課程名稱︰普通物理學甲上
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）︰99/01/14
考試時限（分鐘）：13:20~15:10
是否需發放獎勵金：是 謝謝
（如未明確表示，則不予發放）
試題 :
1. A large block P executes horizontal simple harmonic motion as it slides
across a frictionless surface with a frequency f = 1.50 Hz. Block B rests
on it as shown in Fig. 1, and the coefficient of static friction between
the two is μs = 6.00. What maximum amplitude of oscillation (最大振幅)
can the system have if block B is not to slip (不滑動) ? (10%)
▋ μs ┌─┐
▋ ↘│Ｂ│
▋ ┌┴─┴┐
▋﹌﹌﹌│ Ｐ │
▆▆▆▆▆▆▆▆▆
Figure 1
2. A 1.00-mol sample of a monatomic ideal gas is taken through the cycle shown
in Fig. 2. At point A, the pressure, volume, and temperature are Pi, Vi,
and Ti, respectively. In terms of the gas constant R and Ti, find (a) the
total energy entering the system by heat per cycle, (5%) (b) the total
energy leaving the system by heat per cycle (5%), and (c) the efficiency of
an engine operating in this cycle (5%).
P Q2
│ ↓
3Pi├ － ┬─→┐
│Q1→↑ →│Q3
│ │ ↓ ↓
Pi├ － ┼←─┤
│ ： Q4 ：
│ ： ：
└──┴──┴──V
Vi 2Vi
3. Maxwell's speed distribution (速率分佈) law is given by p(v) =
M 1.5 2 ( -Mv^2 / 2RT )
4π(───) v e , where p(v) is the probability
2πRT
distribution function, M is the molar mass of the gas, R is the gas
constant, T is the gas tmperature, v is the molecular speed, respectively.
∞
∫vp(v)dv
0
It is known that the average speed of the gas is given by ─────, and
∞
∫p(v)dv
0
∞ 2
∫ v p(v)dv
0
the average of the square of the speed is given by ──────. Please
∞
∫p(v)dv
0
∞
prove (證明) that (a) ∫p(v)dv = 1 (10%) (b) the average speed of the gas
0
8RT 0.5
is given by (───) (5%) (c) the average of the square of the speed is
πM
3RT x x
given by ── (5%). [Hint: de / dx = e ]
M
4. Please prove (證明) that the general solution (通解) of the following
equation kx + ma = 0. Where k is the spring constant, m is the mass of the
block attached to the spring, x is the displacement of the block,
2 2
a = d x / d t is the linear acceleration of the block, and t is time, is
given by C1 cos[(k/m)^1/2] + C2 sin[(k/m)^1/2]. Here C1 and C2 are
iθ -iθ
constants. (15%) [Hint: e = cosθ + i sinθ and e = cosθ - i sinθ,
where i = √-1]
5. A solid sphere (實心球) of mass m and radius r rolls without slipping (沒有
滑動) along the track shown in Fig. 3. It starts from rest with the lowest
point of the sphere at height h above the bottom of the loop of radius R,
much larger than r. What is the minimum value (最小值) of h (in terms of R)
such that the sphere completes the loop? (10%) [Hint: I = 2/5 mr^2 ; you
need to consider both translational energy and rotational kinetic energy]
▕╲.m
▕↑╲
▕ h ╲ ╭─╮
▕│ ╲│ R│
▕↓ ╲─╯
▇▇▇▇▇
Figure 3
6. A toy rocket moves at a speed of 242 m/s directly toward a stationary pole
while emitting sound waves at frequency f = 1250 Hz. (a) What frequency f'
is sensed by a detector that is attached to the pole? (10%) Some of the
sound reaching the pole reflects back to the rocket, which has an onboard
detector, what frequency f" does it detect? (10%) [Hint: The speed of sound
is given by 343 m/s]
7. A hypodermic syringe contains a medicine having the density of water
(Fig. 4). The barrel of the syringe has a cross-sectional area
A = 2.50 ×10^(-5) m^2, and the needle has a cross-sectional area
a = 1.00 ×10^(-8) m^2. In the absence of a force on the plunger, the
pressure everywhere is 1 atm. A force of magnitude 2.00 N acts on the
plunger, makng medicine squirt horizontally from the needle. Determine the
speed of the medicine as it leaves the needle's tip. (10%)
▁▁▁▁▁▁▁▁▁▁▁▁▁
▋ ▕ ╲
▋ ▋ ▕ A ╲▁▁▁▁
F→ ████▋ ▕↙ a→▏ →v
▋ ▋ ▕ ╱▇▇▇▇
▋ ▕ ╱
▇▇▇▇▇▇▇▇▇▇▇▇▇
Figure 4
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