課程名稱︰力學上
課程性質︰必修
課程教師︰陳義裕
開課學院：理學院
開課系所︰物理系
考試日期（年月日）︰2010/11/17
考試時限（分鐘）：110分鐘
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
→
1.(20pts) In plane polar coordinates (r,θ) the velocity v and the acceleration
→
a are given by
→ .^ .^
v = re + rθe
r θ
→ .. . ^ .. . ^
a = (r -rθ^2)e + (rθ+2rθ)e
r θ
(a)(10pts) Please prove the above two relations.
(b)(10pts) On a horizontal turntable that is rotating at a constant angular
speed ω, a bug is crawling outward on a radial line such that its
distance from the center increases quadratically with time:
r = bt^2 , θ = ωt .
1.(5pts) Find the radial component of the acceleration of the bug and
explain why it becomes very nagative when t is large.
2.(5pts) Find the angular component of the acceleration of the bug and
explain why it becomes very positive when t is large.
2.(15pts) A particle of mass m moving in a horizontal straight line is subject
to a velocity-dependent resisting force f = -Av^(3/2) , where v is the speed
of the particle, and A is some constant. If the initial speed of the
particle is v , please answer the following:
0
(a)(5pts) Will it stop in a finite time? If the answer is yes, then please
compute this time.
(b)(10pts) Is there a limit to the distance it can travel? If yes, then
please compute it.
3.(30pts) The Lorentz transformation is given by
t' = γ(t - vx/c^2),
x' = γ(x - vt),
y' = y,
z' = z,
if the prime system is moving in the direction of positive x-axis of the
unprimed system at a velocity v. Here γ≡ 1/√(1-(v/c)^2), c being the
speed of light in vacuum, and it is known that v＜c.
(a)(10pts) A particle has a velocity ( u , u , u ) relative to the
x y z
unprimed system.
(b)(10pts) If we know that 0＞u ＞-c and u = u =0, then please show that
x y z
the particle's speed relative to the primed frame is less then c.
(c)(5pts) If we have u = c and u = u =0, then show that the particle also
x y z
has a speed c relative to the primed frame.
(d)(5pts) If we know that u ＞c and u = u =0, then please show that the
x y z
particle's speed relative to the primed frame is still greater than c.
4.(25pts) This problem shares the same two inertial frames of Problem 3.
(a)(10pts) A ruler at rest relative to the unprimed frame is placed along
the x-axis. Its length is l. Please compute its length as measured by the
primed observer.
(b)(10pts) Peter sits at the origin of the unprimed frame and watches TV for
a period of time τ. How much TV time does this appear to the primed
observer?
(c)(5pts) In another "experiment" the TV is carried along by the primed
obserever Tom. Thus instead of watching (and enjoying) the TV program, this
time Peter can only observe a "fly-by" TV set for a period of time τ. To
the primed observer Tom (who now always has the TV with him), how much time
does Peter spend observing the "run-away" TV?
5.(10pts) Consider the following one-dimensional motion (see the picture below
→
): Two blocks of respective mass m and M are placed side by side. A force F
is applied to the block with mass m. Being placed side by side, it then
→
exerts a force f to the block with mass M so that the two blocks move
m→M
together as a whole. The block with mass M also exerts a "reaction" force
→ → →
f on the first block. Nancy proceeds to find the force f and f
M→m m→M M→m
using the following reasoning.
＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿
Nancy's reasoning:
→ →
Let a denote the acceleration of the composite system (m+M), then a =
→
F/(m+M). Applying Newton's second law to the block of mass m, we have
→ → → →
F + f = ma = mF/(m+M).
M→m
Similarly, we have
→ → →
f = Ma = MF/(m+M).
m→M
if we apply Newton's second law to the block of mass M.
But if we add up the above two formulas we will obtain
→ → → → → →
F + f + f = mF/(m+M) + MF/(m+M) = F ,
M→m m→M
which then implies
→ → →
f + f = 0
M→m m→M
and we have
SUCCESSFULLY PROVED NEWTON'S THIRD LAW OF MOTION USING A THOUGHT EXPERIMENT!
＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿
Question: Did Nancy actually succeed in proving Newton's third law? Explain
in detail if you think not.
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