/5 cos(37)≒4/5]
▉＼
▉ ＼
▉ ＼
▉ ＼
▉ ＼
▉ ＼
▉ ＼
Figure 4 ▉ ＼
▉ ＼
▉A 37.0 ( ＼
▉═══╤══════ B
▉ ←x→│
▉ │
▉ ○ Fg
--
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