課程名稱︰普物甲期中考
課程性質︰
課程教師︰趙挺偉
開課系所︰
考試時間︰Tuesday November 18 2003, 10:20-12:20
試題 :
Instructions:
Closed books and notes.
Answer all questions.
You can only use one dictionary and one calculator without any prestored
formulas.
1. (15 pts.) A particle of mass M moves in the x-y plane with its x and y
coordinates as the following functions of time:
x = e!bt cos(!t)
y = sin(!t)
where b and ! are constants.
(a) Describe the trajectory of the particle.
(b) What is the velocity of the particle as a function of time ?
(c) What are the x and y components of the force acting on this particle ?
(d) What is the condition under which the force acting on the particle is
conservative ? Find the potential energy of the particle under this condition,
and show that the mechanical energy of the particle is conserved.
2. (15 pts.) A gun is located on the ground at a distance b from a °agpole of
height h. The gun is aimed at a ¯xed angle of elevation μ (assume tan μ > h=b).
(a) What is the muzzle velocity of the shell in order to hit the top of the
°agpole ?
(b) If the °agpole were removed, at what distance from the gun would the
shell strike the ground ?
3. (15 pts.) An object is ¯xed at the surface of a planet which is identical
in mass and radius of the earth. If the object experiences zero gravity at the
equator of that planet, what is the length of a day on that planet ? (Useful
data : mass of the earth = 5:98 ￡ 1024 kg, radius of the earth = 6:37 ￡ 106 m,
Gravitational constant = 6:67 ￡ 10!11 N.m2=kg2)
4. (15 pts.) An electron of mass m collides head-on with an atom of mass
M, initially at rest. As a result of the collision an amount of energy E is
stored internally inside the atom. What is the minimum initial velocity v of the
electron ?
1
5. (20 pts.) A thin string is wrapped around a solid cylindrical spool of
radius R and mass M. One end of the string is ¯xed, and the spool is allowed
to fall vertically, starting from rest, as the string unwinds. (Recall that the
moment of inertia of the solid cylinder is MR2=2)
(a) Determine the angular momentum of the spool about its center of mass
as a function of time.
(b) What is the tension in the string as a function of time ?
6. (20 pts.) Consider an idealized chain piled up at a point next to a hole
in a table. Let the chain have mass ? per unit length. If one end of the chain
is allowed to slip through the hole, it will start to fall and pull the rest of the
chain through. How does the lower end of the chain move as a function of time ?
What is the acceleration of the chain ? Is the total mechanical energy conserved
in this process ? Explain your answer.
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