97-Final

General Physics (I) 期 末 考 Jan. 12, 2009
試 卷 請 註 明 、 姓 名 、 班 別 、 學 號 , 請 遵 守 考 場 秩 序
I. 計 算 題 (50points) ( 所 有 題 目 必 須 有 計 算 過 程 , 否 則 不 予 計 分 )
1. (10 points) The rotor (flywheel, Fig. 1) of a toy gyroscope ( 陀 螺 儀 ) has mass m. Its moment of
inertia about its axis is ½mr 2 . The gyroscope is supported on a single pivot ( 支 點 ) (at point
P) with its center of mass at a horizontal distance l from the pivot. The gyroscope is precessing
in a horizontal plane at the rate of one revolution ( 一 圈 ) in 2.5 seconds. (A) (1 points) Find the
upward force exerted by the pivot (at point P). Write your answer in terms of the mass of the
gyroscope and the gravitation constant g. (B) (3 points) Find the precessing angular speed Ω (in
unit of radian per second) and the torque on the gyroscope due to the gravitational acceleration
g
g. (C) (6 points) Let ω be the angular speed of the flywheel. Show that ω = A , and find r
2
the constant A.
2. (15 points) A solid uniform disk of mass M and radius R ( I CM
= ½ MR 2 ) is oscillating
about an axis through P (Fig. 2). The axis is perpendicular ( 垂 直 ) to the plane of the
disk. Friction at P is negligibly small and can be ignored ( 忽 略 ). The distance from P
to the center C of the disk is b. The gravitational acceleration is g.
(A)(3 points) When the displacement angle is θ, what is the torque relative to point P?
(B) (3 points) What is the moment of inertia for rotation about the axis through P?.
(C) (4 points) The torque causes an angular acceleration about the axis through P. Write
down the equation of motion in terms of the angle θ and the angular acceleration
d 2 θ/dt 2 .
(D) (5 points) As the disk oscillates, the maximum displacement angle θ max , is very
small and the motion is a near perfect simple harmonic oscillation. What is the
period of oscillation?
Circular motion
of flywheel axis
(precession)
Path followed
by free end of
axis
Fig.1
P
Ω
l
Flywheel
ω
Flywheel
axis
Rotation of
flywheel
P
Fig. 2
C
θ
b
R
3. (10 pints) A temperature controller (Fig. 3), designed to work in a steam environment,
involves a bimetallic constructed of brass and steel, connected at their ends by rivets (
鉚 釘 ). Each of the metals is 2.0 mm thick. At 20°C, the strip is 10.0 cm long and
straight. Find the radius of curvature r of the assembly at 100°C.
(Coefficient of linear expansion: α brass
= 19×10 -6 °C -1 , α steel
= 12×10 -6 °C -1 )
Steel
Brass
Figure 20.28
II. 申 論 題 (9 points)
1. 請 問 為 何 由 水 龍 頭 流 出 的 水 流 量 固 定 時 , 隨 著 落 下 的 距 離 增 加 而 水 柱
的 寬 度 會 變 小 ( 如 圖 五 )?
2. 當 汽 車 壓 過 路 障 , 請 問 在 汽 車 滿 載 重 物 還 是 空 車 時 , 汽 車 避 震 器 的 彈
簧 回 彈 時 間 比 較 短 ? 請 解 釋 。
3. 是 否 有 可 能 在 不 加 熱 的 情 況 下 於 室 溫 中 (20 ℃) 使 水 沸 騰 ? 請 解 釋
該 如 何 做 ( 參 考 圖 六 )。
(a) Fig. 3 Fig. 4
4. (15 points) A gas turbine ( 渦 輪 引 擎 ) operates under the Brayton cycle, which is
depicted in the PV diagram (Fig.4). In process ab the air-fuel mixture undergoes an
adiabatic compression. This is followed, in the process bc, with an isobaric (constant
pressure) heating, by combustion. Process cd is an adiabatic expansion with
expulsion of the products to the atmosphere. The return step, da, takes place at
constant pressure. If the working gas behaves like an ideal gas, show that the
efficiency of the Brayton cycle is
e = 1 – (P b
/P a
) (1-γ)/γ
圖 五
圖 六 , Phase diagram for water
(note that the scales are not linear).

III. 證 明 題 (6 points)
Please show that in adiabatic process for ideal gas (Fig.7),
γ
PV
= const.
