課程名稱︰普通物理學甲上
課程性質︰
課程教師︰李尚凡
開課系所︰機械系
考試時間︰2006-1-12
試題 :
1. Suppose our Sun is about to explode. In an effort to escape in a spacecraft
at v=0.800c and head toward the star Tau Ceti, 12.0ly away. When we reach
two thirds of our journey from the Earth as seen in a frame of reference
in which the Sun and Tau Ceti are at rest, we see our Sun explode and,
unfortunately, at the same instant we see Tau Ceti explode as well. In
the spacedcraft's frame of reference, which occurred first? By how many
years?
2. A string is wound around a uniform disk of radius R and mass M. The disk is
released from rest with the string vertical and it's top end tied to a fixed
bar. Show that(a) the tension in the string is one-third the weight of the
disk. (b)the magnitude of the acceleration of the center of mass is 2g/3, and
(c)the speed of the center of mass is (4gh/3)^1/2 after the disk has descended
through distancd h.
3. Suppose a solid disk of radius R is given an angular speed ωi about an axis
through its center and then lowered to a horizontal surface and released.
Furthermore, assume that the coefficient of friction between disk and surface
is μ. (a)Show that the time interval before pure rolling motion occurs is
Rωi/3μg . (b)Show that the distance the disk travels before pure rolling
occurs is R^2ωi^2/18μg.
4. A ward of sticky clay with mass m and velocity v is fired at a solid cylinder
of mass M and radius R. The cylinder is initially at rest, and is mounted on
a fixed horinzontal axle that runs through its center of mass. The line of
motion of the projectile is perpendicular to the axle and at a distance
d<R from the center. (a)Find the angular speed of the system just after the
clay strikes and sticks to the surface of the cylinder.
5. Two stars of masses M and m, seperated by a distance d, revolve in circular
orbits about their center of mass. Show that each star has a period given by
T^2 = (4π^2d^3) /〔G(M+m)〕
Proceed as follows: Apply Newton's second law to each star. Note that the center
of mass condition requires that Mr2 = Mr1, where r1+r2=d.
6. The water supply of a building is fed through a main pipe 6.00cm in diameter.
A 2.00cm diameter faucet tap , located 2.00m above the main pipe, is
observed to fill a 25.0L container in 30.0s. (a)What is the speed at which
the water leaves the faucet? (b)What is the gauge pressure in the 6cm main
pipe? (Assume the faucet is the only "leak" in the building.)
7. A rope of total mass m and length L is suspended vertically. Show that a
transverse pulse travels the length of the rope in a time interval
Δt = 2〔(L/g)^(1/2)〕
8. Imagine that a tunnel is drilled through the Earth , Mass M, as shown in
Figure. An object of mass m at a distance r from the center of the Earth
experiment gravitation attraction only by the mass within the sphere of
radius r (the reddish region in the Figure). (a)Write Newton's law of
gravitation for an object at the distance r from the center of the Earth,
and show that the force on it is of Hooke's law form, F=-kr, find k .
G is the gravitational constant. (b)Show that m follows simple harmonic
motion if it moves without friction, and follows the differential equation
d^2x/dt^2 + (g/r)x = 0, where g = GM/R^2. (c)A sack of mail is dropped into
the tunnel. How many minutes will it arrive at the other side? R=6370km,
g=9,80m/s^2
9. A train whistle (f=400Hz) sounds higher or lower in frequency depending on
whether it approaches or recedes. (a)prove that the difference in frequency
between the approaching and receding train whistle is
Δf= 〔(2u/v)/(1-u^2/v^2)〕f
where u is the speed of the train and v is the speed of sound. (70%)
(b)Calculate this difference for a train moving at a speed of 130km/h.
Take the speed of sound in air to be 340km/h.(30%)
10.When a metal pipe closed at one end is cut into 2 pieces, the lowest
resonance frequency for the air column in one-end-closed piece is 256Hz
that for the other is 440Hz . What resonant frequency would have been
produced by the original length of pipe?
11.A cylinder is closed at both ends and has insulating walls. Starting with
the expression w = -∫PdV and using the conding PV^γ = constant, we can
show that the work done on the gas is W=〔1/(γ-1)〕(PfVf-PiVi)
for an adiabatic process, which is also equal to nCv(Tf-Ti). When the
cyclinder is devided into two compartments by a perectly insulating partition
that is perpendicular to the axis of the cyclinder. Each compartment contains
1.00mol of oxygen, which behaves as an ideal gas with γ=7/5. Initially the
two compartments have equal volumes, and their temperatures are 550K and
250K. The partition is then allowed to move slowly until the pressures on its
two sides are equal. Find the final temperatures in the two compartments.
12.A sample consisitng of n mol of an ideal gas undergoes a reversible isobaric
expansion from volume Vi to volume 3Vi. Find the change in entropy of the gas.
13.An idealized idesel engine operates in a cycle known as the air-standard
diesel cycle, shown in Figure. Fuel is sprayed into the cyclinder at the
point of maximum compression, B.Combustin occurs during the expansion B→C
which is modeled as an isobaric process. Show that the efficiencyof an engine
operating in this idealized diesel cycle is
e = 〔1-(1/γ)〕〔(Td-Ta)/(Tc-Tb)〕
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