IV. 選 擇 題 (40points)
1. A massless rope is wrapped around a uniform cylinder that has radius R and mass M,
as shown in Fig.8. Initially, the unwrapped portion of the rope is vertical and the
cylinder is horizontal. The linear acceleration of the cylinder is
(a) (2/3)g (b) (1/2)g (c) (1/3)g (d) (1/6)g (e) (5/6)g
2. In Fig. 9, assume the top of the water is static and is subjected to an pressure P 2 ,
which of the following is the correct expression for v 1 ? The atmosphere pressure is P 0
2P0
(a) v1 = + 2g( y2
− y1
) (b) 2( P2
− P0
)
v
( )
ρ
1
=
+ 2g
y2
− y (c) v ( )
1
1
= 2g
y2
− y1
ρ
(d)
v
2
( P − P )
2 0
1
=
− 2
ρ
g
( y − y )
2
1
P
( V , P)
C
γ =
C
( V+Δ V, P+ΔP)
Fig. 7
P
V
V
3. A traffic light hangs from a pole as shown in Fig. 10. The uniform aluminum pole AB
has length L and mass m. The mass of the traffic light is M. Which of the following
statement is incorrect ?
(a) The vertical component of the force F A is (M+m)g.
L ⎛ m ⎞
(b) The tension T in the rope is cosθ
⋅⎜
M + ⎟
H ⎝ 2 ⎠
L ⎛ M + m
(c) The horizontal component of the force F A is cosθ
⋅⎜
H ⎝ 2
(d) T is equal to the horizontal component of the F A .
4. A physical pendulum consists of an uniform rod, which has mass m and length L. It is
hung from a nail near one end (Fig. 11). For motion with small amplitude, the angular
displacement of the rod can be written as θ = Acos
ωt
Which of the following
is correct:
g
3g
2g
2g
(a) ω =
(b) ω =
(c) ω =
(d) ω =
L
2L
3L
L
TT
H
F A
θ
L
⎞
⎟
⎠
Fig.8 Fig. 9
Fig. 10
Fig. 11
5. An asteroid of mass m traveling at a speed of v relative to the Earth (mass M e , radius
R e ), hits the Earth at the equator tangential in the direction of Earth rotation (angular
velocity ω), and remains on earth after impact ( 碰 撞 ). Which in the following is the
correct expression of the Earth rotational angular velocity ω f after the impact.
mv
mv
v
ω +
ω +
M
eRe
ω
Re
(a) ω
f
=
ω
f
=
M
eRe
(b) (c)
5 m
5 m
ω
(d) ω
f
=
f
=
1+
1+
5 m
5 m
1+
1+
2 M
e
2 M
e
2 M
2 M
e
e
6. One mole of ideal monatomic gas is taken from A to B as show in Fig. 12. Calculate
the heat transferred in terms of P 0 and V 0 .
9PV 0 0
15PV 0 0
17PV 0 0
19PV 0 0
21PV
0 0
( a) ( b) ( c) ( d) ( e)
2 2 2 2 2
P
P
b
3P 0
c
2P 0
B
a
P 0
P
A
0
d
V 0 3V
Fig. 12 0
V Fig. 13
V 0 5V 0
V
7. One mole of ideal monatomic gas is taken through the cycle shown in Fig 13. In terms
of P 0 and V 0 , what is the work done by the gas per cycle
( a) −8 PV ( b) −4 PV ( c)4 PV ( d)8 PV ( e)12PV
0 0 0 0 0 0
8. A red-hot piece of iron ( mass m) at temperature T 1 is thrown into a huge lake whose
temperature is T 2 . Assume the lake is so large that its temperature rise is insignificant
( 可 忽 略 ). The change in entropy of the iron is
⎛ 1 1 ⎞
2 1
( ) ( ) (
1 2) ( ) (
2 1) ( ) ln T
T
a mc⎜
− ⎟ b mc T −T c mc T −T d mc ( e) mc ln
⎝T2 T1 ⎠
T1 T2
9. An box at a temperature 127 o C is placed in a room whose dark walls are at a
temperature of 27 o C. Estimate its rate of heat loss by radiation.
−8 2 4
( The area of the box is 0.1 m 2 ; e= 0.8; σ = 5.67× 10 W / m ⋅K
).
( a)60 W ( b)80 W ( c)100 W ( d)120 W ( e)140W
( )
16
10. A tank is filled with 50% (by volume) pure oxygen O and 50% pure Argon
8
at a temperature of 27 o 2
2
C. is the average value of
v O2
v O
v
2
2
What is the value of ?
2 Ar
( a)0.45 ( b)0.89 ( c)1.24 ( d)2.80 ( e)3.13
v O2
40
( A r 18 )